The Physics of Gravitational Redshifts in Clusters of Galaxies

Nick Kaiser Institute for Astronomy, U. Hawaii

LBL, Berkeley 2016/09/15 Clusters of galaxies

14 15 • Largest bound virialised systems ~10 -10 Msun • Velocity dispersion σv~1000 km/s (~0.003c) • Centres - often defined by the brightest galaxy (BCG) • Usually very close to peak of light, X-rays, DM n • σBCG ~ 0.4-0.5 σv so much “colder” than general pop Clusters in the Millenium Simulation (Y. Cai) Gravitational redshift & uRSD 5

Figure 1. Top row: particle distributions within 10 Mpc/h radius from the main halo centre projected along one major axises of the simulation box. n in the label of the colour bars is the number of dark matter particles in each pixel. Middle row: the same zones but showing the potential values of all particles. Sub-haloes and neighbouring structures induce local potential minima. Bottom row: the gravitational redshift profileswithrespecttotheclustercentres.The dashed lines shows the spherical averaged profile, Φiso,whichisthesameasequal-directionalweightingfromthehalo centres. Sub-haloes and neighbours cause the mass weighted profiles Φobs to be biased low compared to the spherical averaging. This is similar to observations where the observed profiles are weighted by galaxies. ⃝c 0000 RAS, MNRAS 000,000–000 Wojtak, Hansen & Hjorth (Nature 2011)

0.8 Wojtek, Hansen & Hjorth stacked 7,800 0.7 0blue-shift) -4 -3 -2 -1 0 1 2 3 4 • 3 corresponding to v = -10 km/s vlos [10 km/s] 5 rv 2rv 3rv c.f. ~600km/s l.o.s velocity dispersionFigure 1 Velocity distributions of galaxies combined from 7, 800 SDSS galaxy clusters. The line- • 0 of-sight velocity (vlos)distributionsareplottedinfourbinsoftheprojectedcluster-centric distances R.Theyaresortedfromthetoptobottomaccordingtotheorderof radial bins indicated in the Interpreted as gravitational redshift effectupper left corner-5 and offset vertically by an arbitrary amountforpresentationpurposes.Redlines • present the histograms of the observed galaxy velocities in the cluster rest frame and black solid lines show the best fitting models. The model assumes a linear contribution from the galaxies which do not-10 belong to the cluster and a quasi-Gaussian contribution from the cluster members right order of magnitude, sign (see SI for more details). The cluster rest frames and centresaredefinedbytheredshiftsandtheGR [km/s]

• Δ positions of the brightest cluster galaxies. The error bars represent Poisson noise. f(R) • “Confirms GR, rules out TeVeS” -15 -20 TeVeS • Had been discussed before (Cappi 1995; Broadhurst+Scannapiaco, ....) -25 0 1 2 3 4 5 6 R [Mpc]

Figure 2 Constraints on gravitational redshift in galaxy clusters. The effect manifests itself as a blueshift ∆ of the velocity distributions of cluster7 galaxies in the restframeoftheirBCGs.Velocity shifts were estimated as the mean velocity of a quasi-Gaussian component of the observed velocity distributions (see Fig. 1). The error bars represent the range of ∆ parameter containing 68 per cent of the marginal probability and the dispersion of the projected radii in a given bin. The blueshift (black points) varies with the projected radius R and its value at large radii indicates the mean gravitational potential depth in galaxy clusters. The red profile represents theoretical predictions of general relativity calculated on the basis of the mean cluster gravitational potential inferred from fitting the velocity dispersion profile under the assumption of the most reliable anisotropic model of galaxy orbits (see SI for more details). Its width shows therangeof∆ containing 68 per cent of the marginal probability. The blue solid and dashed lines show the profiles corresponding to two modifications of standard gravity: f(R) theory4 and the tensor-vector-scalar (TeVeS) model5, 6. Both profiles were calculated on the basis of the corresponding modified gravitational potentials (see SI for more details). The prediction for f(R) represents the case which maximises the deviation from the gravitational acceleration in standard gravity on the scales of galaxy clusters. Assuming isotropic orbits in fitting the velocity dispersion profile lowers the mean gravitational depth of the clusters by 20 per cent. The resulting profiles of gravitational redshift for general relativity and f(R) theory are still consistent with the data and the discrepancybetweenpredictionofTeVeS and the measurements remains nearly the same. The arrows showcharacteristicscalesrelatedto the mean radius rv of the virialized parts of the clusters.

8 SDSS Redshift Survey r 2r 3r 4 v v v

3

2

1 km/s] 3 0 [10

los -1 v -2

-3

-4 0 1 2 3 4 5 R [Mpc]

Supplementary Figure 1 Velocity diagram combined from kinematic data of 7800 galaxy clusters 11 detected in the SDSS Data Release 7. Velocities vlos of galaxies with respect to the brightest cluster galaxies are plotted as a function of the projected cluster-centric distance R.Bluelinesare the iso-density contours equally spaced in the logarithm of galaxy density in the vlos R plane. − The arrows show characteristic scales related to the mean virial radius estimated in dynamical analysis of the velocity dispersion profile. Data points represent 20 per cent of the total sample.

20 The physics of cluster gravitational redshifts • Einstein gravity • gravitational "time dilation" • Weak field limit • δν/ν = -Φ/c2 • Measured by Pound & Rebka

Is that it?

cluster Is there any more to the physics of cluster grav-z?

• Is Pound & Rebka relevant? • equipment lashed to tower in Harvard phys. dept • where is the tower here? • What about the principle of equivalence? • where is the rocket? • where is the earth? The nature of astronomical redshifts • Homogeneous cosmological models: • Wavelength scales as a(t), or δλ/λ = δD/D, but why? • Symmetry → standing wave solutions with λ ~ a(t) • Analogy with expanding cavity • lots of little redshifts as ‘photons’ bounce off walls • Maxwell’s equations written in “expanding” coordinates • extra cosmic ‘damping’ or ‘friction’ term in EOM • So expansion of space stretches wavelengths?

Misner, Thorne and Wheeler

redshift as an effect on standing waves....

But is this a standing wave? Expanding space and redshifts in textbooks..... • E.R. Harrison (2000)

5

1.0

0.8

0.6 a 0.4

0.2

0.0 Wolfgang Rindler (1970) 0.0 0.2 0.4 0.6 0.8 1.0 • t

FIG. 2: The scale factor of a hypothetical “loitering” universe as a function of time (measured in units of the present time). At the time indicated by the dot, a galaxy emits radiation, whichisobservedatthepresenttime.Atthetimesofbothemission and observation, the expansion is very slow, yet the galaxy’sobservedredshiftislarge.

Let us begin by reviewing a standard derivation8,13 of the cosmological redshift. Consider a photon that travels from a galaxy to a distant observer, both of whom are at rest in comoving coordinates. Imagine a family of comoving observers along the photon’s path, each of whom measures the photon’s frequency as it goes by. We assume that each observer is close enough to his neighbor so that we can accommodate them both in one inertial reference frame and use special relativity to calculate the change in frequency from one observer to the next. If adjacent observers are separated by the small distance δr,thentheirrelativespeedinthisframeisδv = Hδr,whereH is the Hubble parameter. This speed is much less than c,sothefrequencyshiftisgivenbythenonrelativisticDoppler formula

δν/ν = −δv/c = −Hδr/c = −Hδt. (4)

We know that H =˙a/a where a is the scale factor. We conclude that the frequency change is given by δν/ν = −δa/a; that is, the frequency decreases in inverse proportion to thescalefactor.Theoverallredshiftisthereforegivenby ν(t ) a(t ) 1+z ≡ e = o , (5) ν(to) a(te)

where te and to refer to the times of emission and observation, respectively. In this derivation we interpret the redshift as the accumulated effect of many small Doppler shifts along the photon’s path. We now address the question of whether it makes sense to interpret the redshift as one big Doppler shift, rather than the sum of many small ones. Figure 2 shows a common argument against such an interpretation. Imagine a universe whose expansion rate varies with time as shown in Fig. 2. A galaxy emits radiation at the time te indicated by the dot, and the radiation reaches an observer at the present time t0.Theobservedredshiftisz = a(t0)/a(te) − 1=1.5, which by the special-relativistic Doppler formula would correspond to a speed of 0.72c.Atthetimesofemissionandabsorptionoftheradiation,the expansion rate is very slow, and the speedar ˙ of the galaxy is therefore much less than this value. We can construct models in which botha ˙(te)and˙a(t0)arearbitrarilysmallwithoutchangingtheratioa(t0)/a(te)andhencewithout changing the redshift. Upon closer examination this argument is unconvincing, because the calculated velocities are not the correct veloc- ities. We should not calculate velocities at a fixed instant ofcosmictime(eithert = te or t = t0). Instead we should calculate the velocity of the galaxy at the time of light emission relative to the observer at the present time.Afterall, if a distant galaxy’s redshift is measured today, we wouldn’texpecttheresulttodependonwhatthegalaxyisdoing today, nor on what the observer was doing long before the age ofthedinosaurs. In fact, when astrophysicists talk about what a distant object is doing “now,” they often do not mean at the present value of the cosmic time, but rather at the time the object crossed our past light cone. For instance, when astronomers measure the orbital speeds of planets orbiting other stars, the measured velocities are always of this sort. There is an excellent reason for this convention: we never have information about what a distant object is doing (or if the object even exists) at the present cosmic time. Any statement in which “now” is used to refer to the present cosmic time at the location of a distant object is not about anything observable, because it refers to events far outside our light cone. In summary, if we wish to discuss the redshift of a distant galaxy as a Doppler shift, we need to be willing to talk about vrel,thevelocityofthegalaxythen relative to us now.Talkingaboutvrel is precisely the sort of thing that Î / v 9 À > È œ Þ > i Ê “ Õ Ê v - µ

y ½

> À Õ

 i

i À i >

> ` Ê ˜ ˜ / À Ê Ê   L  œ Ì i   Þ Ì

° Þ Ã Ê Ê Ê Ê

> Ê Ê Ì  }

 > 

 œ i ° i Û , Ê  ˜ Ì  Ê ˆ

Ê i - à ˆ i ˆ
˜ Ì  Ý

°

Ê iÜ Ê Ü « Ê

> Û À œ ˜ i i

> ˜ à ˜ " Û } ˆ Ê œ > i ˜ à À Ê Ê Ì Ê > œ À " ˜ v œ 1 Ê

` Ì ˜

Ê œ Ê i “ Ê Õ

i / ˜ˆ À Ã Û Ê i
Ê À Ã /

i ¶ Ê 

*  /  "

,

-  Ê Ó ä ä x Ê

*°Ê 
   -Ê i˜ÌiÀÊvœÀÊ
ÃÌÀœ« ÞÈVà É-/-VÉ
-
,  /Ê ˜ } à « Ü Ì œ > “ Ì ˆ « ˆ - Ì Ì Ì V V i
V « L L à  Ì «  ˆ T - i `  Ì œ  º ` Ì œ Ì  Û Ì } > £ œ ˜ ˆ ˜ ˜ ˆ i

Õ >

> ˆ « Ü> À œ i œ Û > Ì ˆ > > ˜ ˜

] v v œ   œ v > i } iÀ iÀ } H V Ü Ì Ã “ ˜ Ê } i i i Õ V ˜ ä Ê } > ˜V Ê i À i ˜ Ê i Ì > > À iÀ > œ  œ   ˜Ì Ó ` à œ œ œ

Ã Ê Ì

Ê i ` À À i Õ ˜ Ê À Ã Ì i > > Ê Ê V Þ Ê Ê Û Ü Ì L } Ì Ì i à ˆ E ä Ê V Ì Ê

/ À «> > “  i  « i Ã

Ê > Ì Ì i À Ì i ] Ì Ê ˜

ˆ “i i L “ Ê Ê Ê i Ê Ê « Ì Ã i Ê Ý Ý Û Ã > ˆ v i ˆ i i L  œ à œ ä ‡ Ê ° “iÀ œ œ œ Ã Ê ˜ ä  > “ } Ý > i ˆ à ˜

Ê Ê ˜ Ê ˜ v Ê i Ê Þ œ œ à P i œ > i ` Ê ] ° À Ì ˆ i Ì œ Ü ˆ ˆ ˜ œ Ì  ˜ ˜ > V Ê ˆ v ä ˜ Û À V v i « ` Ê > i Ã Ê V Ê « i i Ê

Ê ˆ V > Ê i i  } i Õ V œÌ Ì Ã v V Ž À Ê i Ì Ê > ˆ R > à / « « Ê œ V i Ý i à ˜ v Ã Ü iÀ œ à à V À ` >  Ê ä  à à i ` Ê > ˆ œ ˜ ½ Õ > > ˆ i i ˜ ˆ Ê Ì ˆ L Ã Ê Ì Ì Ê Ê >  ˆ Ê Ê ` ˆ Ã Ã Û œ ]

V « ˜ Ê Ì Ì Ì ] > « > «> I L

V â ` Ê

Ì

ˆ ˆ i » } Ê Ê Ê V

}  L i ˆ ˜ Õ Ê Ê ˜ À Ê > ˜ Ì Ê Ì Ê « Ì L Ê

Ì Ã Ž Û Ì Ý Û i ˜ Ê ˆ ˆ œ Ü Ã À Ì M œ ˆ   i “ Ê  v “ À à i Ê œ > ˜ ˆ i

Ê > ˆ i Ê

œ > Ì Ê i Û À i ` w ° Ì i ` > Ì À

ˆ Ê i i i  } i ˜ Õ ˆ “ Û œ L “ œ Ê ˆ i ° “ Ê Ì ` ` Ì Þ Ã ˜ Ê i Þ Õ Ê i ˆ i ` ˜ À œ Þ > Ž ˜ i ˜ i À Ì i ° / Ê Ü Ê ` V Ê À Õ œ ˆ i > à ˆ A i Ý ˆ

Ê À Ê Ã > V Þ Ì  Ê ­ V ˆ ¶ Ê ˆ ½ i Ê Ê Ã Ê Ì ` ` Ì Û œ  Ì V œ Õ Ã Ã “  ` i i Ê ˆ œ L L i Ã É À > Ê Ê  à i i à À > L ˜Ã  i œV i Ê } Ì

> « Ì Ê ˆ Ê > Ê Ê œ > i Ê w R Ê Ã œ i ] i Ê ˆ > i Ã Ê Ã ˜ v Ã Ê v Ì / > Ì À Ê â Ê ` v L ˆ i  œ  Õ Õ i Ì Ã ˜ «  Ê L Ã Ì Ì Ì > ˜ i ] Ê Ê > “ Ê Ê > Ê ˆ

Ã Õ i ] > à V Ì V v Ê> ˆ V > i Ê Ý Ì Y i Þ Ì ` ` L à i “ Ê > ] Ê Ì Ê i ˆ Ê v œ Ã Ê Ì L V

 } À Ê ˜  Ê Ê > i Ã

> À ˜ À iÀ À Ì Ã Ê

Ê  Ü v Ê ˆ ÝV À Ê À Ü Ì Ü Ê Ã “ à ˆ > > Ê i Ê ˆ Ê Ê Ê Ê > Ê > V L i Ì Ü Õ ˆ Ì À i Õ ˆ L i à V Ì œ Ì Ì > Ê ˜ ˆ } ˆ L Ã Ì  i i ˜ } i ˆ ˜ ` œ > O > ˜ } ½ œ Ê ˆ À > V > œ à V iÀ ` i ˆ Ê

 ˜ Þ >

} « ˆ Ì Ã À Ê i  À i Ê Ê ˜ à « Ê ˜ i i Þ ` ` « } i Ì “i > ˆ œ } Ì Ã i À V Ì

Ê Ì Ž i i V Û ° ˜ ˜ ` Ê À Ê ˜ i à B ˆ i ` à ] Ê Þ } Ã Ê µ > i à ˆ Ê >

 Ê ` Ê > > ] Ê

i « Û

iÀ Ê i Ê ˆ Ê Ì i Ž V Ê i Ê œ Ì Ì i œ i œ } > / Ê Ê ° > V À i Ê Ê ° ˜ Ì Ã V p Ê Ê V  Ì ˜ > Õ L

`ˆ À Ü Ê S Ã Ê Ê Ê Ê Ê “ i ° Ê À i ` Û Ê ˆ Ì Ì  Ì `

i ˆ i v i À Ê “ Ü œ i À Ê v Ü ˜ œ Û Ì L
/ Ê > ˆ Ê Ê Ã i Ê Ã Ì

L Ê Ê Ã Ê ˆœ } i i > i  Ê E ˜œ £ / Ê ` i  œ Ê « Ê > Ì

V L Ã Ê Ü Ã > i Ì À Ê > “ > Ê i > > Ã Ì Ê Ì ` ˜ Û Ê L Õ œ Ê Ê Ã V

à >

i œ œ

Ê œ Ì ˆ à ™ Ã

i i ` œ R œ Õ Õ Ì  ˆ Õ > À ˜ « i

˜ ˜ Ì i

Ì  ˆ ˜ Ã Ì ˆ Ê Ê

Þ Ã Ê V Ì Ê Õ ˜ Ì Û ˆ >

` À Õ œ > ˜ V i ˆ Ê Ã > ˜ œ £ Ê Ì ˜ Ì > i À > ˆ ˜ ˜ > ˜ ˜ à À i œ œ > i V i Ê Ì } œ À œ œ Ê

Ê V “ œ i Û œ À Ã

} Ì Ê Ì œ ˜ Ì i  > ˆ Þ i ˜ à i ˜ ° ä > Ê Ì L À œv Ì Ã œ i “ Ê Ê ` œ } ˆ i À ˜ Ì ` œ Û  ˆ œ iÀ

˜ A ˆ

Ê Ã Ì Ê À i Õ Ê V ˆ  Ê ˜ Ê Û }  Ã Ì Ê ˜ ˆ ` ] ` Ì `  œ ` ˜ Ê œ

> Ê Ý Ì œ ` i œ ¶ ˜ œ Ê Ê > Ì À V i i Ê œ Ê ˜ > ˜ Ã Ê Õ i ˆ Ê ˆ À i Ê À ˜ Þ } i } T œ ° Ê Þ ˆ˜ Ê Û ` Ü Ê “ Ê Ã Ê Ã Ã Ã ˜ “ Ì œ “iÀ ° œ ˜ à ` ` À i ½ Ì Õ « ˜ œ

