Gravitational Redshift, Inertia, and the Role of Charge

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Gravitational redshift, inertia, and the role of charge

Johannes Fankhauser,∗ University of Oxford.

(May 6, 2018)

I argue that the gravitational redshift effect cannot be explained purely by way of uniformly accelerated frames, as sometimes suggested in the literature. The concepts in need of clarification are spacetime curvature, inertia, and the weak equivalence principle with respect to our understanding of gravitational redshift. Furthermore, I briefly discuss gravitational redshift effects due to charge. Considering charge and mass together seems to give rise to a way of (locally) shielding gravity.

Contents space and time. There exist some misconcep- tions, as evidenced in the literature, regarding the nature of the gravitational field in Einstein’s 1 Introduction 1 General Theory of Relativity (GR) and how it relates to redshift effects. My aim is to give 2 Gravitational redshift 1 a consistent analysis of the gravitational red- 3 Uniformly accelerated frames and shift effect, in the hope of thereby advancing in the equivalence principle 3 some small measure our understanding of GR. Moreover, I will show that when charge is taken 4 Equivalence and gravitational red- into account, gravitational redshift is subject to shift 6 further corrections. In the first part of this paper I shall de- 5 Redshift due to charge 7 rive and discuss the gravitational redshift in the 5.1 The weight of photons . . . . .7 framework of GR, from the equivalence princi- 5.1.1 Einstein’s thought exper- ple, and from energy conservation principles to iment ...... 7 then compare and relate the different results. 5.1.2 Inertia of energy . . . .8 In the second part of this paper I shall examine 5.2 Reissner-Nordströmmetric . . .9 effects on the redshift due to charge with some 5.3 Shielding gravity ...... 9 remarks on the relationship between GR and electromagnetism, and the possibility of locally 6 Conclusion 9 shielding gravity with charge.

1 Introduction 2 Gravitational redshift In 1911 Einstein foresaw a phenomenon thereafter known as ‘gravitational redshift’ It is a straightforward task to derive the rela- [Einstein, 1911]. His thought experiment ini- tive shift in coordinate time of two clocks in a tiated the revolutionary idea that mass warps given gravitational ﬁeld with metric gµν. Since ∗[email protected]

1 p b c we will employ some alternative approximate −gbcξ ξ are obtained by contraction with the approaches to derive the gravitational redshift metric. We let the observers O1 and O2, whose in the following sections, we shall choose to clock rates we wish to compare, describe their present the exact and most general derivation world-lines. The diﬀerence in the world-lines’ from GR ﬁrst, variants of which are standard lengths in spacetime consequently determines fare (see for example, [Wald, 2010, p. 136]). the amount of gravitational redshift. Figure 1 illustrates the thought experiment. Recall that for a given energy-momentum 4- vector pa = mua of a particle, with respect to a local inertial frame, the energy observed by an observer that moves with 4-velocity va is

a 1 E = −p va. (2.1)

Therefore, for the frequency νi of the photon observed by Oi, which moves with 4-velocity ua, we find the relation hν = E = −k ua| i i k a i Pi (compare Equation 2.1), where Ek is the energy of the photon. By definition of the vector field ξa, we have ξ ξa| = g | since ξa has van- a Pi 00 Pi ishing spatial components. It would involve a fair amount of work to derive the gravitational redshift by finding the geodesic equation. How- ever, this can be avoided by taking advantage Figure 1: Two observers at different heights ex- of a useful proposition. Light travels on null perience a time dilation effect in Earth’s gravi- geodesics (in the geometrical optics approxima- tational field. Emitter O1 on the surface of the tion, i.e. the spacetime scale of variation of the Earth sends a train of electromagnetic pulses electromagnetic field is much smaller than that from point P1 with energy momentum 4-vector of the curvature), from which it follows that the ka to a receiver O , placed at point P , at height a 2 2 inner product kaξ is constant along geodesics, h above P . We assume O and O are static, a a 2 1 1 2 that is kaξ | = kaξ | . a a P1 P2 i.e. their 4-velocities u1 and u2 are tangential Spacetime around Earth (if considered as a ∂ a to the Killing field ξ = ∂t . generated by a point mass M at r = 0) can be modelled by the Schwarzschild metric r An emitter O1 on the surface of the Earth ds2 = g dxµdxν = − 1 − S c2dt2 sends a train of electromagnetic pulses from µν r a r −1 point P1 with energy momentum 4-vector k + 1 − S dr2 to a receiver O2, placed at point P2, at height r 2 2 2 2 h above P1. We assume the two observers O1 + r (dϑ + sin ϑdϕ ), and O2 to be static, which is to say, their 4- (2.2) a a velocities u1 and u2 are tangential to the static where Killing field ξa = ∂ a. Since the 4-velocities 2GM ∂t r = (2.3) of the two observers are unit vectors pointing S c2