˜ ˆ Ê v > Ê v

˜ > Ê À } } L Ê i Ê

Ê Ã Ê Ê ] Ã Ê Ê i Ì L œ « Ê ˆ I i ] - > ] Ê i V Ì  v à  v Ü } Ê  Ã Ã Ê  ° Ì ` V “ Ê Ê > £

Û  > > ` « Ì Ê Ê « Ê

œ Þ  Ì } > Ì  O Ê

Ê Ê > > Ûi v Ì i Ê Ê À i œ Ê  i

œ à > V œÌ Ê œ œ i i À ` v

à à ˆ ˆ ] i Ê “ « œ ˜V Ì

Ã Ê i i Ê Ê i  Õ ` ˆ Ì V Ì > V ˜ Ê Õ > £ Ì > ä > i à i i Ã Ý “i V Ê ˆ

N

Ì Ê “

V Ì À Ã Ì ® i ˆ

À i i Ü > > V Þ À  v Ì

ˆ Ì Ê Ê «> À

Ì À L

à > > “

Ì ˜ ™ à À Ê i i Ê ä ˆ ˆ « > Ê œ œ > i “ > ` À Ê ] Ì œ i à œ ° ˆ Ì Ì `ˆ ˆ ˜

ˆ Ã Ì ˜ ˜ À Ê i « Ê Ì > Ì i Ê i L œ Î Ê i Ê Ê Ã Ê Ì Ì ` Û ‡ Ê ä

i v > Ì œ Ü Ã  ] V Þ  Ê Ê V i } Õ  Ê ˜ ˆ i ˆ

Ì i > À ½ Ê

Õ À Ê ` Ì > V œ Ê Ê Ã À Ê i à i } Ê  i ˜ ð œ Ì ä À à  Ì Ì Ž ˜ ˆ ˆ i ] V « > y Ê œ À À L i Ã

Ã Þ } i Ê i Û v > i > i ˆ Ê V Ê Ã À Ê Ì > > V > À Ê Ì Ê  v > Õ Ì Ê Ã Ã ˜ ° i Ê

Õ « Õ ` > Ê Ì ˆ ˜ Ê  Õ « Þ i ] ½ > œ à V ˆ Ê i Ì i

À i ˜ ° Þ > Ì Û > Ê  Ì

` ­ Ì Ã Ã Ã œ œ  > i Ê  L  i
Ê i > Ê œ V à À « Ê “ Ì Ã ˆ Ê Õ i Ê œ i ˜ Ì Ê  à ˆ > ˜

ˆ  Õ ` Ê Ê > i Ê Ê Õ ˜ Ã Ì œ ˆ

Ê i ˆ Ì ˜ Ê Ê i à `

˜ Ê i  } ` L

i Ã Ì Ã Ì Ì ˜V i ˜ À à V Ì À >

Ê i Ü Ê Ê Ý À Ì Ê Ì À ˆ Ì > > Ê Ê À Ã Ê v ‡ Ã

Ê À > Ì « ˆ Ì ` Ã

° } `

> Ê >

i ` Ê ˆ œ œ « à «  œ ° ˆ À Ý ˆ ˆ i Ì Ê ­ Ê i «  ˜ i Ê i

Ê Ã Ü ˆ Ê ˆ ˜ Ê À i > Ê V Û > Û Ã

Ì > Ê ˆ ˆ i > Ê ˜ i Ê œ Û ˜ ˆ Ê i i  > Ì  ` Ê À “ i À ˜ Ê Ã Ì « ` ` à i œ L > > À Ê i ½ Ê Û œ } ˆ Õ L } i  Ê  ° ˜ v ˆÌ Ê Ì œ i V œ v ] Þ À > à ® Ã Ê Ì i v Ê Û ` à « Ì Ê Ã Ê Ê Ì V Ê Ê > ˆ ˆ Ã Ê > Ê Ê ˜ Ê ˆ Ê Þ i Ê

i v œ Ü Ê œ

° ˜ > Ê  Ê Ì i V à i L Ì Ê Ì Ê Ê Ì À L ˆ Ê Ü Ã > ˜ V

i Ì Ì À Þ À Ì œ >  Ì œ Ü Ê i ˜ }  Ã Ì À Ê ˜  ˜

Ê ]

i  ° ˆ “ i Ê Ì Õ ˆ

> i ° œ ˆ Ê Ì i / i  Ê Ê i œ i Þ Þ ½ Ê Ê > i Ü ˆ > Ê V V Ê ˆ à Õi “ } L Ê Û ˆ i Ì Ê Ì Þ `° Ê L à “ Ì

iÀ  Ê / i > Ê i v Ü / > Ã Ì V ˜ “

i Ì À ˜ } Ê Ê i Ê ˆ i L V Û

Ì   > Ê

˜ i

«

Ã Ê Ê Ì Ì Ê œ Ã Ê Ã i Ü « Ã Ê i Þ i ˆV Ê > ˆ Î Ê “ } Ê œ Õ ] “

> ˆ Õ Ê i Ê Ê Ì

« À i i œ Ì

Ê > Ê } V Ê } i iÀ Õ « « i Û Ì Ê > ˆ  i Ê ] Ì > ` } ˜ « L « Ì

Ì ] Ê ˆ Ãi  i Ì Ì Ê À  Ì i Ê ˜ v ˜ i i > Ê Ì i i Õ À Þ > Ü i

Õ i ä ˜

Û Ì Ã i > « i à i Ê œ  i Ê > i

Ê i ° œ À Ý Û  œ i ˆ Ã Ì > > Ê > Ê i à À

Õ Ê Ê ˆ  Ê ˆ  Ì ˜ ] Ì V œ ˆ i ˜ ˜ ˆ ä > Ê Ê Ê > > i i Û > À

Ý i i i Û Ê œ ˆ Û i ˆ À > V >V ˆ Ê «> Ì À v ˜ Ê i  Ê Ã ˜ à “i i Õ Ã ˆ Ì ˜ ` Ê £ à À } ` Ê Ì ˜ Ã Ì Ý Ê i v à iÀ Ê v ˆ Þ Õ À ä Ü iÀ ÝV i > iÀ Õ Ì i ˆ “ > Ì

V Ã Ì À

Ê i Û Ê ` Ê Ê Ê > œ À Õ L Ê { i œ  } Õ Õ Ê Ì i Ê Ê ˆ ˆ i à À À « Ý ˜ Ê Ì Ê À i Ì ˆ ` L  L à œ i œ ˆ  Ž ˜ œ Ê Ì V V

> > Ê œ Ã Ì À œ à L Ý Ê Ê i Ã Ê i À i Ê “ “ i ˆ Ì ˜ œ À >  v Ê > i Ì > > œÌ Õ Ì L À œ > ˆ ˆ œ Ê Ì V V Ê i “ Ê i Ì v > à i ˆ À i à  à > i Ê ` Ì ˜

À  «

à i L à i  à “ ˆ œ Ã Ã Ì “ “ ˜ i i Ê Ê À Ê ˆ ˆ  ` ˆ v i œ Ê i Ê Ê ` « Ê ` Ì Û ° ˆ Ã Þ Ì i i  ` ۈ ` Ì  Ã

à œ i Õ Ê Ì Õ  `  `

> i Ê i > Ã Ê ` Ê v ˆ Ê “ à ` Ì Õ > ˆ ˆ Þ / Õ  \ i > Ã Ê Þ i Ê Þ Ê `

` Ê Ê Ê i Û « i Ê œ œ œ œ ˜

œ ˜ } } œ p à À Ê

à ˆ ˆ } ˆ ˆ > Ì Ì i i Ê Ì À ˜ ˜ À Ê Ê À

Ê ˜
œ œ Ê ˜ > ° ˜} ˜ L ˆ ˆ Ê ˆ i ˆ i 

ˆ À ˆ À Õ Ì Õ Õ Ì `

˜

> Ê > Ý Ã ˆ i à À > > > i à à à i v ð ˆ Ã Ì Ì

` œ ˜ Ž ˜ L ˜ ˜ ` œ ` Þ Þ } > i i i i i i Ã Ã Ã Ã Ã Ì Ì Ì Ì ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ® Ì ° ° ] Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê ` L œ À À œ Ì Ì «  « Ì > Ã Ì L > Ì i  > œ } V

ˆ œ -

˜ >

œ

À

i i L 1 

Õ œ

L L

˜ >

L }

ˆ >

À À v

> i Û

Þ > Ê Õ Ã Õ Ã Ê Ü > ` ` i i œ À i à * à à « i Ê œ  à à V œ i Û

œ «

Ê Ì Ã ˆ œ Ì > Ì À Ê Ê i ˜ « Ê Ã Ã ˜

“ « Ž À i i i ] i i

Ã Õ œ

7 À ˆ /  - Ì œ > à ˜ Ì Õ Ê Ý ˜ À ˜

Þ > Ê } œ > , ˆ Ê i } Ê > À À

Õ À L à V Ì Ì Ì Ê ˜ œ Û V L i Ì œ ˆ } Ã Ê ˜ ˆ ˆ Ê Û Û } ˜ Ê Þ

À Ü ˜

À i i V ˆ « i Ì Ã Ì Ì i

Ü v v v Ì Þ i Ê  i

Ì œ > > i > œ œ ˆ Ã Ê Ü i Ê ` Ì Ì " Ê i Ý œ Ê Ã i ` i Ê œ à « “ } i Ì i i ° à ˜ œ Ê Ê Ê Õ v ` Ì Ì Ê Û Ì Ê Ã « Ê Ã £ ä Ê Ê Ê 6 } À ˆ À } Ê « œ * 

« Ê « > ˆ > ˜ > ˆ ˆ Ü i Ê À Ì } ˜ ˆ > ° ˜ Û Ã ˆ i
Ü « œ œ > ° ] Þ i Ê À œ > Ž Ê i ˆ i œ  à À ` Ê Ê > Ü Ê x

> Õ > Ã ` Ê ]

> i L œ Ê œ v > ˜ ˜ ˜ i ] > Ì V i - i ˜ ` Ê Ê À Ì Ê  œ ˆ Ê « Ê Õ « V ] Ã Ì Ê i Ì Ê i Û Õ Ý > > > V à à `

i œ Ê Ì À ˜ à V i Ê Ê ‡ Ì ˜ à ˆ Õ > à À i > ˆ Ì ˆ > Ê Ž > i À Ý Ê  Ì i Ê Ê Ê « œ > ˜ ˆ > ˆ L Ì Ž Ê ˜ ` i œ V ˜ i ˆ Ê Õ Ê œ ˜ ˜ ˆ À À Ê i ˆ Õ w  Ê Ã Ž À ` } Ì ° ˜ « ˜  à « i À œ à i Ê œ Ì } ˆ à v ` V Õ Þ Ì  Ü i } Þ ˜

Ê Ž à À à ˆ ˆ ˆ

Ê Ê i Ã Õ i v i à “ ° Ã Ê ˜

Ì ° Ê i > } > ˆ “ À Ê Ã Ì Ì Ê > Ì Ê i œ Ê i i V Ã

} Ì Ã ° > ] Ê Ê Ã v œ Ê Ã œ V i w ˆ

 Ì Ã > Ê Ê Ì Ê ˆ Ì Ê À œ à ` w  Ê ˆ « Ì i à œ i

L À > ˆ ˆ Ê Õ ˜ Ì

Ê Ê > À Ê À Ã Ê Ì œ œ œ à « “ i Ì > À à œ Ê œ Ž > ˜ V Ã Ê œ Þ i Õ ˜ “ i œ L Ã Ê i à 

Ê “ Ê Ì ˜ ˜ À À Ê Ì   À Ê Ê Ê Ì Ã V  ` À Ì « Ì ˜ > Þ


} > Þ «  ` Û Þ œ i i Ê

Ê Ê ° Þ Ê Ê Ê œ i ° i Ì > i Ê i œ Ì ˆ œ Ê « Ê Ã “ i Ê i Ê i Ê Ê } “ « Ê Ê / i L Ì Ã >  œ ­

À Ê ]

œ Ý ] Ý ˜ Ê  v ˆ Ê Ê Ì Ê Ì À Ê V Ê Õ Ì œ Ê

> i > Ê œ Õ > > > } Ì

ˆ œ Ê > i « L i Ì

œ à > `

À Ã Ì Ê Ã ˜ à V i ˜ Ì Õ Þ ` à ˜ ˜

Ê Ì > œ “ À i Ê Ã Ê i ` ˆ À Ê Ê i « Ê Ã Ê

i > œ i  ˜ ˆ Ž i Þ ` Ì i « Ê Ì Ì ˜ ` Ì Õ Ü > i à “ i V À

À Ì Ê Ê «  ` Ê i v ` Ê ˜ v Ê Ê Ê ` Ã Ì i Ê } ­ “ À Ì i ` œ ˆ > à > Ì À  Ê Ì i v œ

 Ã Ì Ê Ã ˆ i ] œ ˆ ° « œ Û

Ê i Ì

i L i V i v Õ ˜ i Ê œ ˜ ˜

Û Þ ˆ Ì Ê Ê À À i

Ê ` i > Ü œ > i Ì i Ì Ì « i ` ] V Þ À } œ v ˜ Ê } i i i Ê « V Ê Ê ˆ i À Ì Ê À à V Ü ˜ v  « ˆ “ À

Ì Ê Ã > ˜ ˜ à } Ê À i Ê Ã Ê V  ˆ Ê Ã

Õ v à  > i ° Ê « i Ì ] Ê œ

˜ Ì Õ œ Ü Ê

i Ê ] à ` > Ê À Ê Ê œ Ê

> ˜ i
Ü Õ L ˜ œ Ê ˆ  L ° à œ œ ˆ } « Ì Ê Û Ì Ã i ˆ Ê Ü Ê À Ì Þ Ê i Ž Ì `

v œ À V  œ “ i Þ œ à ˆ i “ Ê i Ì i œ ˆ œ Ê ˆ ˜ Ì Ã º ˜ v V Ê i « - > ° œ Û ` à œ Ì Õ Ì ˜ Ê Õ V Ê i Ê > Ê Ì À Ê À Ê Ì Ê Ì Ã œ / i } Ê > « Ê Ã œ Ê > ˜ i Þ

˜ œ À ˆ œ

˜ « L Õ Ã i Ž

V p À

Ã Ê Ê ˆ Ê Ì À À Ì  Ê Ã y i i  ˆ ˜ i Ã Þ « à ° i œ ` à à à ˜ i > i Ê “ À ˜ i

Ê Õ

V Ê / œ Ê Ê ˜ à ˆ “ >  À À Æ Û Ã ˆ ` Ì p Ê } >

`  Ì V ` Þ Ê L i Ê > à Û

œ i

À ˆ ˜ > Ì Ê œ Ê i Ê > i À i Ê > V } » « ˆ Ì ˆ ˆ Ê Ì œ À à ] i Õ Ì ` à À i À œ ` i ˜ À V / V Ê Ê Ê Ê i à ˜

˜ ˆ > œ

À ˆ œ i Ì ˆ Ê Ê Ê i à } i } ˆ Ã Ê i Û > œ Ì Ê œ  i Ì Ã p œ Ê

Ã Ì “ i V ] Ü Ê Ý  ˜ > Ü Ê  Ê > i ˆ Ê Þ

Ê ˆ > > Ý Ê Ì À œ ˜ Ê Ê i Ê V > > œ « ˆ Ì Ã Ã À ˆ > Ê ® Ì

> } à À ˜  ® ˆ ˆ « ˆ ˜ Ì Ê v V ˆ Ã Ê i ˜ i Ê > Ê Ã ˜ ˆ Ì i œ V Ì Ì

ˆ Ê Ê œ Ì Ê Ì ˜ ° Ž Ê Õ ˜ V Ì `  ˜ Ê L Ê Ì 

à `  à ˜ i Ê > ˆ

> À Ã Ê “ œ > À Ê ˜ ˆ ˆ œ ˆ i Ì i V i Ê Ê ˆ Õ
œ Ê } Ì Ê > Ã Ì Ã V ‡ À > ˆ Ê Ê Ã ° i ` Û  Ü À à ` à } à à  Ê

 Ê  i } œ ˆ Ì Ü Ê Ê

« Ì Ê i Ì ˆ ˆ Ã Ì Ê ˆ Ì “ i À } œ Ê > œ Ê Ã i Ê i œ ˜ } °

` Ê Ì ˜ Þ

 À Ì

À i Ì Ç Ê Ê > ˜ à > œ

œ

Ê Ê Ê  > Ý Ê

œ

Ê œ Ê œ > Õ 6 œ ˜ Ê } ° Ü > Ì > ` Ì x > Ê ° Û Ê ˜  ˜ > , « ˜ >  ˜ i ,

Ì ˆ Ê « œ > Ì v > Ê œ / Ý œ Ì Ã À à i Ê i Ê } Ê Ê Ê Ì > Þ ` Ã Ê œ  > Ê Ê i } > 

Ì Ý } Þ i

Ì  Ê Ê Ê i Ê

i Ì V Ê ˜ Ê Ê i œ “ œ > Ì V } i Þ Þ > w / ˜ Ê ˆ

Ž œ À Ê >
Ê > v à à ° i ˆ Ž Ê > Ê Ê Ì Ì Ê Ê i œ À i Ã Ì v ˜

À ˆ i Ê ˜ ˆ i Ê Ì Ê Ã œ À Þ À ] v Ã Ê i ° Ê Ê Ã Ì Ê Ê i à Ê

˜ Ê Ê Ê Ã Ê Ê Ê { Ê ° Ê Ê £

Mathematics of radiation in expanding space • In non-expanding space, extremizing the action 3 • S = ∫dt d x L(∂tφ,∇xφ) 2 2 • with e.g. L = ((∂tφ) - (∇xφ) )/2 (i.e. massless scalar) • Gives the wave equation with propagation speed c (unity here) 2 2 • ∂t φ - ∇x φ = 0 In expanding universe, physical coordinate x = a(t) r, where r is • ‘comoving’ coordinate, so 3 • δS = δ∫dt d r L’(∂tφ,∇rφ) = 0 3 • with L’ = a(t) L (and ∇rφ = a(t) ∇xφ) 2 -2 2 • Gives ∂t φ + 3 H ∂tφ - a ∇r φ = 0 • expansion gives extra ‘damping’, or ‘friction’, term 3 H ∂tφ involving the expansion rate H = (da/dt)/a. • energy not conserved (Lagrangian formally time dependent) 2 -4 • admits damped standing wave (in r) solutions (∂tφ) ~ a • And this does give the right answer for photons between FOs Expanding Space: the Root of all Evil?∗