a in the direction of ξa, we have ua = √ ξ is the so-called Schwarzschild radius, r the dis- 1 b −ξ ξb tance from the Earth’s centre, G the gravita- P1 a tional constant, c the speed of light, and M the and ua = √ ξ . The lengths p−ξbξ = 2 b b −ξ ξb mass of the Earth. This yields P2 1 a a a 2 In particular, if u = v , i.e. the particle’s 4-velocity aligns with the observer’s, then E = −mv va = mc . 2For a detailed proof see for instance, [Wald, 2010, p. 442]

2 However, it is clear that Einstein was well q aware of the mere linearly approximate validity p−ξbξ 2GM b 1 − c2r of the equivalence principle when he wrote: ν1 P2 2 = = q ν2 p b 2GM −ξ ξb 1 − 2 c r1 ‘...we arrive at a principle [the P1 GM 1 1 gh equivalence principle] which, if it is ≈ 1 + − ≈ 1 + , (2.4) c2 r r c2 really true, has great heuristic im- 1 2 portance. For by theoretical consid- or eration of processes which take place ∆ν GM 1 1 relative to a system of reference with ≈ 2 − , (2.5) ν c r1 r2 uniform acceleration, we obtain in- GM formation as to the behaviour of with g := c2r2 the gravitational constant at r1, 1 processes in a homogeneous gravi- ν = ν1, ∆ν = ν1 − ν2, and r2 − r1 = h. For the last approximation in the second last line tational field. ... It will be shown 1 1 r2−r1 h in a subsequent paper that the grav- we have used r − r = r r ≈ r2 if r1 ≈ r2 1 2 2 1 1 itational field considered here is ho- and r1, r2 h. Moreover, we used the approximations mogeneous only to a first approximation.’ [Einstein, 1911, p. 900] √ 1 1 + x ≈ 1 + x 2 The principle, thus, only holds in a ‘small neigh- 1 1 √ ≈ 1 − x. (2.6) bourhood’ of a point-like observer. Nonethe- 1 + x 2 less, a treatment of the redshift effect in a uniform static gravitational field proves instruc- Experimental tests of the gravitational tive, insofar as it shows that certain conse- redshift were first conducted by Cranshaw, quences of GR can be explained by means of ge- Schiffer and Whitehead in the UK in 1960 ometry without resorting to gravitational fields. [Cranshaw et al., 1960]. It was not clear Dealing with uniform accelerations to derive whether significant conclusions could be drawn the gravitational redshift, however, is a deli- from their results. In the same year, the ex- cate business, and we shall see that the field, periments by Pound and Rebka in Harvard suc- resulting from uniform (proper) acceleration, is cessfully verified the gravitational redshift effect not uniform if we demand a constant (proper) [Pound and Rebka Jr, 1960]. distance between emitter and observer! We consider a spaceship that is uniformly 3 Uniformly accelerated frames accelerated. An emitter E and receiver R inside and the equivalence principle the spaceship, separated by a height h, compare frequencies of signals ascending the spaceship. Einstein’s equivalence principle (also called the For an illustration, see Figure 2. weak equivalence principle) assumes that any As in the derivation of the gravitational red- experiment in a uniform gravitational field shift from the Schwarzschild metric, we let the yields the same results as the analogous exper- observers describe their world-lines. It suffices iment performed in a frame removed from any to consider only one spatial dimension x. Accel- source of gravitational field but moving in uni- eration a is measured in an inertial frame S with form accelerated motion with respect to an in- momentary velocity v relative to the inertial ertial frame [Norton, 1985].3 frame S0 outside the spaceship, inside of which 3Note that [Brown and Read, 2016] use ‘Einstein equivalence principle’ to refer to what is often called the ‘strong equivalence principle’. 4It is implicitly assumed that the proper time of co-moving clocks depends only on velocity and is independent of acceleration. This assumption is often called the Clock Hypothesis (see for example, [Brown and Read, 2016, Section 3]).

3 the acceleration is measured to be a0.4 Rela- constant we integrate Equation 3.1 twice to ﬁnd tivistic transformation of 3-acceleration gives the trajectory — so-called Rindler hyperbola — of a uniformly accelerated spacetime point as a = γ3a0, (3.1) observed by the inertial frame S0:

1 5 where γ = q 2 is the Lorentz factor. s 2 1− v c2 at c2 x(t) = 1 + + C, (3.2) a c with C a constant from integration. The second constant from the ﬁrst integration was set to zero such that v(0) = 0. Without loss of gen- erality we can also set C = 0. The result rep- resents a hyperbolic path in Minkowski space, i.e.

c4 x2 − c2t2 = , (3.3) a2 from which the term ‘hyperbolic motion’ is derived. We assume the back of the spaceship be subject to this motion. Note thatx ˙ t→∞→ c, as expected. We recover uniform acceleration in the New- tonian sense for t 1. That is,

2 Figure 2: The gravitational redshift experiment at x(t) = x0 + , (3.4) in a uniformly accelerated spaceship. The red- 2 shift effect can be explained by the equivalence 2 with x0 = c /a the position at t = 0. principle (to first order). For an exact derivation, it would lead to inconsistencies to assume emitter and re- Note that the acceleration of the spaceship ceiver be subject to the same Rindler hy- needs to be measured in the (momentary) in- perbola with only an additional spatial dis- ertial frame with instantaneous velocity v such tance h in the coordinate x. For if we main- 0 dv that a = dt (proper acceleration). With re- tained a constant height between E and R rel- spect to the accelerated frame, sure enough, the ative to the inertial observing frame S0, length ship’s acceleration is zero. However, the princi- contraction, as predicted by special relativ- ple of relativity — the requirement according to ity, would stretch the spaceship and eventually which the laws of physics take the same form in tear it apart (cf. also Bell’s spaceship paradox any inertial frame — no longer holds in acceler- in [Dewan and Beran, 1959] and [Bell, 1987, ated, hence non-inertial, frames. Therefore, as Chapter 9]). This is key. As was also pointed expected, the two observers in the spaceship are out by [Alberici, 2006], assuming the gravita- going to feel a (pseudo)force F = m0a, where tional acceleration to be the same for the top m0 is the rest mass (invariant mass) of an ob- and bottom observers leads to all kinds of para- ject in the spaceship. doxes. Most notably, it is not possible in this We want the (proper) acceleration a of the case to define a globally freely falling inertial spaceship to be constant. The right hand side frame because the corresponding metric would d of Equation 3.1 is equal to dt (γv). Since a is lead to a non-vanishing Riemann tensor, and 5To find the transformation of acceleration, one has to differentiate the spatial coordinates of the Lorentz transformation with respect to the time coordinates to first find the 3-velocity transformation (velocity-addition formula). Another differentiation of the velocities yields the transformation law for 3-acceleration.