Matthew J. Francis1,4,LukeA.Barnes1,2,J.BerianJames1,3 &GeraintF. Lewis1

1Institute of Astronomy, School of Physics, University of Sydney, NSW 2006, Australia 2Institute of Astronomy, Madingley Rd, Cambridge, UK 3Institute for Astronomy, Blackford Hill, Edinburgh EH9 3HJ,U.K. 4Email: [email protected]

Abstract: While it remains the staple of virtually all cosmological teaching, the concept of expanding space in explaining the increasing separation of galaxies has recently come under fire as a dangerous idea whose application leads to the development of confusionandtheestablishmentofmisconceptions. In this paper, we develop a notion of expanding space that is completely valid as a framework for the description of the evolution of the universe and whose application allows an intuitive understanding of the influence of universal expansion. We also demonstrate how arguments against the concept in general have failed thus far, as they imbue expanding space with physical properties not consistent with the expectations of general relativity. Keywords: cosmology: theory

1Introduction is far from uncontroversial. Recent attacks on the physical concept of expanding space have centred on When the mathematical picture of cosmology is first the motion of test particles in the expanding universe; introduced to students in senior undergraduate or ju- Whiting (2004), Peacock (2006) and others claim that nior postgraduate courses, a key concept to be grasped expanding space fails to adequately explain the motion is the relation between the observation of the redshift of test particles and hence that it should be abandoned. of galaxies and the general relativistic picture of the ex- But what, exactly, is at fault? Crucially, these claims pansion of the Universe. When presenting these new rely on falsifying predictions made from using expand- ideas, lecturers and textbooks often resort to analo- ing space as a tool to guide intuition, to bypass the full gies of stretching rubber sheets or cooking raisin bread mathematical calculation. However, the very meaning toExpanding allow students to Space: visualise the how Root galaxies of are all moved Evil?∗ of the phrase expanding space is not rigorously defined, apart, and waves of light are stretched by the “expan- despite its widespread use in teaching and textbooks. sion of space”. These kinds of analogies are appar- Hence, it is prudent to be wary of predictions based entlyMatthew thought to J. be Francis useful1 in,4,LukeA.Barnes giving students a men-1,2,J.BerianJameson such a poorly1,3 defined&GeraintF. intuitive frameworks. tal pictureLewis1 of cosmology, before they have the ability In recent work, Barnes et al. (2006) solved the test to directly comprehend the implications of the formal particle motion problem for universes with arbitrary general1 relativistic description. However, the academic asymptotic equation of state w and found agreement Institute of Astronomy, School of Physics, University of Sydney, NSW 2006, Australia argument surrounding the expansion of space is not as between the general relativistic solution and the ex- 2 clearInstitute as standard of Astronomy, explanations Madingley suggest; Rd, an Cambridge, interested UK pected behaviour of particles in expanding space. We student3Institute and reader for Astronomy, of New Blackford Scientist Hill, may Edinburgh have seen EH9 3HJsuggest,U.K. that the apparent conflict between this work Martin4 Rees & Steven Weinberg (1993) state and others, for instance Chodorowski (2006b), lies pre- Email: [email protected] arXiv:0707.0380v1 [astro-ph] 3 Jul 2007 ...how is it possible for space, which is ut- dominantly in differing meanings of the very concept of terly empty, to expand? How can noth- But see alsoexpanding Weinberg, space. 1st This 3 Minutes, is unsurprising, given that it is aphraseandconceptoftenstatedbutseldomdefined Abstract:ing expand?While it The remains answer the stapleis: space of virtually does allp31: cosmological “One can te thinkaching, of thethe concept wave crests of expanding with any rigour. spacenot in expand. explaining Cosmologists the increasing sometimes separation talk of galaxiesbeing h aspulled recently farther come and under farther fire asapart a dangerous by ideaabout whose expanding application leadsspace, to but the development they should of confusiothe expansionnandtheestablishmentofmisconceptions.In of this the paper, universe.” we examine the picture of expand- In thisknow paper, better. we develop a notion of expanding space that ising completely space within valid as the a framework framework for of the fully general rela- whiledescription being told of the by Harrisonevolution of(2000) the universe that and whose applicationtivistic allows cosmologies an intuitive and understanding develop it into a precise def- of the influence of universal expansion. We also demonstrateinitionhow arguments for understanding against the the concept dynamical in properties of generalexpansion have failed redshifts thus far, are as produced they imbue by expanding the spaceFriedman-Robertson-Walker with physical properties not consistent (FRW) spacetimes. This withexpansion the expectations of space of general between relativity. bodies that framework is pedagogically superior to ostensibly sim- are stationary in space pler misleading formulations of expanding space — or Keywords: cosmology: theory What is a lay-person or proto-cosmologist to make of more general schemes to picture the expansion of the this apparently contradictory situation? Universe — such as kinematic models and approxi- Whether or not attempting to describe the obser- mations to special relativity or Newtonian mechanics, 1Introduction is far from uncontroversial. Recent attacks on the vations of the cosmos in terms of expanding space is a since it is both clearer and easier to understand as well physical concept of expanding space have centred on Whenuseful the goal, mathematical regardless picture of the of devices cosmology used is first to do so,the motionas being of test a particles more accurate in the expanding approximation. universe; In particu- introduced to students in senior undergraduate or ju- Whiting (2004), Peacock (2006) and others claim that nior postgraduate courses, a key concept to be grasped expanding1 space fails to adequately explain the motion is the relation between the observation of the redshift of test particles and hence that it should be abandoned. of galaxies and the general relativistic picture of the ex- But what, exactly, is at fault? Crucially, these claims pansion of the Universe. When presenting these new rely on falsifying predictions made from using expand- ideas, lecturers and textbooks often resort to analo- ing space as a tool to guide intuition, to bypass the full gies of stretching rubber sheets or cooking raisin bread mathematical calculation. However, the very meaning to allow students to visualise how galaxies are moved of the phrase expanding space is not rigorously defined, apart, and waves of light are stretched by the “expan- despite its widespread use in teaching and textbooks. sion of space”. These kinds of analogies are appar- Hence, it is prudent to be wary of predictions based ently thought to be useful in giving students a men- on such a poorly defined intuitive frameworks. tal picture of cosmology, before they have the ability In recent work, Barnes et al. (2006) solved the test to directly comprehend the implications of the formal particle motion problem for universes with arbitrary general relativistic description. However, the academic asymptotic equation of state w and found agreement argument surrounding the expansion of space is not as between the general relativistic solution and the ex- clear as standard explanations suggest; an interested pected behaviour of particles in expanding space. We student and reader of New Scientist may have seen suggest that the apparent conflict between this work Martin Rees & Steven Weinberg (1993) state and others, for instance Chodorowski (2006b), lies pre- arXiv:0707.0380v1 [astro-ph] 3 Jul 2007 ...how is it possible for space, which is ut- dominantly in differing meanings of the very concept of terly empty, to expand? How can noth- expanding space. This is unsurprising, given that it is ing expand? The answer is: space does aphraseandconceptoftenstatedbutseldomdefined not expand. Cosmologists sometimes talk with any rigour. about expanding space, but they should In this paper, we examine the picture of expand- know better. ing space within the framework of fully general rela- while being told by Harrison (2000) that tivistic cosmologies and develop it into a precise def- inition for understanding the dynamical properties of expansion redshifts are produced by the Friedman-Robertson-Walker (FRW) spacetimes. This expansion of space between bodies that framework is pedagogically superior to ostensibly sim- are stationary in space pler misleading formulations of expanding space — or What is a lay-person or proto-cosmologist to make of more general schemes to picture the expansion of the this apparently contradictory situation? Universe — such as kinematic models and approxi- Whether or not attempting to describe the obser- mations to special relativity or Newtonian mechanics, vations of the cosmos in terms of expanding space is a since it is both clearer and easier to understand as well useful goal, regardless of the devices used to do so, as being a more accurate approximation. In particu-

1 Arguments against “expanding space” • Space-time is locally flat (a.k.a. ‘Minkowskian’) • Whiting, 2004; Peacock, 2009; Bunn & Hogg 2009, Chodorowski, 2007, 2011, Rees & Weinberg....

2 2 • Writing ∂t φ - ∇x φ = 0 in re-scaled coordinates does not change the local physics. • despite apparent damping term • Example: the Milne model • Limiting case of Ω →0 FRW model • Different families of FOs → different a(t) • → Red-shifting or blue-shifting fireball solutions • multiple families of uniformly expanding (or contracting) observers and radiation fields - even in same region of space • The expansion rate H is defined by the radiation itself! • Determined by the initial conditions. Observers and photon paths in the Milne Model

A 3-surface of constant proper time since the explosion

Geometry of these 3- Minkowski spacetime surfaces is hyperbolic Same as open FRW models Standing waves (in expanding coords) in Milne’s model Peebles (’71) explanation of cosmological redshift • The redshift λrec/λem is the product of a lot of small shifts between a set of FOs along the look-back path In the vicinity of a neighouring pair of • FOs • space-time is locally flat, so incremental redshifts are Doppler • shifts • Yields differential equation • dλ/λ = da/a with solution λ ∝ a(t) • So fractional change in proper separation is the same as the fractional change in λ • i.e. δlog(λ/D) = 0 What about our lumpy universe? • Bondi (1947): Spherical models: • for low-Z, redshift is product of Doppler and gravitational redshift • also in homogeneous case • But Synge (1960) shows that all redshifts are given by Doppler formula • “In attributing a cause to this spectral shift, one would say .... that the spectral shift was caused by the relative velocity of the source and the observer''. Synge, 1960; General Relativity • Observed (or emitted) energy is dot product of observer 4- velocity and the photon 4- momentum. • → wavelength shift is given by Doppler’s formula with “relative velocity” being the l.o.s. component of the difference of the receiver 4- velocity and a parallel transported version of the emitter 4-velocity “Not a gravitational redshift as • the Riemann tensor does not appear in formula” • But parallel transport does know about gravity Bunn & Hogg, 2009 • Like Peebles they break photon path into a set of intervals • set of intervening observers along line of sight • local flatness →product of Doppler shifts • Intervening observers need not be freely falling • Any incremental shift can be considered to be either Doppler or gravitational gravitational redshifts are just Doppler shifts viewed • from an unnatural coordinate system - so an enlightened cosmologist would never try to draw any distinction All redshifts can (and should!) be considered to be • Doppler, or ‘kinematic’ in nature. (much like Synge) • Does this mean Δln(λ/D) = 0 is universal? ideas about redshifts in astronomy - summary

• The redshift of light in cosmology • redshift is caused by the expansion of space? • standing waves in a cavity • Maxwell's equations in expanding space: "Hubble damping" • counter arguments (e.g. "how can space expand?") • Peebles' picture - lots of little Doppler shifts • The redshift of light in general • Bondi ('47): combination of gravity & Doppler • Synge ('60) redshifts "caused by the relative velocity..." • Bunn & Hogg ('09): "gravitational redshifts are just Doppler shifts viewed from an unnatural coordinate system" Interpretation of redshifts: why do we care? • The equivalence principle: • locally gravity does not exist • "transformed away" for freely falling observers • special relativity rules • Lesson from cosmology: • wavelength is tied to the expansion of space • we don't seem to see the effect of gravity • redshift appears to be purely kinematic • The "kinematic picture" may suggest this is general • But clusters are not expanding • why would we see any gravitational redshift? • is Einstein/Newton/Pound+Rebka complete? Back to the Wojtak et al. measurement

0.8 Gravitational redshift for light 0.7 0

at ~2.5 sigma level 5 r 2r 3r • v v v Figure 1 Velocity distributions of galaxies combined from 7, 800 SDSS galaxy clusters. The line- 0 of-sight velocity (vlos)distributionsareplottedinfourbinsoftheprojectedcluster-centric distances Claimed to conflict with TeVeS R.Theyaresortedfromthetoptobottomaccordingtotheorderof radial bins indicated in the • upper left corner and offset vertically by an arbitrary amountforpresentationpurposes.Redlines -5 present the histograms of the observed galaxy velocities in the cluster rest frame and black solid modified gravity lines show the best fitting models. The model assumes a linear contribution from the galaxies which do not-10 belong to the cluster and a quasi-Gaussian contribution from the cluster members (see SI for more details). The cluster rest frames and centresaredefinedbytheredshiftsandtheGR [km/s]

positionsΔ of the brightest cluster galaxies. The error bars represent Poisson noise. descendent of Milgrom f(R) • -15 theory -20 TeVeS But OK with GR or e.g. f(R) • -25 0 1 2 3 4 5 6 modifications R [Mpc]

Figure 2 Constraints on gravitational redshift in galaxy clusters. The effect manifests itself as a blueshift ∆ of the velocity distributions of cluster7 galaxies in the restframeoftheirBCGs.Velocity shifts were estimated as the mean velocity of a quasi-Gaussian component of the observed velocity distributions (see Fig. 1). The error bars represent the range of ∆ parameter containing 68 per cent of the marginal probability and the dispersion of the projected radii in a given bin. The blueshift (black points) varies with the projected radius R and its value at large radii indicates the mean gravitational potential depth in galaxy clusters. The red profile represents theoretical predictions of general relativity calculated on the basis of the mean cluster gravitational potential inferred from fitting the velocity dispersion profile under the assumption of the most reliable anisotropic model of galaxy orbits (see SI for more details). Its width shows therangeof∆ containing 68 per cent of the marginal probability. The blue solid and dashed lines show the profiles corresponding to two modifications of standard gravity: f(R) theory4 and the tensor-vector-scalar (TeVeS) model5, 6. Both profiles were calculated on the basis of the corresponding modified gravitational potentials (see SI for more details). The prediction for f(R) represents the case which maximises the deviation from the gravitational acceleration in standard gravity on the scales of galaxy clusters. Assuming isotropic orbits in fitting the velocity dispersion profile lowers the mean gravitational depth of the clusters by 20 per cent. The resulting profiles of gravitational redshift for general relativity and f(R) theory are still consistent with the data and the discrepancybetweenpredictionofTeVeS and the measurements remains nearly the same. The arrows showcharacteristicscalesrelatedto the mean radius rv of the virialized parts of the clusters.

8 0.8 0.7 0

Figure 1 Velocity distributions of galaxies combined from 7, 800 SDSS galaxy clusters. The line- of-sight velocity (vlos)distributionsareplottedinfourbinsoftheprojectedcluster-centric distances R.Theyaresortedfromthetoptobottomaccordingtotheorderof radial bins indicated in the upper left corner and offset vertically by an arbitrary amountforpresentationpurposes.Redlines present the histograms of the observed galaxy velocities in the cluster rest frame and black solid lines show the best fitting models. The model assumes a linear contribution from the galaxies which do not belong to the cluster and a quasi-Gaussian contribution from the cluster members (see SI for more details). The cluster rest frames and centresaredefinedbytheredshiftsandthe positions of the brightest cluster galaxies. The error bars represent Poisson noise.

7 Jimeno, Broadhurst, Coupon, Imetzu, LazkosSDSS Clusters 20152005 Downloaded from http://mnras.oxfordjournals.org/ at University of California, Berkeley on September 14, 2016

Figure 6. GMBCG (top), WHL12 (middle) and redMaPPer (bottom) phase-space diagrams before (left) and after (right) removing statistically the foreground and background contribution of galaxies. Black contours represent iso-density regions. The asymmetry between the positive and negative vlos region can be particularly clearly seen in the redMaPPer case. This difference disappears after the statistical interloper removal. We also plot as red dashed lines the boundaries at 1, 2.5 and 4.5 Mpc that will determine the radial bins we will use in Section 3. In these diagrams, the position of the BCG is fixed at r 0Mpc 1 ⊥ = and vlos 0kms by definition, and the density is determined by the number of galaxies with spectroscopic redshift measurements around them. = − Also, this will help us identify which of the BCGs have the best proximation, where galaxies not belonging to clusters are not iden- spectroscopic measurements, so, taking a conservative approach, tified individually in each cluster, as in the direct method, but taken we will only work with those BCGs identified in our ‘high quality’ into account statistically once all the cluster information has been SDSS galaxy sample, discarding this way BCG redshift measure- stacked into one single distribution of galaxies. See Wojtak et al. ments obtained from ‘bad’ plates. This leaves us with a total sample (2007) for a detailed study of different direct and indirect foreground of 19 867 BCGs in the GMBCG catalogue, 52 255 in the WHL12 and background galaxies removal techniques. case, and 10 197 in the redMaPPer one. We compute the pro- In our case, we apply the following procedure: first, jected transverse distance r and the line-of-sight velocity vlos we bin the whole phase-space distribution in bins of size ⊥ = 1 c (zgal zBCG)/(1 zBCG)ofallSDSSgalaxieswithrespecttothe 0.04 Mpc 50 km s− . After that, we take all those bins lying in two − + × 1 1 BCGs, and keep those that lie within a separation of r < 7 Mpc stripes 4500 km s− < vlos < 6000 km s− , where we assume that 1 ⊥ | | and vlos < 6000 km s− from these. It should be noted that, as all the galaxies there belong either to the pure foreground or to the we are| working| mainly in a low redshift region, the impact of the pure background sample. Then, we fit a quadratic polynomial de- cosmological parameters used is not significant. Stacking all the ob- pendent of both vlos and r to the points in both stripes, and use the tained pairs into one single phase-space diagram, we get the density interpolated background⊥ model to correct the ‘inner’ phase-space distributions shown on the left-hand side of Fig. 6. region bins. We use a function that depends not only on r , but also ⊥ To remove the contribution of foreground and background galax- on vlos; this is because at high redshifts, and due to observational ies not gravitationally bound to clusters, we adopt an indirect ap- selection, we may have more spectroscopic measurements of those

MNRAS 448, 1999–2012 (2015) Jimeno, Broadhurst, Coupon, Imetzu, Lazkos 2015 Jimeno, Broadhurst, Coupon, Imetzu, Lazkos 2015 4 Sadeh et al 2014 3 ×10 + 0.2 + 0.3 + 0.4 + 0.5 gal rgc [Mpc] : 0.5 1.0 1.6 2.1 n 14 - 0.2 - 0.2 - 0.3 - 0.3 12 20 wsw = 0.5 × r200 GR only 10 wsw = 1.0 × r200 GR + kinematic [km/s] 3 gc w = 1.0 r (mix) WHH 10 sw × 200 × v 10 7 ∆ clst n 6