4 hence curvature! The receiver R in the bow lying higher by height h with respect to the τ g g h emitter E must follow the hyperbola R E E = = 1 + 2 , (3.7) τE gR c c2 2 x2 − c2t2 = + h , (3.5) a yielding the exact gravitational redshift formula for uniform acceleration. Alternatively, we can for the proper height (relative to S) to be con- write stant. These are the two desiderata to sim- ulate reasonably the gravitational redshift by νE gRh uniform acceleration: First, the ship must have νR = = νE 1 − (3.8) gE h c2 a constant acceleration; and second, the ship 1 + c2 must have a constant proper height. The world- lines of emitter and receiver are denoted in Fig- for the corresponding observed frequencies, to ure 3. highlight the dependence on the two diﬀer- ent proper accelerations of emitter and receiver (cf. also the results in [Alberici, 2006]). From the preceding derivations we readily ﬁnd for the (Rindler) metric of an accelerated frame

g x2 ds2 = g dxµdxν = 1 + E c2dt2 − dx2. µν c2 (3.9) Thus, the gravitational redshift according to this metric reads Figure 3: The world-lines of emitter E and receiver R are Rindler hyperbolae when experi- encing constant proper acceleration. νE ∆tx=h gEh = = 1 + 2 , (3.10) νR ∆tx=0 c Due to relativistic length contraction, the receiver’s proper acceleration needs to be which is consistent with the first order approx- slightly greater. By comparing the two hyper- imation of the gravitational redshift from the bolae it immediately follows that the accelera- Schwarzschild metric in Equation 2.4. Clocks tion gR of the receiver is related to the emitter’s at E and R, whose rates one wishes to compare, acceleration gE by are permitted to describe their world-lines, i.e. Rindler hyperbolae, with respect to the inertial gE gR = . (3.6) frame, and the value for the redshift is obtained 1 + gE h c2 by comparing the lengths of their world-lines in Compare also the treatment and related para- spacetime. Therefore, the treatment here is ex- doxes in [Fabri, 1994]. Therefore, the gravita- act. The Rindler metric is, in fact, a solution tional field is not constant over the extended to the vacuum Einstein field equations and has region of the spaceship. That is, however, not vanishing curvature (Rµνρσ = 0). a surprise, for we would not expect the equiva- In the experiments of Pound-Rebka to con- lence principle to hold globally in the first place. firm gravitational redshift, the emitter sends a Further, it follows that proper time intervals signal at equal intervals on a clock at the sur- along two different Rindler hyperbolae between face of the Earth. The receiver measures the two events having the same coordinate velocity time interval between receipt of the signals on are in a fixed proportion, an identical clock at height h (see Figure 4).

5 modern textbooks on GR, a single gravitational redshift experiment does not require an explanation in terms of curvature. Rather, it is only multiple such experiments, performed at appropriately different lo- cations in spacetime, that suggest curvature, via the notion that inertial frames are only defined locally, ...This “redshift” effect follows directly from the claim that the emitter and absorber are accelerat- ing vertically at a rate of g m/s2 relative to the (freely falling) inertial Figure 4: The Pound-Rebka experiment. frames.’ [Brown and Read, 2016, p. 327, 329] Merely when the experiment is taken to be Here, Brown and Read assume the ‘redshift’ ef- at rest in a Rindler field, the equivalence princi- fect to be independent of ‘tidal effects’ (which ple implies that the relation between the clock is what they refer to as curvature). We have times of emitter and receiver must be the same in fact already shown such a derivation is lim- as if a spaceship were to accelerate vertically ited and does not fully account for gravitational upwards in free space, as shown in Figure 2. redshift. There are indeed tidal effects in a sin- The signals at the back are received at longer gle redshift experiment as outlined above in the intervals than they are emitted because they most general derivation. Moreover, as we have are catching up with the accelerated bow of seen, assuming both emitter and absorber to ac- the spaceship and thus exhibit a Doppler shift. celerate at the same rate is impossible given the Note that the equivalence principle is a local two desiderata mentioned. However, they ac- law. Thus, in a field like that of the Earth it knowledge there is nonetheless a connection be- holds only approximately (to first order) for a tween spacetime curvature and redshift exper- small spacetime region. iments. This connection, to Brown and Read, amounts to the fact that redshift experiments 4 Equivalence and gravitational carried out at different places on Earth reveal redshift ‘geodesic deviation’ due to the spherical shape of the planet. That is, relative to a global freely Although GR is a well-established framework, falling frame at the site of one redshift experi- it often occurs that its application amounts to ment, a freely falling frame at another site is an analysis that renders conclusions equivocal. not moving inertially. Multiple gravitational This, in particular, happens to be the case for redshift experiments thus require for their joint gravitational redshift. For instance, Brown and explanation the rejection of the global nature of Read comment on the gravitational redshift ef- inertial frames. Brown and Read think it is only fect as follows: geodesic deviation that reveals curvature. How- ever, one experiment is sufficient to detect tidal ‘The second possible misconception effects of Earth’s gravitational field. After all, relates to the notion that gravi- an extended body will experience a stronger at- tational redshift experiments pro- traction on its nearest side than on its furthest vide evidence for spacetime curva- side. ture. They do, but contrary to What Brown and Read deem to be a mis- what is claimed in some important conception, that is

6 ‘An explanation for the results of a of the results of the Pound-Rebka experiment single gravitational redshift experi- with first order calculation because higher order ment of Pound–Rebka type will ap- effects are beyond their measurement accuracy, peal to a notion of spacetime cur- they show that the qualitative explanation of vature.’ [Brown and Read, 2016, p. the result requires to invoke spacetime curva- 330], ture and an exact treatment of accelerations. is in fact one. However, this results not from an absence of curvature. Rather, since the Pound- Rebka experiment was solely designed to verify 5 Redshift due to charge the first order effects predicted by GR, in this case a derivation via accelerated frames gives 5.1 The weight of photons the desired result. Brown and Read’s proposal holds if the What Pound and Rebka call the ‘weight of pho- gravitational field of the Earth was assumed tons’ in their experiments, in fact, aptly de- to be uniform, that is, independent of the ra- scribes how Einstein originally had thought of dial distance from the centre of the earth, and gravitational redshift and what he had termed gh the inertia of energy. also if c2 1. In experiments involving larger spatial separations or stronger gravitational field variations, it is necessary to use the 5.1.1 Einstein’s thought experiment exact Schwarzschild solution of GR. By means of fully formed GR, of course, all approxima- Let us go back to the thought experiment al- tions are bound to disappear. Incidentally, the luded to in the introduction. Einstein fore- second and higher order contributions to the saw the gravitational redshift on the basis of redshift effect amounts to a correction of a mag- a thought experiment using the ‘inertia of en- −9 nitude of 10 relative to the first order result. ergy’ he had discovered in 1905 [Einstein, 1905], Clearly, in the case of the Earth’s gravitational six years before his famous paper on relativity field, a freely falling observer does not accel- [Einstein, 1911]. A variant of this I shall spell erate uniformly relative to a static frame on out here. Earth, but rather her acceleration increases as Consider a test body of mass m0 at rest at she gets closer to the surface since the gravita- 2 a height h, with a total energy m0c + m0gh. tional field strength increases. The mass subsequently is dropped, and when I conclude with two clarifications. First, if 2 it reaches the ground the total energy γm0c the equivalence principle is to be used to explain 1 is obtained, where γ = q 2 , and v the ve- 1− v the gravitational redshift, then it is important c2 to realise that this can only be done to first or- locity of the mass at the ground (such that 2 2 der. Second, the quantitative results of Pound- m0c + m0gh = γm0c ). The mass is then Rebka can indeed be justified without appeal- transformed into a packet of radiation of energy ing to spacetime curvature, but one should be hν1, which is then sent from the ground back aware that a complete theoretic description has to height h, where the mass m0 initially had to take into account the inhomogeneous gravi- been. There, the packet is transformed back tational field of Earth. After all, more sophis- into a mass m. By energy conservation, m must ticated experiments with higher accuracy than equal the mass m0, which amounts to saying 2 those used by Pound and Rebka are in fact able that hν2 = m0c , where ν2 is the frequency of to measure effects due to curvature in a single the packet at height h. See steps 1–4 in Figure redshift experiment.6 Although my considera- 5. tions do not inhibit the successful comparison From this we regain the first order approxima-

6 Note that for the Schwarzschild metric R = 0 and Rµν = 0, but not all entries of the Riemann curvature tensor Rµνρσ vanish.