5 0

2.3 -10 clst

/ n 2.2 gal n 2.1 -20

0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 r / r gc 200 rgc / r200 (a) (b)

FIG. 1. (a):Dependenceofthenumberofgalaxies,ngal,ofthenumberofassociatedclusters,nclst,andoftheratio,ngal/nclst, on the separation between BCGs and associated galaxies, rgc.Binsofrgc are defined by a sliding window with a width of 0.5r200,whereeachdata-pointisplacedinthecenter-positionofthe corresponding bin. (b):DependenceofthesignaloftheGRS,∆vgc,onrgc,wherethewidthoftheslidingwindowisdenotedbywsw.Theshaded areas around the two nominal results (circles and squares) correspond to the variations in the signal due to the systematic tests − +7 described in the text, combined with the uncertainty on the model-fit. On average, ∆vgc = 11−5 km/sfor1masses is based on extracting the GRS signal from the distri- used in our analysis is comparable to that of the WHH bution of the velocities of galaxies in the rest frame of sample. We therefore refer to the corresponding esti- corresponding BCGs. We focus on optimizing the selec- mate of Kaiser of 9 (GR only) or 12 km/s (including tion procedure of clusters and of galaxies, and take into kinematic effects)− [11]. Our results− are in good agree- account multiple possible sources of systematic biases not ment with this prediction for rgc >r200, while at smaller considered by WHH. values of rgc, the profile of ∆vgc is steeper in the data. We find an average redshift of 11 km/s with a stan- − Additionally, we observe that it is not possible to distin- dard deviation of +7 and 5km/sfor1future spectroscopic surveys, such Hjorth, we present a new measurement with a larger as Euclid and DESI [19], we will have access to larger, The calculational framework Zhao, Peacock & Li, 2012 • δz is not just gravitational redshift • Sources are moving, so we also see • transverse Doppler effect: 1st order Doppler effect averages • to zero, but.... • to 2nd order <δz> = /2 can be understood as time • dilation Generally of same order of magnitude • as gravitational redshift from virial theorem, Jeans eq... • (And this doesn’t really test GR • see also Bekenstein & Sanders, 2012 • more later.....) • Is that the full story? No - there is another effect of same order

• Light cone effect • we will naturally tend to see more objects moving away from us than towards us in any observation made using light as a messenger • this gives an extra red-shift effect • again of the same order of magnitude as the gravitational redshift Light-cone effect • Light cone effect • we will see more galaxies moving away from us in a photograph of a swarm of particles • past light cone of event of our observation overtakes more galaxies moving away than coming towards us • just as a runner on a trail sees more hikers going the other way... • So not Lorentz-Fitzgerald contraction effect • phase space density contains a factor (1-v/c) 2 • <δz> = <(vlos/c) > • same sign as TD effect • 2/3 magnitude (for isotropic orbits) Quasar absorption lines Light-cone effect - more particles moving away! Another way to look at LC effect

• Particle oscillating in a pig-trough • r(t) = a cos(ωt + φ) • v(t)/c = -(aω/c) sin(ωt + φ) • v(t) averages to zero average could be over phase or • time • but vobs = v + (r/c) dv/dt + ... • where r/c is the look-back time and the extra term does not • average to zero ~ same as Einstein prediction for • Pound & Rebka • δz ≈ / c2. Why is the transverse Doppler effect a redshift?

• Transverse Doppler redshift effect: • first order Doppler shift ~v/c is large but averages to zero • residual is a quadratic ~(v/c)2 effect which caused randomly moving objects appear redshifted on average • can also be understood as a time dilation effect • But moving objects have more energy per unit mass (in the observer frame) • So if they convert their rest mass to photons we should see a blue-shift of photons on average Transverse Doppler Effect: Blue is the new Red • Moving objects appear redshifted on average • But moving objects have more energy per unit mass (in the observer frame) • So photons should be blue-shifted on average • Averaging over objects vs averaging over photons • e.g. unresolved objects => blue shift (e.g. BCG or low resolution HI of total cluster z) • here we have a hybrid situation: • redshifts measured for objects • but objects are selected according to flux density • surface brightness modulation => net blueshift Surface brightness modulation

• Line of sight velocity changes surface brightness • relativistic beaming (aberration) plus change of frequency • but doesn’t change the surface area • so velocities modulate luminosity depends on SED: δL/L = (3 + α)v/c • Gravitational Redshifts in Clusters 3 • α ~= 2, so big amplification • spectroscopic sample is flux limited at r=17.8 dN/dZ

• Δn/n = - d ln n(>Llim(Z))/d ln L * ΔL/L • opposite sign to LC, TD effects, but larger because the sample here is limited to bright end of the luminosity function

Figure 1. Spectral index vs. redshift for representative galaxy Figure 2. The dot-dash curve is the logarithmic derivative of types observed in Sloan r-band the comoving density of objects above the luminosity limit asa function of redshift. The bell-shaped curve is dN/dZ = Z2n(Z) and the solid curve is that truncated at the minimum redshift imposed by the parent cluster catalogue. The mean of the log- derivative, averaged over the redshift distribution turns out to be ≃ 2.0. luminosity function does not, and the parameters are not very different from the field galaxy luminosity function, so we will use the latter, as determined by Montero-Dorta & Prada (2009), as a proxy. Their estimate of the LF obtained 4 EFFECT OF SECULAR INFALL from the r-band magnitudes K-corrected to Z =0.1 has M∗ − 5 log10 h = −20.7 and faint end slope of α = −1.26. The discussion so far has focused mostly on the stable, viri- The resulting d ln n(>L)/d ln L, computed using the flux alised regions. Clusters, however, are evolving structures and limit r = 17.77 appropriate for the SDSS spectroscopic sam- the mass within a fixed physical radius M(L)/d ln L over the galaxies used. The 7,800 clus- which would reduce the mass and would have an associated ters used by WHH were selected by applying a richness outflow. limit to the parent GMBCG catalog (Hao, J., et al. 2010) The combination of infall and the associated M˙ will re- that contains 55,000 clusters extending to Z =0.55. These sult in a positive offset of the mean line of sight velocity clusters were derived from the SDSS photometric catalog since the density will be slightly higher in front of the clus- that is much deeper than the spectroscopic catalog. Con- ter where we see the galaxies later and these galaxies will sequently, at the low redshifts where the spectroscopically be moving against us. There is also a potentially larger ef- selected galaxies live, this parent catalog is essentially vol- fect from the fact that along any line of sight we observe ume limited for the clusters used, so the redshift distribu- galaxies that lie in a cone that will be wider at the back tion for the cluster members used is essentially the same of the cluster, and at the same order, we need to allow for as that for the redshift distribution for the entire spec- the bias caused by the fact that the more distant galaxies troscopic sample, save for the fact that the GMBCG cat- will be fainter. These geometric and flux limit effects, whose alog has a lower redshift limit Zlim =0.1, which is very effects on the foreground and background galaxies was dis- 2 close to the redshift where dN/dZ = Z n(Z) peaks. This is cussed by Kim and Croft (2004), will cause a back/front the bell shaped curve in figure 2. Combining these we find anisotropy in the number of galaxies within the clusters 2 2 ⟨d ln n/d ln L⟩ = dZ Z n(Z)d ln n/d ln L/ dZ Z n(Z) ≃ ∆N/N ∼ 2H∆r(1 − δ(Z))/cz while the change in the phys- 2.0 with integrationR range 0.1 L)/d ln L⟩≃10. This may be a slight asymmetry ∆N/N ∼ (r/c)(M/M˙ ) ∼ Hr/c, where we have overestimate, as the cluster catalogue is not precisely vol- assumed that the mass within radius r for the ensemble av- ume limited and the actual dN/dZ may lie a little below the erage cluster is changing on a cosmological timescale. Evi- solid curve in figure 2 at the highest redshifts. dently, for low redshift clusters, the geometric and flux limit With this value, the surface brightness modulation ef- effects will tend to be the largers. fect is roughly a factor 10 larger in amplitude than the light- To order of magnitude, the mean offset induced is 2 2 2 cone effect, but has opposite sign. For isotropic orbits the ⟨βz⟩∼H r /c z. The gravitational potential, for compar- combination of these gives a blue-shift 6 times as large as ison, is Φ/c2 ∼ (δρ/ρ)H2r2/c2 where δρ/ρ ≃ 200 at the the TD effect so the overall effect is therefore similar in am- virial radius. Thus, within the virialized region, this geo- plitude to the TD effect but with opposite sign, so it causes metric term is small compared to the gravitational redshift, the total observed effect to be larger than the gravitational but further out at the turnaround radius where δρ/ρ ∼ 5, effect alone rather than smaller. this is a substantial correction.

⃝c 0000 RAS, MNRAS 000,000–000 More implications of the transverse Doppler red/ blue-shift dichotomy • Contribution to cluster grav-Z from motions of stars in the BCG velocity dispersions are smaller than in cluster, but not • negligible • stars are unresolved so we get a transverse Doppler blue- shift • 21cm radio observations of galaxies sees mostly galaxies falling into cluster for first time as gas is • stripped within virial region • should have a large potential difference relative to BCG • but the prediction for δZ is highly dependent on whether one makes unresolved single dish (e.g. Aricebo) measurements or resolved (e.g. Westerbork, ASKAP) Corrected grav-z measurement

Fairly easy to correct for TD • 620 isotropic 7 anisotropic isotropic +LC+SB effects anisotropic 6 • TD depends on vel. disp. 600 5 v c

anisotropy [km/s] 580 los • LC+SB directly measured σ 4 560 3 • net effect is a blue-shift 540 2 ~-9km/s in centre, falling 0 0.2 0.4 0.6 0.8 1 1.2 -2.5 -2.4 -2.3 -2.2 -2.1 • R [Mpc] α to ~-6km/s at larger r 0.8 0.7 0mass distribution Substantial change in α (right 0.2 panel) from fitting the velocity dispersion profile with an isotropic (blue) or anisotropic • (red) model 0.1 of galaxy orbits. The solid lines in the left panelshowthebest-fittingprofilesofthe measured grav-z term 0 velocityprobability distribution dispersion profile. The contours in the right panel are the boundaries of the 1σ and 2σ -0.1 but still consistent with confidence-0.2 regions of the likelihood function. The error barsintheleftpanelrepresenttherange • containing-0.3 68 per cent of the marginal probability. dynamical mass estimate -4 -3 -2 -1 0 1 2 3 4 3 vlos [10 km/s]

Figure 1 Velocity distributions of galaxies combined from 7, 800 SDSS galaxy clusters. The line- of-sight velocity (vlos)distributionsareplottedinfourbinsoftheprojectedcluster-centric distances R.Theyaresortedfromthetoptobottomaccordingtotheorderof radial bins indicated in the upper left corner and offset vertically by an arbitrary amountforpresentationpurposes.Redlines present the histograms of the observed galaxy velocities in the cluster rest frame and black solid lines show the best fitting models. The model assumes a linear contribution from the galaxies which do not belong to the cluster and a quasi-Gaussian contribution from the cluster members (see SI for more details). The cluster rest frames and centresaredefinedbytheredshiftsandthe positions of the brightest cluster galaxies. The error bars represent Poisson noise.

7

21 Gravitational Redshifts in Clusters 7

WHH figure 1 appear to decrease by about 15% between the inner-bin and the outer points. This is reasonably con- − sistent with the expected σ2 ∝ r 0.2 trend predicted if galaxies trace mass, but this is perhaps fortuitous since the outer points are well outside the virial radius. Regardless of whether the galaxies at large radius are equilibrated or not, we can use the change in the observed velocity disper- sion with radius to obtain the differential TD+LC+SB effect which is shown, added to the GR effect, as the solid line in figure 3. The kinematic effects flatten out the predicted pro- file, so the prediction is quite different from the gravitational redshift alone. The situation is clearly rather complicated, especially when using BGCs as the origin of coordinates since the ef- fects depend on things like the relative velocities of the top ranked pair of cluster galaxies, and on the BCG halo prop- Figure 3. Data points from figure 2 of WHH and prediction based erties, that are quite poorly known. However, those factors on mass-traces-light cluster halo profile and measured velocity only influence the prediction for the innermost data point. dispersions as described in the main text. The dashed line is the The empirically based theoretical prediction for the profile gravitational redshift prediction, which is similar to the WHH of the redshift offset for the hot population as a function model prediction. The dot-dash line is the transverse Doppler of impact parameter at r⊥ > 0.6Mpc is the most robust; effect. The dotted line is the LC effect. The triple dot-dash line if galaxies are reasonable tracers of the mass then profile is the surface brightness effect. The solid curve is the combined should be very flat, quite unlike the GR effect from a NFW effect. profile. The predicted GR and total effects are shown in fig- ure 3. However, this analysis ignores the effect of secular would appear to be discrepant, but only at about the 1.5- infall and out-flow which we consider next. sigma level. − The NFW model predicts δz ≃−10 km s 1/c for the outer measurements r ≃ 3.3, 5.3Mpc, and the measurements straddle this value. While this model may provide a reason- 6 EFFECT OF INFALL AND OUTFLOW able description for isolated clusters in the virialised domain, it is not at all clear that it is appropriate to describe the com- The discussion so far has focused mostly on the stable, posite cluster being studied here. Tavio et al. (2008) have virialised regions. Clusters, however, are evolving structures claimed that beyond the virial radius the density in numer- and the mass within any fixed physical radius M(r)willin ical LCDM simulations actually falls off like ρ ∼ 1/r rather general be changing. Outside of the virial radius (generally than the ρ ∼ 1/r3 asymptote for the NFW profile, and the considered, inspired by the spherical collapse model, to be extended peculiar in-fall velocities found by Cecccarelli et the radius within which the mean enclosed mass density is al. (2011) also argue for shallow cluster profiles, but it is not 3π/Gt2) we expect to see net infall, and the enclosed mass at clear that these results are widely accepted. those radii will be increasing with time, while at still larger An alternative, and possibly more reliable, approach is radii there will be outflow tending asymptotically toward the to assume that galaxies trace the mass reasonably well, in Hubble flow. In the spherical collapse model the transition which case the density profile of the stacked cluster has the from inflow to outflow takes place at the turnaround radius 2 same shape as the cluster-galaxy cross correlation function where the mean enclosed mass density is ρt =3π/32Gt . (e.g. Lilje & Efstathiou, 1988; Croft et al, 1997). This has a This is for a matter dominated Universe; allowing for a cos- − power-law dependence ρ ∼ r γ with γ ≃ 2.2, i.e. intermedi- mological constant makes only a small change (Lokas & Hoff- ate between the NFW and Tavio et al. model predictions. man, 2001). −γ −γ For space density ρ(r)=ρ0(r/r0) ,wherer0 is an ar- For the empirically motivated ρ = ρ0(r/r0) 2−γ bitrary fiducial radius, the potential is Φ(r)=Φ0(r/r0) model the mean enclosed mass is ρ(r) = 3(γ − −1 2 γ−2 −γ and the 1-D velocity disperson, for isotropic orbits, is 1)(2πG) σ0 r0 r and the nominal virial radius is rvir = 2 2 2−γ 2 2 γ−2 2 2 1/γ σ (r)=σ0 (r/r0) with Φ0 = 2((1 − γ)/(2 − γ))σ0 . ((γ − 1)σ0 r0 t /2π ) ≃ 1.8Mpc using γ =2.2, r0 = −1 The projected velocity dispersion measured is related to 1Mpc, σ0 = 545 km/s and t ≃ 1/H =1/(70 kms /Mpc) 2 2 2−γ the 3-D velocity dispersion by σ (r⊥)/σ (r)= dy y (1+ and turnaround is at rt ≃ 8.7Mpc. − y2) γ/2 but the projected potential is relatedR to the 3- In the centres of clusters there may be softening of the D potential in the same way, so the projected quantities cores which would reduce the enclosed mass and would have 2 are related by Φ(r⊥) = 2((1 − γ)/(2 − γ))σ (r⊥). This an associated outflow. is the potential relative to infinity. The difference in pro- In any single cluster, the density may be changing jected potential between two projected radii r1 and r2 is rapidly — on the local dynamical timescale — especially 2 0.2 Φ(r2) − Φ(r1) ≃ 12σ (r1)(1 − (r1/r2) ) for γ =2.2. The during mergers and as clumps rain in, but for a compos- resulting GR effect is shown as the dashed line in figure 3 ite cluster such as considered here these rapid changes will and is actually quite similar to the shape of the profile for average out and the mass can only change on a cosmologi- the WHH NFW composite model. cal timescale: M˙ ∼ HM. For power law profile with γ ≃ 2 The FWHM of the bell-shaped velocity distributions in M ≃ 4πρr3 and M˙ ≃ 4πρr2v,wherev is the mean infall ve-

⃝c 0000 RAS, MNRAS 000,000–000 What does it all mean?

• Effect is very small - and hard to measure measuring 10km/s mean offset with 600km/s velocity • dispersion is truly impressive requires careful modeling of background & cluster velocity • distribution function f(v) • and predicting potential from kinematics is not trivial rather sensitive to assumed velocity distribution for the • brightest cluster galaxies (BCGs) used as centres but it probably is a real measurement of gravitational • redshift (+ special relativistic nuisance factors) What else does it mean? • Probe of curvature of space in GR? • matter tells space how to curve • space tells matter how to move.... • Sadly no.... • Effect does not rule out any most metric theories of gravity • motion non-relativistic matter & gravitational redshifts are determined only by ‘gtt’; the spatial metric is irrelevant • It is really a test of the equivalence principle Provides a test of theories that invoke long-range non- • gravitational forces in the “dark sector” • e.g. Gradwohl & Frieman 1992; Farrar & Peebles 2004; Farrar & Rosen 2007; Keselman, Nusser & Peebles 2010; and many, many more.... and (maybe) f(R) gravity. though such theories are already constrained by X-ray • temp. vs galaxy motions in clusters.... Future prospects...