7 tion in Equation 2.4; Bear in mind that neither the derivation ν m c2 + m gh gh by means of uniformly accelerated frames nor 1 = 0 0 = 1 + . (5.1) the derivation by means of energy conserva- ν m c2 c2 2 0 tion yield the correct value for the gravitational Equation 5.1 again involves an approximation redshift in the first line of Equation 2.4. The of the exact redshift formula, for we assume a former holds in virtue of the inhomogeneity of uniform gravitational field. Hence we use m0gh Earth’s gravitational field and the merely local for the energy of the test body. If we were to validity of the equivalence principle. The lat- 1 take into account the r -dependence of the grav- ter is true because the Newtonian central body itational potential, we would obtain force law is an approximate limit of GR.

r1 m c2 + R F dr 0 N 5.1.2 Inertia of energy ν1 r2 = 2 ν2 m0c The approach of describing the redshift effect GM 1 1 as a result of energy conservation suggests the = 1 + 2 − c r1 r2 following idea: gh ≈ 1 + , (5.2) c2 Any ‘source’ of energy causes clocks at different distances from the ‘source’ to exhibit time with FN Newton’s gravitational force of a massive central body. dilation effects. As one example, charged particles attracted by a charged source should likewise be expected to give rise to redshift effects. We can, however, not follow the procedure from above and play the same game with charged bodies, replacing the Newtonian potential with the Coulomb potential. Consider a charged source Q and a test particle of charge q and mass m0. We assume the mass of the source to be negligible. The charged particle falls under the attraction of the source according to the Coulomb force. When it reaches height r1, a photon is created out of it and sent back to the particle’s initial position, where it is transformed back into a mass m with charge q. For this process to happen, we can imagine annihilating the descending charge Figure 5: Gravitational redshift as a conse- by an anti-charge of size −q to create a pho- quence of energy conservation. A test body ton (or actually at least two photons, which we of mass m0 at rest at a height h is dropped. can think of a single photon for the discussion). When it reaches the ground the total energy The photon is sent back, and when it reaches 2 γm0c is obtained. The mass subsequently is the top, the initial charge q plus its anti-charge transformed into a photon of energy hν1, which −q is created via pair-production. We assume is then sent from the ground back to height h. the two particles have the same mass m. The There, the photon is transformed back into a anti-charge −q subsequently is brought back to mass m. By energy conservation, m must equal the bottom to restore the initial situation. It the mass m0, from which it follows that the is precisely the energy contribution of this last photon’s frequency must have decreased at its step that cancels a redshift effect in the calcu- ascent. lations, which is not further analysed here.