• Can expect immediate improvements in measurement • 3x increase in number of redshifts available (BOSS) • and more to come: • optical: big-BOSS • radio: FAST, ASKAP-Wallaby+WNSHS • interesting to compare unresolved radio and optical • Extension to larger scales.... • Lots of rich material in the front-back asymmetry of the galaxy correlation function. What's wrong with the "kinematic picture"? • Recall Bunn & Hogg: "A gravitational redshift is just a Doppler shift viewed from an unnatural coordinate system" • Confusion of gravity and acceleration • Einstein rocket thought experiment: • the redshift caused by rocket motor is just a Doppler shift • From this perspective Pound & Rebka did not measure a gravitational redshift; they measured the non-gravitational redshift caused by the laboratory being accelerated • In GR the gravitational field is the Riemann tensor • or the tidal field in the Newtonian limit • So is there a truly gravitational component to the redshift? Are gravitational and Doppler shifts the same? 4 Nick Kaiser • Heavy lines are pair of accelerated observers the observers and the photon path then whatever happens t ✻ there can hardly be said to be a gravitational redshift. Similarly, while the velocity of an object depends on • with same constant the frame from which it is observed, the relative velocity acceleration of two objects in their centre of velocity frame is another s absolute quantity. Accelerated observers know that they are Light lines are a pair of being accelerated. Once they allow for this the accelerated • observers here would be in full agreement with the cop as freely falling observers to how fast the motorist was approaching. It is true that in the Pound & Rebka (1959) experiment • These pairs perceive the the wavelength shift ∆λ/λ = gh/c2 they measured is the same redshift for the photon same as the (constant) relative velocity of a pair of hypo- thetical freely-falling observers launched so as to be tangent (wiggly line) to the world-lines of the actual emitter and receiver at the s interaction events (this being the relative velocity in the • Redshift only depends on ‘lab’ or in the centre of velocity frame – the difference be- instantaneous velocity, not ing negligible – but not the difference in velocities at times of the actual events). But that is just telling us that this on the path before or after experimental result is fully accounted for by the fact that the interaction event. the real apparatus is being accelerated by non-gravitational ✲ stresses in the instrument supports and in the planet that is x But is this gravity? standing in the way of its natural free fall. From a Syngean • Figure 1. Schematic space-time diagram for exchange of a pho- perspective, Pound & Rebka did not measure a gravitational ton in flat space-time between a pair of freely-falling observers redshift at all as their experiment was simply not sensitive (thin lines) and between a pair of observers being subject to non- enough to measure the gravitational curvature or tide. gravitational acceleration (thick lines). Relative motions and ac- Accelerated observers are interesting, but are something celerations are assumed here to be aligned with the photon path. of a distraction. For redshifts between galaxies there are no Bunn & Hogg (2009) pointed out that for any such photon path non-gravitational forces to worry about; all real sources and and freely falling observers the emission and reception events can observers are freely-falling. Knowledge of the tidal field in be taken to lie on the world lines of a pair of observers who live the vicinity of the observers and along the photon path is on opposite ends of a uniformly accelerating rod with those world then all that is needed to calculate how the observers’ mo- lines being tangent to those of the freely falling observers.Thisis possible since one can choose the initial position and velocity of tions evolve and how photons exchanged between them get the rod to make the observer at one end of the rod be co-located redshifted. It does not matter that the gravity g is only de- and co-moving with the freely falling emitter at the emissionevent termined by local measurements up to an additive constant and one can then choose the length of the rod and its acceleration vector as that has no effect on any measurements made by so that, by the time the photon reaches the freely falling receiver observers in free-fall in the region where the tide has been the other end of the rod has caught up with it. The accelerated determined.2 observers perceive the rod to have fixed length, though in the So there is no ambiguity in defining the gravitational ‘lab-frame’ the rod will appear progressively foreshortened. The field, or in calculating its influence on photons or observers’ freely falling observers view the redshift as a Doppler effect with trajectories. The only possible ambiguity here is that if there ∆λ/λ = ∆v/c (for ∆v ≪ c)causedbytheirrelativemotion.The is non-vanishing tidal field and if one tries to decompose the accelerated observers would note that the redshift is related to their acceleration a and the rod length l by ∆λ/λ = al/c2. redshift into a 1st order Doppler effect and a gravitational effect then the latter will depend, possibly quite sensitively, on the time at which one choses to compare velocities to name, GR is an absolute theory since whether or not there is obtain the first order term. This is analogous to the inter- a gravitational field in some region of space is unambiguously pretation of the Bondi gravitational term as a correction of measurable from geodesic deviation of freely-falling test par- the Doppler term from final time to average time (see also ticles (though the values of the components of the curvature Chodorowski 2011). But the redshift itself is not ambigu- tensor are coordinate system dependent). The curvature, or ous, and if the relative velocity is chosen to be either at the tidal field, is unaffected by the presence of any observers time of emission, reception or, say, half way along the pho- (real or imaginary) who might be accelerated by rockets.1 If ton path there is no ambiguity. And, as we shall see, if we the curvature vanishes in the region of space-time containing compare the redshift to the change in separation D –which involves an average of the velocity over the photon travel time – there is no ambiguity either. 1 Rindler (1970) gives an interesting argument, which he at- tributes to Dennis Sciama, that the weight of objects sensed by an accelerated observer in a rocket can be thought of, in a Machian 2 For example, while it is widely believed that the dipole sense, as gravity arising from the relative acceleration of the rest anisotropy of the microwave background is the result of our be- of the Universe. That argument cannot be applied here, since the ing accelerated by large-scale structure, it is possible that some of acceleration of the imaginary intervening observers is determined the dipole is generated by a large-scale specific entropy gradient by the arbitrary choice of their velocities; this is generally varying (Gunn 1988), but this indeterminacy of the local value of g has along the photon path and the gradient of this is not equal to the no effect on local dynamics within the milky way or within the real tidal field. local supercluster say.

⃝c 0000 RAS, MNRAS 000,000–000 Equivalence principle & the Pound + Rebka experiment

• Einstein’s Equivalence Principle: Observers on earth (being accelerated by the stress in the ground under them imparting momentum to them) will see light being red-shifted (and all other local physics being modified) exactly as would a pair of astronauts in empty space being accelerated by a rocket motor. • Pound and Rebka (1959, 1960): He was right. • But if you replace non-inertial apparatus by freely falling kit with same instantaneous velocities then B&H say Doppler formula will apply. They are right too - almost exactly.... Why is the gravitational-z hidden in cosmology? • Cosmological redshift a la Bondi • Consider expanding sphere of dust and source A sending photon to receiver B • Photon suffers gravitational red-shift climbing up the potential and then a A Doppler red-shift on reception • For source B sending to A the photon has a Doppler red-shift (as seen in our B frame) then enjoys a gravitational blue-shift • But the net effect is the same. • The opposite gravitational shifts are cancelled by the Doppler shift change Whereas in general..... • Consider pair of freely-falling observers 1,2 in arbitrary gravitational field who exchange a photon. Use rigid, non-rotating lattice picture to calculate changes in • wavelength and proper separation (work in CoM frame) • work to 2nd order in v/c and 1st order in φ/c2 2 • Δλ/λ = n . (v1 - v2)t1 / c + ∫dr . (g2 - g(r)) / c (1) 2 • ΔD/D = n . (v1 - v2)t1 / c + Δr . (g2 - g1) / 2c (2) • Both are 1st order Doppler (with initial Δv) plus ‘tidal’ term • Spatially constant tidal field stretches λ just like D • includes Minkowski spacetime and FRW • but that's because of special symmetry of FRW • does not apply for a galaxy cluster • extra intrinsically gravitational term (gradient of tide) 8 Nick Kaiser it is if the relative velocity is taken to be in the rest-frame, ′ ✻W0(r)=−W1(r) or whether it also involves the gravity, as is the case if the ✲ velocity difference is at different times. This is at odds with r 8 NickSynge’s Kaiser statement that the curvature does not appear in the redshift. What about the change in the separation during the ′ it is if the relative velocity is taken to be in the rest-frame, W0(r)=−W (r) light-propagational time? Letting the centre of velocity ✻ 1 or whether it also involves the gravity, as is the case if the frame separations, in the absence of gravity, at reception ′ ✲ velocity difference is at different times. This is at odds with ✻W1(r)=−W2(r) r and emission8 be DR,DE =(D ± ∆D/2) we see from the Synge’scaption statement of Fig.Nick that 3 that the Kaiser curvature∆D/D is not does precisely not appear the same in the as +d redshift.the flat-space ∆λ/λ =2µβ.Butthedifference is of order ✲ What3 aboutit is if the the change relative in velocity the separation is taken to during be in the the rest-frame, −d ′ r β , so to our precision goal we can take them to be equal. ✻W0(r)=−W1(r) light-propagationalSwitchingor on whether gravity, time? it the also Letting fractional involves the the change centre gravity, of the of as separation velocity is the case if the ✲ velocity difference is at different times. This is at odds with ′ r framebetween separations, the particles in the as absence measured of by gravity, lattice based at reception observers W1(r)=−W (r) Synge’s statement that the curvature does not appear in the ✻ 2 and emission– i.e. the be changeDR,D inE the=( properD ± ∆ separationD/2) we in see the from centre the of 2 2 W2 =(r − d )θ(|r| − d)/2 captionvelocity of Fig.redshift. frame 3 that – is ∆D/D is not precisely the same as ✻ ✲ +d r the flat-space ∆λWhat/λ =2 aboutµβ.Butthedi the changeff inerence the separation is of order during the ✲ 3 light-propagational time? Letting the centre2 of velocity r β , so to our∆D/D precision= n · ( goalv2 − wev1) cant1 /c + take∆r them· (g2 − tog1 be)/2c equal.(4) −d frame separations, in the absence of gravity, at reception ′ Switching on gravity, the fractional change of the separation ✻W1(r)=−W2(r) and emission be D ,D =(D ± ∆D/2) we see from the betweenwhich the is particles also a simple as measured NewtonianR byE latticelooking based result. observers Unsurpris- caption of Fig. 3 that ∆D/D is not precisely the same as ingly, to first order in the relative velocity the fractional +d – i.e. the changethe flat-space in the proper∆λ/λ separation=2µβ.Butthedi in theff centreerence of is of order 2 2 ✲ changes in wavelength and separation are identical. Both ✻W2 =(r − d )θ(|r| − d)/2 velocity frameβ3 –, so is to our precision goal we can take them to be equal. −d ✲ r contain an additional gravitational term that is, to lowest Figure 4. The upper plot shows, schematically, the dimensionless r Switching on gravity, the fractional change of the separation order, a tidal effect. In general, these gravitational2 effects weight function W0(r)=θ+(r)θ−(r) − d(δ(r − d)+δ(r + d))/2 ∆D/Dbetween= n · (v the2 − particlesv1)t1 /c as+ ∆ measuredr · (g2 − byg1 lattice)/2c based(4) observers ′ are not precisely equal – g2 − g1 is the integral of the tide that, when multiplied by the gravity φ (r)givesthedifference – i.e. the change in the proper separation in the centre of∆λ/λ − ∆D/D.TheDiracδ-functions are shown2 as the2 narrow while the wavelength shift involves the integral of the grav- W2 =(r − d )θ(|r| − d)/2 which is alsovelocity a simple frame Newtonian – is looking result. Unsurpris- box-cars at r = ±d and together have (minus)✻ the same weight as ✲ ity – so the ‘cosmological relation’ that wavelengths vary r ingly,precisely to first inorder proportion in the to relative the source-observer velocity the proper fractional sepa- the central box-car. As described in the text, this difference can 2 also be computed as a weighted average of the tide φ′′(r)using changesration, in wavelength does not∆D/D hold and= inn general. separation· (v2 − v1)t1 are/c + identical.∆r · (g2 − Bothg1)/2c (4) the weighting function W1(r)showninthecentreplot,whichis contain anBut additional if the tide gravitational is spatially termconstant that – is,i.e. to the lowest poten- Figure 4. The upper plot shows, schematically, the dimensionless which is also a simple Newtonian looking result. Unsurpris-(minus) the integral of W0(r), and also has zero net weight. The order,tial a tidal has no effect. spatial In derivatives general, these higher gravitational than second eff –ects then weightthird way function to computeW0(r)= the diθ+fference(r)θ−( isr) averaging− d(δ(r − thed)+ gradientδ(r + odf))/2 ingly, to first order in the relative velocity the fractional ′′′ ′ are notthe precisely gravity varies equal linearly – g2 − withg1 is position the integral then weof the can tide write that,the tide whenφ multiplied(r)withtheweightfunctionshowninthebottom by the gravity φ (r)givesthedifference changes in wavelength and separation are identical. Both whileg the(r)= wavelengthg1 +(g2 − shiftg1)|r involves− r1|/|r2 the− r1 integral| and the of gravitational the grav- ∆plot.λ/λ − ∆D/D.TheDiracδ-functions are shown as the narrow contain an additional gravitational term that is, to lowest Figure 4. ity –terms so the are ‘cosmological readily found relation’ to be identical. that wavelengths Thus the relation vary box-cars at r = ±dTheand upper together plot have shows, (minus) schematically, the same the weight dimensionless as seen inorder, cosmology a tidal is of eff widerect. In generality, general, these and applies gravitational for an effectsthe centralweight box-car. function As describedW0(r)=θ+ in(r the)θ−( text,r) − d this(δ(r di−ffderence)+δ(r can+ d))/2 precisely in proportion to the source-observer proper sepa- ′ arbitraryare pair not of precisely particles equal moving – g in2 − ag field1 is the that integral has a spa- of the tidealso beSecond, computedthat, taking when as multiplied the a weighted difference by average the of gravity (3) of and theφ (4),(r tide)givesthedi weφ have′′(r)usingfference ration, does not hold in general. ∆λ/λ − ∆D/D.TheDiracδ-functions are shown as the narrow tially constantwhile the tide. wavelength This includes, shift involves as a special the integral case, a pairof the grav-the weighting function W1(r)showninthecentreplot,whichis But if the tide is spatially constant – i.e. the poten- box-cars at r = 1±d and together have (minus) the same weight as of particlesity – with so the relative ‘cosmological motion along relation’ their thatseparation wavelengths in a vary(minus)∆ thelog( integralλ/D)= of W0(dr),n and· (g1 also+ g2 has) − zerodr net· g weight.(6) The the central box-car.c2 „ As described in theZ text, this« difference can tial has no spatialprecisely derivatives in proportion higher to the than source-observer second – then proper sepa-third way to compute the difference is averaging the gradient′′ of quadratic potential as in a FRW model containing matter also be computed as a weighted average of the tide φ (r)using the gravity variesration, linearly does not with hold position in general. then we can write the tide φ′′′(r)withtheweightfunctionshowninthebottom and/or dark energy. Note that there is no need for the parti- where dthe≡ weighting|r2 − r1|/ function2. TakingW1 the(r)showninthecentreplot,whichis origin of coordinates to g(r)=g1 +(g2 −Butg1)| ifr − ther1 tide|/|r2 is− spatiallyr1| and the constant gravitational – i.e. the poten- cles to be co-moving with the matter density, though again plot.lie at (r1(minus)+ r2)/2 the for integral simplicity, of W0 this(r), is and a weighted also has zero average net weight. of The ′ termsthis are result readilytial does has found apply no spatial to in be that identical. derivatives situation. Thus higher the than relation second – thenthe gravitythird− wayφ : to compute the difference is averaging the gradient of ′′′ seen in cosmologyThisthe is gravity one is of of the wider varies two linearlygenerality, main results with and of position this applies paper: then for a we con- an can write the tide φ (r)withtheweightfunctionshowninthebottom ′ g(r)=g1 +(g2 − g1)|r − r1|/|r2 − r1| and the gravitational 1 arbitrarystant pair tide of stretches particles wavelength moving of in radiation a field that just has as it a changes spa- Second,plot. ∆ takinglog(λ/D the)= differencedr Wof0 (3)(r)φ and(r) (4), we have (7) c2 Z tiallythe constant separationterms tide. are of This test-particles. readily includes, found Arguably as to a be special identical. this case, ‘explains’ Thus a pair the the relation seen in cosmology is of wider generality, and applies for an 1 of particlesapparent with stretching relative of motion wavelength along of their light separation by the expansion in a with∆ dimensionlesslog(λ/D)= weightingd n · (g1 function+ g2) − Wd0r(r· )g ≡ (6) arbitrary pair of particles moving in a field that has a spa- Second,Subtracting takingc2 „ the diff erence(1) - of(2) (3)Z gives and (4), « we have quadraticof space potential in FRW as models. in a FRW model containing matter θ+(r)θ−(r)−d(δ(r−d)+δ(r+d))/2; where φ(r) ≡ φ(r = rn); tially constant tide. This includes, as a special case, a pair ± and/or darkTo energy. highlight Note the that differences there is between no need separationfor the parti- and whereand whered ≡ θ|r2(r−) r≡1|θ/(2.±r Taking− d)with1 theθ( originr) and ofδ( coordinatesr) denoting to 8 Nick Kaiser∆ log(λ/D)= d n · (g1 + g2) − dr · g (6) of particles with relative motion along their separation inthe a Heaviside function and the2 Dirac delta function respec- or cles towavelength be co-moving changes with – the what matter one density, might though call the again ‘non- lie at (r1 + r2)/2 for simplicity,c „ this is a weightedZ average« of quadratic potential as in a FRW model containing matter ′ tively.it is The if the weight relative′ velocity function is takenW to0 be(r in) the is rest-frame, shown schematically W0(r)=−W (r) this resultkinematic’ does apply component in that of the situation. redshift – and to see how this the gravity −φ : ✻ 1 and/or dark energy. Note that there is no need for the parti-as theor whetherwhere upper itd also plot≡ involves|r in2 − Fig. ther1 gravity,|/2. 4. Taking asThe is the product case the if the origin of Heaviside of coordinates to ✲ depends on the tide (and its derivatives) we note the follow- velocity difference is at different times. This is at odds with r This is onecles of to the be two co-moving main results with the of matter this paper: density, a con- though again lie at (r1 + r2)/2 for simplicity, this is a weighted average of ing: functionsSynge’s is statement zero for that the|r′| curvature>d1so does the not appear range in of the ′ integration is stant tide stretchesthis result wavelength does apply of radiation in that situation. just as it changes redshift.the gravity∆ log(λ−/Dφ :)= 2 dr W0(r)φ (r) (7) First, we can write now unrestricted. The integralc ofZ W0(r) over all r vanishes, the separation ofThis test-particles. is one of the Arguably two main this results ‘explains’ of this paper: the a con- What about the change in the separation during the so welight-propagational can immediately time? Letting integrate the centre by1 parts of velocity to eliminate′ the apparent stretchingstant tide of stretches wavelength wavelength of light of by radiation the expansion just as it changeswithframe dimensionless separations, in∆ thelog( absence weightingλ/D of)= gravity,2 at function receptiondr W0(r)φW(r0)(r) ≡ (7) ′ ∆r ′′ gravity and write (7) as an integralc of the tide: ✻W1(r)=−W2(r) and emission be DR,DE =(D ± ∆D/2) we seeZ from the of space in FRWthe∆D/D separation models.= n · (v of2 − test-particles.v1)t1 /c − Arguably2 drφ ( thisr) ‘explains’ (5) theθ+(r)θ−(r)−d(δ(r−d)+δ(r+d))/2; where φ(r) ≡ φ(r = rn); 9 2c Z caption of Fig. 3 that ∆D/D is not precisely the same as Redshifts and the Expansion of Space apparent stretching of wavelength of light by the expansion with± dimensionless1 weighting′′ function W0(r) ≡ +d ✲ To highlight the differences between separation and and wherethe flat-spaceθ (r∆)λ/λ≡=2θ(µ±β.Butthedir − d)withfference1 isθ( ofr) order and δ(r) denoting 3 ∆ log(λ/D)= 2 dr W (r)φ (r)−d (8) r of space in FRW models. β ,θ so+ to(r our)θ− precision(r)−d goal(δ( wer− cancd)+ takeZ δ( themr+d to)) be/2; equal. where φ(r) ≡ φ(r = rn); wavelengthwhere φ changes(r)=φ(rn –) is what the gravitational one might potential call the and ‘non- prime themore HeavisideSwitching by parts on gravity,function to express the fractional and the the change diff Diracerence of the separation delta of these function as a weighted respec- The perfect correlation of ∆ log(λ) and ∆ log(D) in ho- To highlight the differences between separation and betweenand the where particles′′′ θ as± measured(r) ≡ θ by(± latticer − basedd)with observersθ(r) and δ(r) denoting kinematic’denotes component the operator of the∂r = redshiftn · ∇, i.e. – and the to spatial see how derivative this tively.whereaverage TheW1( ofr weight)=φ (r−): functiondr W0(rW)=0(r−)r isθ+ shown(r)θ−(r)whichis schematically mogeneous models can be considered to be a reflection of ′ – i.e. the change in the proper separation in the centre of 2 2 wavelength changes – what one might call the ‘non- the Heaviside function and the Dirac delta function respec- W2 =(r − d )θ(|r| − d)/2 dependsalong on the the photon tide (and path, its so derivatives)φ (r)=n · ∇ weφ( noter)= the−n · follow-g.Thus asshown thevelocity is upper the frame middle – plot is R in plot Fig. in Fig. 4. The (4). product But the integral of Heaviside of the✻ symmetry of the✲ gravitational fields that are allowed in kinematic’ component of the redshift – and to see how this tively. The weight function1 W0(r) is′′′ shown schematically r the gravitational contribution to ∆D/D is the average of functionsW1(r) also is vanishes zero for (so,|r| as>d alreadyso the mentioned range of for integration spatially is ing: ∆ log(λ/D)= 2 dr W22(r)φ (r) (9) these models, in accord with the conjecture of Melia (2012). depends on the tide (and its derivatives)2 we note the follow- as∆D/D the= uppern · (v2 − v plot1)t1 /c + in∆cr · Fig.(g2 − g 4.1)/2 Thec (4) product of Heaviside the tide along the photon path times (∆r/c) /2. nowconstant unrestricted. tide ∆λ/λ The= ∆ integralD/D)Z and of W we0(r can) over integrate all r vanishes, once First, we can write functions is zero for |r| >dso the range of integration is In the Introduction we asked what redshift would be ing: which is also a simple Newtonian looking result.2 Unsurpris-2 2 so we can immediately integrate by parts to− eliminate the withingly, weightnow to first unrestricted. function order in theW relative2 The(r)=( velocity integralr − thed of fractional)θW+(0r()rθ) over(r)/2=( all r vanishes,r − seen by a pair of observers in a cluster who have the same First, we can write 2 ⃝c 0000 RAS, MNRAS 000,000–000 ∆r ′′ gravityd )θchanges(| andsor| − we in writed wavelength) can/2 now (7)immediately and asshown separation an integral as integrate are the identical. bottom of the by Both plotparts tide: in to figure eliminate (4). the separation at emission as at reception. Our analysis shows ∆D/D = n · (v2 − v1) 1 /c − drφ (r) (5) contain an additionalThere gravitational is a non-kinematic term that is, to lowest component of the redshift: it is a t 2 Figure 4. The upper plot shows,that schematically, they do the dimensionlessnot see the effect of any local tidal field. What 2c Z ∆r ′′ Thisorder, isgravity the a tidal second e andffect. In write main general, (7) result these as gravitational an of integral this paper. effects of theweight tide: function W0(r)=θ+(r)θ−(r) − d(δ(r − d)+δ(r + d))/2 n v v 1 ′′ ′ ∆D/D = · ( 2 − 1)t1 /c − 2 drφ (r) (5) are not precisely equal – g2 −measurementg1 is the integral of the tide of thethat, whengradient multiplied by of the happens gravitythe φtide(r)givesthedi is thatfference any gravitational stretching of the wave- 2c Z ∆ log(λ/D)= 2 dr W1(r)φ (r) (8) while the wavelength shift involvesc theZ integral1 of the grav- ∆λ′′/λ − ∆D/D.TheDiracδ-functions are shown as the narrow box-cars at r = ±d and togetherlength have (minus) that the wouldsame weight be as perceived in the frame of the rigid non- where φ(r)=φ(rn) is the gravitational potential and prime ity – so the ‘cosmological∆ log( relation’λ/D that)= wavelengths2 dr vary W1(r)φ (r) (8) precisely in proportion to the source-observerc properZ sepa- the central box-car. As described in the text, this difference can denotes the operatorwhere φ(r∂)=r =φ(nrn·)∇ is, the i.e. gravitational the spatial derivativepotential and primewhere W1(r)=− dr W0(r)=−rθ+(r)θ−also(r)whichis be computed as a weightedinertial average of observers the tide φ′′(r)using is counteracted by the red- or blue-shifts 3ration, DISCUSSION does not hold in general. ′ the weighting function W1(r)showninthecentreplot,whichis along the photondenotes path, the so operatorφ (r)=∂rn=· ∇nφ·(∇r)=, i.e.− then · g spatial.Thus derivativeshown iswhereBut the if the middleW tide1(r is)=R spatially plot− constant indr Fig. W –0 i.e.( (4).r the)= poten- But−rθ the+(r) integralθ−(r)whichis of in boosting from the observers’ frames to the rigid frame ′ (minus) the integral of W0(r), and also has zero net weight. The along the photon path, so φ (r)=n · ∇φ(r)=−n · g.ThusWetial haveshown has nomade spatial is various the derivatives middle simplifying higherR plot than in second assumptions. Fig. – then (4). Butthird The the way results to integral compute the of differenceand is back averaging again the gradient (if, o forf example they were moving apart at the gravitational contribution to ∆D/D is the average of W1(r)the also gravity vanishes varies linearly (so, with as position already then we mentioned can write for spatially′′′ 2 the tide φ (r)withtheweightfunctionshowninthebottomthe emission time they will be moving together again by the the tide alongthe the gravitational photon path contribution times (∆r/c to)∆/D/D2. is the averageconstant ofareg only(r)=W tide1g(1 validr+()g∆ also2 −λ upg/1)λ vanishes|r to−=r1 2nd|/∆|r2D/D− order (so,r1| and) as andthe in already gravitational velocity we can mentioned and integrateplot. 1st order for once spatially in the tide along the photon path times (∆r/c)2/2. theterms potential,constant are readily but tide found this∆ toλ be/ isλ identical.= adequate∆D/D Thus the) to andrelation encompass we can integrate the phe- once time of reception). seen in cosmology is of wider generality, and applies for an nomenaarbitrary that pair of have particles proved moving⃝c in0000 controversial. a field RAS, that has MNRAS a spa- We have000Second,,000–000 ignored taking the difference ofIn (3) an and inhomogeneous (4), we have system such as the solar system, anytially time constant variation tide. This of includes, the as potential, a special⃝c 0000 case, and RAS, a pair we MNRAS have000 also,000–000 ig- 1 or a galaxy, cluster or supercluster, the tidal field neces- of particles with relative motion along their separation in a ∆ log(λ/D)= d n · (g1 + g2) − dr · g (6) c2 „ Z « noredquadratic any potential bending as in of a FRW light model rays. containing These matter are higher order ef- sarily varies with position. There is then a non-kinematic fects;and/or the dark Rees-Sciama energy. Note that there (1968) is no need effect for the is parti- of orderwhere (v/cd ≡)|3r2 −forr1|/2. Takingcomponent the origin of coordinates to the redshift to that violates (1) and which is es- cles to be co-moving with the matter density, though again lie at (r1 + r2)/2 for simplicity, this is a weighted average of instance.this result We does apply can in also that situation. ignore spatial curvature.the In gravity a FRW−φ′: sentially gravitational in nature. Combining this with the This is one of the two main results of this paper: a con- model, for example, the distance between two FOs as mea- 1kinematic′ redshift component, if any, one obtains complete stant tide stretches wavelength of radiation just as it changes ∆ log(λ/D)= dr W0(r)φ (r) (7) c2 Z suredthe separation by observers of test-particles. on a Arguably ruler this is not ‘explains’ precisely the the same as consistency with the conventional view of the gravitational theapparent distance stretching as determined of wavelength of light by by a the chain expansion of FOswith lying dimensionless along weightingredshift function in clustersW0(r) of≡ galaxies and other gravitating systems. of space in FRW models. θ+(r)θ−(r)−d(δ(r−d)+δ(r+d))/2; where φ(r) ≡ φ(r = rn); a geodesicTo highlight in the the 3-space differences of between constant separation proper and timeand where sinceθ±( ther) ≡ θ(±r − dNote)withθ that(r) and ourδ(r) denoting description of components of the redshift is wavelength changes – what one might call the ‘non- the Heaviside function and thediff Diracerent delta from function the respec- terminology of Chodorowski (2011) who big-bang,kinematic’ but component the fractional of the redshift di – andfference to see how is at this mosttively. of order The weight the function W0(r) is shown schematically squaredepends of on the the tideseparation (and its derivatives) in units we note of the the follow- curvatureas the radius upper (or plot in Fig.was 4. The considering product of Heaviside the gravtiational component of the redshift ing: 2 functions is zero for |r| >dinso FRW the range models of integration that is arises if the ‘kinematic’ component is of orderFirst, (v/c we can) ) write so any effect on ∆D/D is of highernow unrestricted. order. The integral of W0(r) over all r vanishes, Similarly the asynchronicity between clocks carriedso we can by immediately our integratetaken by parts to be to eliminate the relative the velocity at emission or reception ∆r ′′ gravity and write (7) as an integral of the tide: ∆D/D = n · (v2 − v1) 1 /c − drφ (r) (5) non-inertial grid-basedt observers2c2 is negligible at our stated rather than the average velocity. Here we consider redshifts Z 1 ′′ ∆ log(λ/D)= dr W1(r)φ (r) (8) level of precision. c2betweenZ observers in FRW models to be purely kinematic where φ(r)=φ(rn) is the gravitational potential and prime in the sense that (1) is obeyed. denotesMore the fundamentally operator ∂r = n · ∇, while i.e. the spatial matter derivative tells space-wheretimeW1(r)=how− dr W0(r)=−rθ+(r)θ−(r)whichis ′ to curve,along the photon it is path, only so φ the(r)= curvaturen · ∇φ(r)=−n of· g.Thus time thatshown tells is the non- middleR plot in Fig.For (4). But emitter/receiver the integral of pair separation that is small com- the gravitational contribution to ∆D/D is the average of W1(r) also vanishes (so, as already mentioned for spatially relativisticthe tide along matter the photon how path to times move, (∆r/c) and2/2. is is also onlyconstant the tide time-∆λ/λ = ∆D/Dpared) and we to can the integrate size once of the gravitating system the difference time part of the metric perturbation that is relevant for the between the fractional change in the wavelength and sep- ⃝c 0000 RAS, MNRAS 000,000–000 calculation of the redshift. So at the specified level of preci- aration is on the order of the gravitational potential well sion we can ignore the spatial part of the metric. depth times the cube of the separation in units of the over- We have asked: what is the domain of validity of the re- all system size. This is seen most easily from (9), and the 2 lationship between wavelengths and emitter/receiver proper fact that W2(r) ∼ d , which together imply ∆ log(λ/D) ∼ separation (1) that we see for FOs in homogeneous mod- (d/R)3φ/c2. So, just as the local gravity is invisible to freely- els? We have shown that a spatially constant tide stretches falling observers, as far as the ratio of wavelength to separa- wavelength in exactly the same way it affects the observers’ tion is concerned, the local tide is also invisible. But if the separation, but if the tide varies with position the relation- path length has a similar size to the entire system the error ship between wavelength and separation is modified. is on the order of the gravitational potential. From this perspective, the perfect correlation seen be- To see better how this relates to Synge’s result that tween changes in wavelengths ∆λ/λ of light exchanged be- the redshift is always given by the Doppler formula, con- tween FRW FOs and the change ∆D/D in the space (be- sider the case of an emitter at the centre of the poten- tween said FOs) is not a causal relationship, rather both tial for a small uniform spherical distribution of matter of the change in the wavelength and the change in the space mass M and radius r and a receiver outside at distance between the observers are ‘caused’, or determined, by a com- D ≫ r who happens to be at rest at the moment of re- bination of the observers’ initial velocities and the tidal field ception. In this situation, the redshift is just the static in which they and the photons propagate. Echoing Whiting gravitational redshift: ∆λ/λ = dr g/c2 ∼ GM/c2r.But (2004), the expansion rate defined by the matter content the more distant the receiver,R the smaller any fractional of the universe is irrelevant (which is a jolly good thing if change in the emitter/receiver proper separation during the universe has a cosmological constant or a scalar field the time of flight: ∆D ! (GM/D2)(∆t)2/2whichimplies to realise dark energy since neither defines either a frame of ∆D/D ! GM/c2D ≪ ∆λ/λ. Evidently the kinematic re- motion or a state of expansion). Rather, on the parabolic po- lation does not apply here. But, following Peebles, we can tential generated by gravitating matter and dark energy one still break the net wavelength ratio down into the product can have emitter/receiver pairs that recede from each other of ratios between a set of pairs of neighbouring particles. or pairs that approach each other, and what determines both We can take these to be particles on a set of radial orbits the changes in the wavelengths and proper separations is a such that each particle is at apogee at the time the photon combination of initial conditions and the curvature, or tidal passes (see Fig. 5). Thus the nth particle has zero velocity field. as the photon passes it, as does the (n+1)th particle. In the

⃝c 0000 RAS, MNRAS 000,000–000 Conclusions • Gravitational redshifts in clusters of galaxies have been measured! • Technically challenging but apparently real and prospects for better measurements and extension to larger scales is promising. • Potentially useful for testing some alternatives to GR • But mainly interesting as a "sand-box" that illustrates some subtleties of simple special relativity + Newtonian gravity • Effect raises some questions of principle about how to think about redshifts in cosmology and astronomy in general. • Redshifts are not entirely kinematic - there is an truly gravitational component - but it is hidden in cosmology

A strange incident in the history of physics (C. Moller, 1967) • 1905 - Einstein establishes SR By 1909, Planck, Einstein, Pauli all concluded that temperature of • a moving body is T(rest frame) / γ • Enshrined in text books (e.g. Tolman) and there it rested • until ‘60s, when Ott (1963) and Arzelies (1965) turned it all around T = γ T(rest frame) • much confusion ensued P.T. Landsberg (2 Nature articles, ’66, 67) “Does a moving • body appear cool” (ans: no!) • largely clarified by Kibble, ’66: Ott, Arzelies were right! • issue reverberates to this day: • Dunkel, Haenggi, & Hilbert 2009 - light-cone effect • But now seems anachronistic....

Biro & Van 2010

ABOUT THE TEMPERATURE OF MOVING BODIES 7

2.0 Doppler blue shift

1.5 Blanusa ! Ott 2 T

! 1.0 1 ! T Planck Einstein Landsberg

0.5 Doppler red shift

0.0 !1.0 !0.5 0.0 0.5 1.0

w2

Figure 1. Ratio of the temperatures of the observed body in its rest frame, T2 to that shown by an ideal thermometer, T1 as a function of the the speed of the heat current in the body, w2 while approaching with the relative velocity v = 0.6. −

v"!0.6

u2 g2 g1 u1

w1 w2

Figure 2. The space-time figure for the Planck-Einstein rule of two thermodynamic bodies in equilibrium. There is no energy a a current in the observed body (w2 =0),thereforetheu2 four- a a velocity (solid arrow) is parallel to the vectors (g1 ,g2 ). The ratio of the temperatures is T1/T2 < 1.

5. Summary We investigated the possible derivation of basic thermodynamical lawsforho- mogeneous bodies from relativistic hydrodynamics. The dependenceofentropy on internal energy is replaced by a dependence on the energy-momentum four- vector, Ea. As a novelty a relativistic heat four-vector has been formulated. For the traditional, energy exchange related temperature, T , a universal transforma- tion formula is obtained. For a general observer four velocities are involved in the equilibrium condition of two thermodynamic bodies in equilibrium. One of them can be eliminated by choosing the observing frame, the physical relation depends only on the relative velocity. Another condition connects the internal heat currents in the bodies in thermal contact. So there remains two velocity like parameters to describe thermal equilibrium: the energy current speed (the velocity related to the integrated internal heat current density) in one of the bodies and their relative Conclusions: 1) redshifts in general • Redshifts in FRW are not caused by the expansion of space • both changes in wavelength and changes in the space between observers are ‘caused’ by the initial relative velocities and the tidal field in which the observers and photon move. • In homogeneous models they happen to be equal because a constant tide stretches λ the same way it stretches D • But neither can they be thought of as being essentially kinematic, or Doppler, in nature in presence of inhomogeneity • that gives the wrong answer for the gravitational redshift component of the redshift • which is an integral of the gradient of the tide along the photon path • Allowance for this largely reconciles conventional view with direct calculation (modulu kinematic TD, LC, SB effects) Conclusions: 2) cluster gravitational-Z • Gravitational redshifts in clusters have been measured! • but the interpretation is considerably more complicated than originally thought • The static gravitational redshift is augmented by 3 other kinematic effects that are generally of the similar magnitude • time dilation + light-cone effect + relativistic beaming • These measurements essentially provide a test of the equivalence principle • i.e. whether light and galaxies “fall” the same way in clusters • constrains “5th force” theories • and we can expect more precise measurements in the near future Gravitational Redshifts in Clusters 3

dN/dZ

Figure 1. Spectral index vs. redshift for representative galaxy Figure 2. The dot-dash curve is the logarithmic derivative of types observed in Sloan r-band the comoving density of objects above the luminosity limit asa function of redshift. The bell-shaped curve is dN/dZ = Z2n(Z) and the solid curve is that truncated at the minimum redshift imposed by the parent cluster catalogue. The mean of the log- derivative, averaged over the redshift distribution turns out to be ≃ 2.0. luminosity function does not, and the parameters are not very different from the field galaxy luminosity function, so we will use the latter, as determined by Montero-Dorta & Prada (2009), as a proxy. Their estimate of the LF obtained 4 EFFECT OF SECULAR INFALL from the r-band magnitudes K-corrected to Z =0.1 has M∗ − 5 log10 h = −20.7 and faint end slope of α = −1.26. The discussion so far has focused mostly on the stable, viri- The resulting d ln n(>L)/d ln L, computed using the flux alised regions. Clusters, however, are evolving structures and limit r = 17.77 appropriate for the SDSS spectroscopic sam- the mass within a fixed physical radius M(L)/d ln L over the galaxies used. The 7,800 clus- which would reduce the mass and would have an associated ters used by WHH were selected by applying a richness outflow. limit to the parent GMBCG catalog (Hao, J., et al. 2010) The combination of infall and the associated M˙ will re- that contains 55,000 clusters extending to Z =0.55. These sult in a positive offset of the mean line of sight velocity clusters were derived from the SDSS photometric catalog since the density will be slightly higher in front of the clus- that is much deeper than the spectroscopic catalog. Con- ter where we see the galaxies later and these galaxies will sequently, at the low redshifts where the spectroscopically be moving against us. There is also a potentially larger ef- selected galaxies live, this parent catalog is essentially vol- fect from the fact that along any line of sight we observe ume limited for the clusters used, so the redshift distribu- galaxies that lie in a cone that will be wider at the back tion for the cluster members used is essentially the same of the cluster, and at the same order, we need to allow for as that for the redshift distribution for the entire spec- the bias caused by the fact that the more distant galaxies troscopic sample, save for the fact that the GMBCG cat- will be fainter. These geometric and flux limit effects, whose alog has a lower redshift limit Zlim =0.1, which is very effects on the foreground and background galaxies was dis- 2 close to the redshift where dN/dZ = Z n(Z) peaks. This is cussed by Kim and Croft (2004), will cause a back/front the bell shaped curve in figure 2. Combining these we find anisotropy in the number of galaxies within the clusters 2 2 ⟨d ln n/d ln L⟩ = dZ Z n(Z)d ln n/d ln L/ dZ Z n(Z) ≃ ∆N/N ∼ 2H∆r(1 − δ(Z))/cz while the change in the phys- 2.0 with integrationR range 0.1 L)/d ln L⟩≃10. This may be a slight asymmetry ∆N/N ∼ (r/c)(M/M˙ ) ∼ Hr/c, where we have overestimate, as the cluster catalogue is not precisely vol- assumed that the mass within radius r for the ensemble av- ume limited and the actual dN/dZ may lie a little below the erage cluster is changing on a cosmological timescale. Evi- solid curve in figure 2 at the highest redshifts. dently, for low redshift clusters, the geometric and flux limit With this value, the surface brightness modulation ef- effects will tend to be the largers. fect is roughly a factor 10 larger in amplitude than the light- To order of magnitude, the mean offset induced is 2 2 2 cone effect, but has opposite sign. For isotropic orbits the ⟨βz⟩∼H r /c z. The gravitational potential, for compar- combination of these gives a blue-shift 6 times as large as ison, is Φ/c2 ∼ (δρ/ρ)H2r2/c2 where δρ/ρ ≃ 200 at the the TD effect so the overall effect is therefore similar in am- virial radius. Thus, within the virialized region, this geo- plitude to the TD effect but with opposite sign, so it causes metric term is small compared to the gravitational redshift, the total observed effect to be larger than the gravitational but further out at the turnaround radius where δρ/ρ ∼ 5, effect alone rather than smaller. this is a substantial correction.

⃝c 0000 RAS, MNRAS 000,000–000 620 isotropic 7 anisotropic isotropic anisotropic 6 600

5 v c [km/s] 580 los σ 4

560 3

540 2 0 0.2 0.4 0.6 0.8 1 1.2 -2.5 -2.4 -2.3 -2.2 -2.1 R [Mpc] α

Supplementary Figure 2 Velocity dispersion profile of the composite cluster (left panel) and constraints on the concentration parameter cv and the logarithmic slope of the mass distribution α (right panel) from fitting the velocity dispersion profile with an isotropic (blue) or anisotropic (red) model of galaxy orbits. The solid lines in the left panelshowthebest-fittingprofilesofthe velocity dispersion profile. The contours in the right panel are the boundaries of the 1σ and 2σ confidence regions of the likelihood function. The error barsintheleftpanelrepresenttherange containing 68 per cent of the marginal probability.

21 Possible resolution of staticRedshifts and and the“kinematic” Expansion of Space view?5

What then of points (ii) andConsider (iii) above, thatparticles one should in r ✻ consider all redshifts to be• Doppler, or kinematic, in nature and that, perhaps with restrictionsequilibrium on recession in velocity potential and ❄ ❜ curvature, one should be confidentwell in applying the Doppler ✻ g(∆t)2/2 formula? photon emitted by The legitimacy of considering• any redshift to be the product of a lot of little Doppler“cold” shifts particle (with the at under- standing that the relative velocitybottom for each of pair potential of particles is in their centre-of-velocity frame)well isand not inreceived question. Norby is Synge’s result that the overallrandomly redshift ischosen given by “hot” the Doppler formula and that, forparticle low redshift, at Synge’s larger velocity radius ❜ ✲ must be equal to the sum of the incremental velocities as de- ✛ ∆t ✲ t fined here. But the burning• questionFor quadratic is: What does potential this net , ‘velocity’ mean physically? AsSynge’s Synge said, velocity arguments is about the Figure 2. Illustration of situation described in text where a pho- whether a spectral shift is a gravitationalaverage velocity or velocity eoverffect ton is emitted by a ‘cold’ particles near the centre of a smooth are just ‘windy warfare’ without analysing the meaning of potential well and is received by a randomly chosen ‘hot’ parti- the time of flight of the cle at large distance. A sample of orbits from the distribution of the terms being employed. So, how does Synge’s velocity photon velocities – here a simple box-car – is shown as curves. The orbit relate to the actual relative velocity of the source and ob- for the average velocity particle is shown as the heavy curve.The server? • Reconciles “kinematic” average radial velocity is zero at the time of reception, but the Bunn & Hogg shy away fromview this with question Δλ and/λ deny = theΔφ average velocity over the time-of-flight of the photon is −g∆t/2 legitimacy of considering the relative velocity or separation and, for a parabolic potential, this velocity, in units of c is equal of observers as we have usedBut the terms is this above. generally They argue to the gravitational redshift. It is tantalising to think that a gen- • eralisation of this reconciles the GR kinematic view of redshifts that one needs to talk about true?vrel, the velocity of a galaxy then relative to us now; that it is hard to define relative ve- with more conventional view of gravitational redshifts. locity of two separated observers in curved space-time; that all one can do is parallel transport, and that the only ‘nat- ural’ choice of path is the null-ray of the photon. That all would normally associate with the phrase ‘relative-velocity’ sounds very reasonable. But if this were the only way to de- (rather than the relatively abstract definition in terms of the fine relative velocity then to say that redshift is a velocity mathematical operation of parallel transport). effect would be circular. Saying that the redshift is given In the weak-field limit, and for stationary non-inertial by the Doppler formula if the velocity can only be deter- observers, Synge’s relative velocity is ∆v = dr · g/c.As mined by taking the inverse Doppler function of the redshift discussed, in the context of the Pound-RebkaR experiment, is not very useful. In contrast, saying that the redshift be- this is the same as the constant relative velocity of a pair tween FOs in FRW models is kinematic is meaningful since of particles launched so as to be tangent at the interac- the fractional change of wavelength really is equal to the tion events. But that is a flat-space (constant gravity) phe- fractional change in proper separation. This is not a mathe- nomenon. More interesting is how the velocities are related matical identity, but a relationship that holds between two when curvature cannot be neglected. To this end, consider distinct physical entities. test particles in dynamic equilibrium in a static potential There are many ways, in principle at least, to directly well. Imagine a dark matter halo that generates a smooth measure the relative velocity of a pair of observers that are bowl-shaped potential well and imagine two families of test independent of the redshift, at least at low redshift. One can particles; a ‘hot’ high energy population and a ‘cold’ low use rate of change of parallaxes or light-echo time delays. It energy population that have have relatively negligible ve- has been suggested that one can directly measure velocities locities and are confined to the very centre of the halo. The on cosmological scales by the shrinking of the angular size conventional view is that photons emitted from a cold parti- of bound objects with time (Darling 2013). And one could, cle and received by a randomly chosen hot particle at larger radius will on average suffer the usual static gravitational in principle, use rate of change of luminosity of standard 2 candles. It may not be practically achievable, but one can redshift ⟨∆λ⟩/λ ≃ ∆φ/c , and that photons emitted by a imagine a very large rigid lattice populated with observers randomly chosen hot particle and received by a cold par- with clocks and rulers who record the rate of motions of ob- ticle will be blue-shifted by the same amount. This is the servers flying past. This, after all, is how we usually imag- redshift that which would be seen by a static non-inertial ine measuring geodesic deviation in order to determine the observer being supported against falling by stress in its sup- gravitational field. The question here is not whether this porting structure, but the hot population is similarly being can be done in practice; only whether it is possible in prin- supported against falling by the stress of its kinetic pressure ciple. As we describe below, we are free to construct the so it should be essentially identical. Though as mentioned lattice such that the velocities of the emitter and receiver above, the redshift is not exactly the same because there will relative to the lattice are equal and opposite. All of these concepts converge, at low redshift, to an unambiguous oper- ational definition of relative velocity in the centre of veloc- ities as legitimate; if there are any problems with the concept of relative velocity of two observers at the same time (i.e. in their ity frame.3 This, of course, is the physical quantity that one centre of velocity frame) they seem to us to pale into insignif- icance compared to the more fundamental problem of defining the difference of velocities at different times in the face of the 3 It is hard to understand the reluctance to consider such veloc- unknown absolute value of g.

⃝c 0000 RAS, MNRAS 000,000–000 Is the ‘kinematic’ picture correct? • B&H (and many others) say that the only way to compare velocities of separated objects is via parallel transport • But that would mean the only way to measure the relative velocity is via redshifts. But then saying ‘redshift is a velocity effect’ is circular • But there are other ways: • this is how we think about measuring geodesic deviation (and hence the gravitational field) • It not vacuous to say that redshifts in flat space-time are ‘Doppler’ because the velocity that gives the redshift really is the same as the rate of change of proper separation. • Same is true for FOs in FRW, and many other situations • Key question: is this a general law? Seems plausible, right? Application to gravitational redshifts in clusters WHH experiment measures the redshift of the general galaxy • population relative to the BCG • In equilibrium, their mean separation D is unchanging • So the 1st order Doppler effect averages to zero. • What is the residual (2nd order effect)? Conventional view: The relative redshift is just the mean • gravitational potential difference. Just as in Pound and Rebka. But doesn’t it matter that the • galaxies are in free fall? • Kinematic picture: D is unchanging, so redshift vanishes? Or alternatively, one might imagine that the redshift between a • pair of galaxies would not be sensitive to the first order potential difference δφ ~ D.g (EP: gravity g is “transformed away” in free fall) but would see higher order effects (i.e. tidal field to lowest order). • Several plausible pictures - which is right? But wait! There’s something fishy here... • Why is the transverse Doppler effect a red-shift? Take a birthday cake; light the candles and put it on a • turntable and spin it. • Detect all the photons and measure their frequency • Compare with non-rotating experiment. • Shouldn’t we see blue-shift λobs = λem/γ? As moving candles have more energy than candles at rest Or what if we have a swarm of moving astrophysical sources • destroying rest mass and turning it into light and we catch all the photons and measure their energy? • Does their motion induce a red-shift? If so, how is can that be compatible with energy conservation? Note: this is SR, so unlike in cosmology, energy is supposed • to be conserved Unresolved sources composed of moving sources have a net transverse Doppler blue-shift • Objects will appear red-shifted (on average at least) And a swarm of objects will have an additional red-shift from • their motions (light-cone effect) But photons from an object composed of moving sources • must, on average, be blue-shifted • if not, energy conservation would be violated The apparent contradiction is resolved once you appreciate • that a source that radiates isotropically in its rest frame is not radiating isotropically in the observer (or lab) frame • It is a mild relativistic beaming effect: • slightly more photons emerge in the forward direction • and these pick up a 1st order Doppler blue-shift • which leads to a 4th effect: What went wrong with the argument4 Nick about Kaiser little Doppler shifts?

— that any non-sophisticated physicist would say are grav- itational — are kinematic in nature. If instead we consider a similar pair of particles in a We break the null path into quadratic potential — where, unlike the Keplerian example • above the tidal field is spatially constant — the redshift is a set of segments with a again just the gravitational redshift, but in this case this is family of observers not inconsistent with the the kinematic interpretation since at the time of emission the receiver was indeed closer to the Each can be assumed to be source such that the fractional change in λ is indeed the • same as the fractional change in separation. turning around as photon Returning to the homogeneous model, the simple anal- ysis provided here helps to clarify the consistency of the pic- passes ture of redshift as the combination of a Doppler effect and a gravitational redshift. In the development above we consider But Δv is in rest-frame of the Doppler term to be that due to the initial velocity at the • time of emission and we need to add an appropriate change the pairs from the tidal field. Had we taken the Doppler term to be the final relative velocity then we would have had to apply a Generally these don’t add gravitational correction with the opposite sign. Had we used • the average velocity over the time of flight there would be up to the rate of change of no gravitational correction. See Chodorowski (2011) for a separation of end-points similar discussion couched in the language of parallel trans- port. but in constant tidal field We gratefully acknowledge stimulating discussions with • Shaun Cole and with numerous fellows of the Cifar Cosmol- they do ogy and Gravitation programme on this subject. Figure 1. Illustration of the example described in the text. Here 3 3 we have a potential with gravity g ∼ r/(r + rc )whichislike that for a uniform density sphere at r ≪ rc and is Keplerian at REFERENCES r ≫ rc.Theredshiftbetweenanemitteratr =0andadistant receiver at R ≫ rc who happens to be turning around at the Bondi H., 1947, MNRAS, 107, 410 instant of reception is just the static gravitational redshift. But Bunn E. F., Hogg D. W., 2009, AmJPh, 77, 688 following Synge & Peebles the wavelength ratio is also equal to Chodorowski M. J., 2007, ONCP, 4, 15 the product of Doppler shifts between pairs of fictitious particles Chodorowski M. J., 2011, MNRAS, 413, 585 along the line of sight which, for simplicity, can be taken to be Harrison E. R., 2000, Cosmology: The Science of the Universe, all on radial orbits (curved lines) that happen also to be turn- (Cambridge University Press, Cambridge) ing around as the photon (dashed line) passes. The velocitiesare Lineweaver C. H., Davis T. M., 2005, SciAm, 292, 030000 supposed all to be small compared to c but have been exagger- Narlikar J. V., 1994, AmJPh, 62, 903 ated here for clarity. Now the pairwise velocity differences have Peacock J. A., 2008, arXiv, arXiv:0809.4573 A diatribe on ex- to be calculated in the rest-frame of each pair; i.e. the differences panding space are between the space-time points connected by the horizontal Peebles P. J. E., 1971, Physical Cosmology, (Princeton University lines. If the velocities were differenced along a continuous path Press, Princeton) —ataconstanttimesay—thesumofthepairwisevelocities Pound, R. V.; Rebka Jr. G. A., 1959, Physical Review Letters 3 and the relative velocity of the end points has to be equal. But (9): 439441 when a null path is chopped up into a set of space-like intervals Rindler W., 1977, Essential Relativity. (Springer-verlag,New like this the connection between the ‘relative velocity’ obtained York) by summing pairwise differences and the true relative velocityis Synge, J. L., 1960, Relativity: the General Theory. (North- broken. For this type of potential, the resulting net ‘relative ve- Holland, Amsterdam) locity’ is dominated by the transition region r ∼ rc,wherethere Whiting A. B., 2004, Obs, 124, 174 are only fictitious particles. For R ≫ rc the true rate of change of the separation of the only two real particles involved is much smaller than that calculated by summing these fictitious velocity differences.

which is just the gravitational redshift for this element of the path, and this δv is also the same as the result of parallel transporting the 4-velocity of the nth particle along the null ray and subtracting it from the 4-velocity of the (n + 1)th particle. Either way, integrating these velocity increments gives the gravitational redshift, so there is no mathematical conflict with Synge. But this ‘velocity’ is not related in any sensible way to the rate of change of the emitter-receiver proper separation which is much smaller. We feel it is mis- leading to say that redshifts in the situation described here

⃝c 0000 RAS, MNRAS 000,000–000 3

In Sec. IV we widen our focus to consider frequency shifts in arbitrary curved spacetimes. In any curved spacetime the observed frequency shift in a photon can be interpreted aseitherakinematiceffect (a Doppler shift) or as a gravitational shift. The two interpretations arise from different choices of coordinates, or equivalently from imagining different families of observers along the photon’s path. We will describe this construction explicitly, and show that the comoving observers who are usually usedBunn to describe & phen Hoggomena in the expanding universe are the ones that correspond to the Doppler shift interpretation.

II. REDSHIFTS OF NEARBY GALAXIES ARE DOPPLER SHIFTS

We begin by returning to the parable of the speeding ticket, mentioned in Sec. I. “A driver is pulled over for speeding. The police officer says tothedriver,‘AccordingtotheDopplershift of the radar signal I bounced off your car, you were traveling faster than the speed limit.’ “The driver replies, ‘In certain coordinate systems, the distance between us remained constant during the time the radar signal was propagating. In such a coordinate system, our relative velocity is zero, and the observed wavelength shift was not a Doppler shift. So you can’t give me a ticket.’ ” If you believe that the driver has a legitimate argument, thenyouhaveourpermissiontobelievethatcosmological redshifts are not really Doppler shifts. If, on the other hand, you think that the officer is right, and the redshift can legitimately be interpreted as a Doppler shift, then you should believe the same thing about redshifts of nearby galaxies in the expanding universe. Why is the police officer right and the driver wrong? Assuming the officer majored in physics, he might explain the situation like this: “Spacetime in my neighborhood is very close to flat. That means that I can lay down space and time coordinates in my neighborhood such that, to an excellent approximation, the rules of special relativity hold. Using those coordinates, I can interpret the observed redshift as a Doppler shift (because there is no gravitational redshift in flat spacetime) and calculate your coordinate velocity relative to me. The errors in this method are of the same order as the departures from flatness in the spacetimeinaneighborhoodcontainingbothmeandyou.As long as I’m willing to put up with that very small level of inaccuracy, I can interpret that coordinate velocity as your actual velocity relative to me.” The principle underlying the officer’s reasoning is uncontroversial. It is no different from the principle that lets football referees ignore the curvature of Earth and use a flat coordinate grid in describing a football field. We now return to the consideration of redshifts in the expanding universe. As noted in Sec. I, it is instructive in this context (as in many others) to start with the zero-density expanding universe (the Milne model).11 The Milne universe consists of a set of galaxies expanding outward from an initial Big Bang, with the galaxies assumed to have negligible mass, so that the geometry is that of gravity-freeMinkowskispacetime.Thisspacetimecanbedescribed either in the usual comoving coordinates of cosmology or in Minkowskian coordinates. Because there is no spacetime curvature and no gravity in this universe, it is clear that theobservedredshiftsshouldbeinterpretedasDoppler shifts. In comoving coordinates, however, the redshifts areeasilyseentobetheusualcosmologicalredshift.Inthis case there is no distinction between the cosmological redshift and the Doppler shift. In our actual universe spacetime is not exactly flat, but we canapproximateitasflatinasmallneighborhood.It might be tempting to think that “approximating away” the curvature of spacetime is the same as approximating away the expansion altogether. (It seems to us that people who believe that cosmological redshifts cannot be viewed as Doppler shifts, even arbitrarily nearby, often believe something like this.) However, this belief is incorrect. When we 2 approximate a small neighborhood of an expanding spacetime as flat, we make errors of order (r/Rc) in the metric, where r is the size of the neighborhood and Rc is the curvature scale (generally the Hubble length). The redshifts of galaxies in that neighborhood are of order (r/Rc), so they are not approximated away in this limit. (The Milne model corresponds to the limit Rc →∞,inwhichcasenoapproximationismade.) In comoving coordinates the spacetime line element for the Robertson-Walker expanding universe is

ds2 = −c2dt2 +[a(t)]2 dr2 +[S(r)]2[dθ2 +sin2 θ dφ2] . (1) ! " Here S(r)=r for a flat universe. For a closed universe with positive curvature K, S(r)=K−1 sin(Kr), and for an open universe with negative curvature −K, S(r)=K−1 sinh(Kr). For realistic models K−1 is at least comparable to the Hubble length, and thus S(r) ≈ r for nearby points in all cases. Assume that an observer at the origin at the present time t0 measures the redshift of a galaxy at some comoving distance r.Assumethatthegalaxyisnearbysothatr/Rc ≪ 1, where Rc is the curvature length scale. Over such a distance scale spacetime can be well approximated as flat. Lettheobserverlaydowncoordinatesthatapproximate The gravitational redshift according to wikipedia The nature of astronomical redshifts - prologue • For photons exchanged between fundamental observers in FRW λ increases with the scale-factor a(t) • Equivalently, wavelength changes in proportion to the emitter-receiver proper separation D so Δln(λ/D) = 0 • Just as for a pair of observers in flat space-time • Widely accepted, but there has been much debate as to why? • There has been a shift away from describing redshifts as being caused by the ‘expansion of space’ and towards a `kinematic’ description in which redshifts --- and perhaps all redshifts --- are thought of as essentially Doppler-like, or ‘kinematic’ in nature. • Here I will review the old picture and the arguments against it, and the arguments for the kinematic picture • but both lead to wrong expectations for cluster grav-Z • Calculation: the intrinsically gravitational component of Z arXiv:0808.1081v2 [physics.pop-ph] 14 Apr 2009 h aueo pc n time. and space of nature us, the before have others as fadsatglx sams ennls.Ide,teinabi spacetime. the Indeed, curved meaningless. spaceti a almost separated of is widely galaxy at distant vectors a compare of apply to we way should unique or no outs can the when ob space,” the at at of clear arrive “stretching not make or does shift,” photon to a that wish equat t given We general-relativistic of interpretation: the purpose or validity. The observations the their space.” either of assess “stretching to the to and due is but shift ymn omlgssadatohscss atclrywh consequ a is particularly redshift observed the astrophysicists, that and and rest, at “really” cosmologists many by ytmaepeetda fte eesaeet bu h uni the about statements relativity. were in descriptio they spacetime these if of In as presented shift. are Doppler system kinematic usual the from hfsdet eaievlcte,adw ilageta fw if that argue other. will any we than and natural velocities, more relative to due shifts etok hsna admfo h okhl foeo h au the of one of bookshelf the from random at chosen textbooks ehv ie a efudee nsm dacdtxbos Fo an textbooks. peculiar by advanced produced some “are in shifts even co Doppler found the that of be states nature can text the cited of have descriptions we clear and careful ularly rubya nihee omlgs ee ssti questi this asks never cosmologist enlightened an Arguably ob the that declare cosmology big-bang of descriptions Many nti ae ewl hwta h esit fdsatobjec distant of redshifts the that show will we paper this In ebgnwt xmlso h ecito ftecosmologica the of description the of examples with begin We oeavne etok fe vi hslnug.Frinst For language. this avoid often textbooks advanced More • • • hog space through “is redshift cosmological The otercsino h galaxies.” formul the Doppler of relativistic recession Einstein’s radial the were to . . they . if st shifts as is Doppler redshifts space express If well.” often as “Astronomers . . stretched . be through will travel waves they light e the space Doppler of the stretching [than the view accurate more “A arXiv:0808.1081v2 [physics.pop-ph] 14 Apr 2009 hf yiaiigasial aiyo bevr ln h p the along natural. observers kinemat more correspo of a is interpretation hence as family kinematic Minkowsk either the suitable universe interpreted a locally expanding be is imagining can that by spacetime shift system any in coordinate a shift accumulat centra choose vi the a to as this obscures valid rather to or interpretation contrary shift, stretching-of-space Doppler that, The a argue as We recessio is a redshift space. for the of evidence as stretching is, the (that of shift Doppler a as viewed h aueo pc n time. and space of nature us, the before have others as fadsatglx sams ennls.Ide,teinabi spacetime. the Indeed, curved meaningless. spaceti a almost separated of is widely galaxy at distant vectors a compare of apply to we way should unique no or outs can the when ob space,” the at at of arrive clear “stretching not make or does shift,” photon to a that wish equat given t We general-relativistic of interpretation: the purpose or validity. The observations the their space.” either of assess “stretching to the to and due is but shift ytmaepeetda fte eesaeet bu h uni the about statements relativity. were in descriptio they spacetime these if of In as presented shift. are wh consequ Doppler system a kinematic is particularly usual redshift observed the the from astrophysicists, that and and rest, at “really” cosmologists many by hfsdet eaievlcte,adw ilageta fw if that argue other. will any we than and natural velocities, more relative to due shifts etok hsna admfo h okhl foeo h au the of one of bookshelf the from random at chosen textbooks ehv ie a efudee nsm dacdtxbos Fo an textbooks. peculiar by advanced produced some “are in shifts even co Doppler found the that of be states nature can text the cited of have descriptions we clear and careful ularly rubya nihee omlgs ee ssti questi this asks never cosmologist enlightened an Arguably ob the that declare cosmology big-bang of descriptions Many nti ae ewl hwta h esit fdsatobjec distant of redshifts the that show will we paper this In ebgnwt xmlso h ecito ftecosmologica the of description the of examples with begin We oeavne etok fe vi hslnug.Frinst For language. this avoid often textbooks advanced More Ac o m m o nb e l i e fa b o u tb i g - b a n gc o s m o l o g yi st h a tt h ec o s m o l o • • • hog space through “is redshift cosmological The otercsino h galaxies.” formul the Doppler of relativistic recession Einstein’s radial the were to . . they . if st shifts as is Doppler redshifts space express If well.” often as “Astronomers . . stretched . be through will travel waves they light the e space Doppler of the stretching [than the view accurate more “A ,w h e r e a sac o s m o l o g i c a lr e d s h i f ti sc a u s e db yt h e eateto hsc,Uiest fRcmn,Rcmn,V Richmond, Richmond, of University Physics, of Department hf yiaiigasial aiyo bevr ln h p the along natural. observers kinemat more correspo of a is interpretation hence as family kinematic Minkowsk either the suitable universe interpreted a locally expanding be is imagining can that by spacetime shift system any in coordinate a shift accumulat centra choose vi the to a as this obscures valid rather to or interpretation contrary shift, stretching-of-space Doppler that, The a argue as We recessio is a redshift space. for the of evidence stretching as is, the (that of shift Doppler a as viewed 1,2,3,4 e okUiest,4Wsigo lc,NwYr,NwYr 1 York New York, New Place, Washington 4 University, York New etrfrCsooyadPril hsc n eatetof Department and Physics Particle and Cosmology for Center Ac o m m o nb e l i e fa b o u tb i g - b a n gc o s m o l o g yi st h a tt h ec o s m o l o 11,12,13,14 h ieai rgno h omlgclredshift cosmological the of origin kinematic The ,w h e r e a sac o s m o l o g i c a lr e d s h i f ti sc a u s e db yt h e npatc,hwvr h nihee iwi a rmunive from far is view enlightened the however, practice, In eateto hsc,Uiest fRcmn,Rcmn,V Richmond, Richmond, of University Physics, of Department 1,2,3,4 e okUiest,4Wsigo lc,NwYr,NwYr 1 York New York, New Place, Washington 4 University, York New etrfrCsooyadPril hsc n eatetof Department and Physics Particle and Cosmology for Center 11,12,13,14 h ieai rgno h omlgclredshift cosmological the of origin kinematic The htsaeet uha hs r ilaigadfse misu foster and misleading are these as such statements that not npatc,hwvr h nihee iwi a rmunive from far is view enlightened the however, practice, In 7 h aea ope hf.Dplrsit r asdb nob an by caused are shifts Doppler shift. Doppler a as same the htsaeet uha hs r ilaigadfse misu foster and misleading are these as such statements that not 7 h aea ope hf.Dplrsit r asdb nob an by caused are shifts Doppler shift. Doppler a as same the .INTRODUCTION I. 6 mr .Bunn F. Emory ff .INTRODUCTION I. ai .Hogg W. David 6 c]o h redshif the of ect] mr .Bunn F. Emory ff ai .Hogg W. David c]o h redshif the of ect] osta ucsflyepante.Rte,orfcsi on is focus our Rather, them. explain successfully that ions osta ucsflyepante.Rte,orfcsi on is focus our Rather, them. explain successfully that ions neo oesr f“tecigo pc, hc sdistinct is which space,” of “stretching of sort some of ence s ttmnsta r riat fapriua coordinat particular a of artifacts are that statements ns, ea r ef o r c e dt oi n t e r p r e tt h er e d s h i f t ,t h i si n t e r p r e t a t i o n iyt opr etr tdi at vectors compare to lity evrcneinl aee Dplrsit”“gravitatio shift,” “Doppler labeled conveniently server mlgclrdhf.Hwvr ttmnssmlrt those to similar statements However, redshift. smological ntligt oseilss sta itn aaisare galaxies distant that is nonspecialists, to talking en neo oesr f“tecigo pc, hc sdistinct is which space,” of “stretching of sort some of ence s ttmnsta r riat fapriua coordinat particular a of artifacts are that statements ns, ea r ef o r c e dt oi n t e r p r e tt h er e d s h i f t ,t h i si n t e r p r e t a t i o n iyt opr etr tdi at vectors compare to lity evrcneinl aee Dplrsit”“gravitatio shift,” “Doppler labeled conveniently server mlgclrdhf.Hwvr ttmnssmlrt those to similar statements However, redshift. smological ntligt oseilss sta itn aaisare galaxies distant that is nonspecialists, to talking en n ntecre pctm fgnrlrltvt,teeis there relativity, general of spacetime curved the In on. dn o tb yr e c e s s i o nv e l o c i t i e s . ” si h xadn nvremyb iwda kinematic as viewed be may universe expanding the in ts n ntecre pctm fgnrlrltvt,teeis there relativity, general of spacetime curved the In on. aa p p l i e st om o t i o nt h r o u g hs p a c e ,s oi td o e sn o ta p p l y dn o tb yr e c e s s i o nv e l o c i t i e s . ” i ae st xmn h enn fsc statements such of meaning the examine to is paper his epit,adhnetento fterltv velocity relative the of notion the hence and points, me si h xadn nvremyb iwda kinematic as viewed be may universe expanding the in ts aa p p l i e st om o t i o nt h r o u g hs p a c e ,s oi td o e sn o ta p p l y hs labels? these i ae st xmn h enn fsc statements such of meaning the examine to is paper his es,rsligi iudrtnig bu h nature the about misunderstandings in resulting verse, epit,adhnetento fterltv velocity relative the of notion the hence and points, me hs labels? these es,rsligi iudrtnig bu h nature the about misunderstandings in resulting verse, ne h ok yPeacock by books the ance, ne h ok yPeacock by books the ance, li d e ao fr e l a t i v i t y ,n a m e l yt h a ti ti sa l w a y s li d e ao fr e l a t i v i t y ,n a m e l yt h a ti ti sa l w a y s thors. thors. evdrdhf fdsatglxe sntaDoppler a not is galaxies distant of redshift served evdrdhf fdsatglxe sntaDoppler a not is galaxies distant of redshift served d oafml fcmvn bevr and observers comoving of family a to nds lr e d s h i f ti nt h efi r s tt h r e ei n t r o d u c t o r ya s t r o n o m y d oafml fcmvn bevr and observers comoving of family a to nds lr e d s h i f ti nt h efi r s tt h r e ei n t r o d u c t o r ya s t r o n o m y re x a m p l e ,al e a d i n ga d v a n c e du n d e r g r a d u a t el e v e l eoiis u h esit fteglxe r not are galaxies the of redshifts the but velocities, re x a m p l e ,al e a d i n ga d v a n c e du n d e r g r a d u a t el e v e l ∗ eoiis u h esit fteglxe r not are galaxies the of redshifts the but velocities, ecigdrn l h ietelgti traveling, is light the time the all during retching ∗ a.W hwta nosre frequency observed an that show We ian. nv e l o c i t y ) ,b u tm u s tb ev i e w e di nt e r m s ecigdrn l h ietelgti traveling, is light the time the all during retching tta eaentsgetn n ob about doubt any suggesting not are we that et o fmn nntsmlDplrshifts. Doppler infinitesimal many of ion a.W hwta nosre frequency observed an that show We ian. nv e l o c i t y ) ,b u tm u s tb ev i e w e di nt e r m s tta eaentsgetn n ob about doubt any suggesting not are we that et o fmn nntsmlDplrshifts. Doppler infinitesimal many of ion so aaisi httewvsaesrthdby stretched are waves the that is galaxies of ts w h otntrlitrrtto of interpretation natural most the ew, xaso fspace of expansion so aaisi httewvsaesrthdby stretched are waves the that is galaxies of ts w h otntrlitrrtto of interpretation natural most the ew, oo’ ah ntecneto the of context the In path. hoton’s c(ope)sito gravitational a or shift (Doppler) ic xaso fspace of expansion oo’ ah ntecneto the of context the In path. hoton’s c(ope)sito gravitational a or shift (Doppler) ic ia esitcno eproperly be cannot redshift gical ia esitcno eproperly be cannot redshift gical rii 23173 irginia Physics, rii 23173 irginia ff .” 0003 rn onsi h definition the is points erent 5 Physics, 10 sl h iwpresented view The rsal. 8 Epai noriginal.) in (Emphasis n Linder and ff nti ae eargue, we paper this In .” 0003 rn onsi h definition the is points erent 5 drtnig about nderstandings 10 sl h iwpresented view The rsal. 8 Epai noriginal.) in (Emphasis n Linder and nti ae eargue, we paper this In 9 ject’s drtnig about nderstandings iepartic- give motion

9 the ject’s nal is iepartic- give e motion the nal is e

My thoughts on P&R, B&H, Synge, Rindler B&H argument that gravitational and Doppler redshifts are in • some way equivalent is misleading. • Accelerated observers know that they are accelerated So when they allow for this they agree that the redshift • measured by the cop was a Doppler shift cause by δv • An acceleration does not create gravity (Rαβγδ = 0) • Pound and Rebka did not measure a gravitational redshift they measured a redshift caused by the non-gravitational • acceleration that their apparatus experienced • Synge is right: gravity = curvature (or tidal field) • Pound & Rebka too insensitivite to measure tidal effects • But Synge is wrong to say that redshift is just Doppler • parallel transport of photon involved the connection Most redshifts are mostly Doppler - but there is an intrinsically • gravitational component - it is the gradient of the tide So why does the ‘kinematic’ law give the wrong answer? • Recap: kinematic law Δlog(λ/D) = 0 holds in a wide range of circumstances any observers in flat space-time, FOs in FRW, freely falling • observers co-moving with Pound & Rebka apparatus • Bunn & Hogg’s argument ⇒ universal • But consider emitter at rest in centre of a cluster and distant receiver turning around at large radius • calculation ⇒ Δλ/λ ≫ ΔD/D

Cluster equivalence principle Principle of Equivalence (from dummies.com)

What is the equivalence principle? • Synge:

• Rindler: Redshifts in homogeneous FRW models

• λ scales with proper separation • analogous to EM waves in an expanding cavity • stretching of λ is caused by the expansion of space • that this should be so is obvious • expansion of space causes damping in Maxwell's equations • cosmological redshifts do not obey special relativity • z is a combination of SR Doppler shift + gravitational z • at low-z at least (Bondi 1947) • overall frequency shift is the product of little Doppler shifts • Peebles Redshifts in general

• all redshifts are Doppler shifts • Riemann tensor Rαβψδ does not appear in the formula • acceleration and gravity are the same thing • principle of equivalence • acceleration creates gravity • Pound and Rebka measured a gravitational redshift • redshifts can be considered as either gravitational or Doppler • simply a difference of coordinate systems • all redshifts can (and should) be considered to be Doppler • all redshifts are kinematic in nature • the only way to measure velocity is through redshift