8 5.2 Reissner-Nordströmmetric Whereas for charged sources, the usual force terms from elecrodynamics need to be consid- In fact, charge does give rise to redshift ef- ered additionally in the geodesic equation. fects — and consequently time dilation — in the standard formalism of GR, though not, in a way analogous to how mass curves spacetime. 5.3 Shielding gravity The Einstein equations for a charged point- It is often considered to be a feature of grav- like (non-rotating) mass with stress-energy ten- ity that shielding an object from the influence sor Tµν reads of a gravitational field is impossible — unlike 1 8πG electromagnetism. Gravitational redshift due R − Rg = T . (5.3) to charge is, indeed, a counterexample to this µν 2 µν c4 µν statement. Recall the Reissner-Nordströmmet- We neglect the cosmological constant Λ. From ric 5.4. There, the two terms in g , one propor- this we obtain the Reissner-Nordströmmetric 00 tional to M, the other to Q, come with opposite (cf. [Reissner, 1916]) signs. This makes it possible to tweak the pa- rameters such that curvature vanishes, at least 2 2 2GM GQ 2 2 locally. If we choose ds = − 1 − 2 + 4 2 c dt c r 4πε0c r 2MG Q2G 2GM GQ2 −1 = , (5.6) 2 c2r 4πε c4r2 + 1 − 2 + 4 2 dr 0 c r 4πε0c r + r2(dϑ2 + sin2 ϑdϕ2), (5.4) then we recover the Minkowski metric for flat space. This equality, obviously, can only be met from which we recover the Schwarzschild met- at one fixed radius r. But it means that gravity, ric in the limit Q = 0. It is worth mentioning in this sense, can at least be ‘shielded’ locally. that the charge term in the Reissner-Nordström This is surprising, since there was thought to be metric affects geodesics of particles even though no way to shield a massive test body from grav- they may be uncharged. For Q 6= 0, this met- itational forces. What’s more, gravitational ric gives rise to an additional gravitational red- shielding is said to be a violation of the equiva- shift. Similar to the derivation of gravitational lence principle and thus inconsistent with both redshift due to mass we obtain Newtonian theory and GR, as was argued in [Bertolami et al., 2008]. r 2GM GQ2 1 − c2r + 4πε c4r2 ν1 2 0 2 6 Conclusion = r ν2 2GM GQ2 1 − 2 + 4 2 c r1 4πε0c r1 There are several lessons to be drawn. Al- ternative derivations of gravitational redshift, gh gC h ≈ 1 + − 2 , (5.5) as well as Einstein’s historical ones, hold ap- c2 c2 proximately only under special circumstances. GQ2 where g defined as before, and gC := 2 3 . Brown and Read used the equivalence princi- 2 4πε0c r The approximations are as in the case without ple, in conjunction with accelerated frames, and charge (first order terms in h and large radii claimed the gravitational redshift to derive from r1, r2). this. Moreover, they supposed only multiple The effect is quadratic in the charge Q, and, Pound-Rebka experiments to reveal curvature. in fact, leads to a blueshift of the photon. Thus, If Brown and Read’s statements about the in- it partly compensates the gravitational redshift ertial nature of gravitational redshift are valid due to mass. Note that gravity is fully ‘ge- in some sense, then only to first order and un- ometrised’ by GR. That is, geodesics of the met- der certain approximations. And if valid in this ric fully describe the motion of test particles. sense, then only in Rindler spacetime, for it

9 is the only framework that consistently treats [Brown and Read, 2016] Brown, H. R. and uniform acceleration (and uniform gravitational Read, J. (2016). Clarifying possible miscon- fields, respectively), in which the Riemann cur- ceptions in the foundations of general relativ- vature tensor vanishes globally. Arguments of ity. American Journal of Physics, 84(5):327– equivalence can be used to infer a general re- 334. lationship between redshift and acceleration, but the details are subtle. Thus, the conclu- [Cranshaw et al., 1960] Cranshaw, T. E., sion that a single gravitational redshift effect Schiffer, J. P., and Whitehead, A. B. (1960). on the Earth’s surface can be derived from the Measurement of the gravitational red shift equivalence principle and uniformly accelerated using the Mössbauereffect in Fe 57. Physical frames should be taken with a grain of salt. In Review Letters, 4(4):163. experiments involving strong gravitational field [Dewan and Beran, 1959] Dewan, E. and Be- variations only the exact Schwarzschild solution ran, M. (1959). Note on stress effects due to is accurate. relativistic contraction. American Journal of If the field generating source is charged, Physics, 27(7):517–518. a blueshift effect arises, which is captured by the Reissner-Nordströmmetric. Interestingly, [Einstein, 1905] Einstein, A. (1905). Ist die this blueshift partly compensates the gravita- Trägheiteines Körpers von seinem Energiein- tional redshift due to the mass of the source, halt abhängig? Annalen der Physik, which could be used to locally shield gravity. 323(13):639–641. It seems not difficult to convince oneself that any ‘sources of energy’ may likewise lead to [Einstein, 1911] Einstein, A. (1911). Uber¨ further redshift effects. Instances of these are den Einfluß der Schwerkraft auf die Aus- the strong and weak force, angular momentum, breitung des Lichtes. Annalen der Physik, magnetic fields, etc. 340(10):898–908. Translated by Michael D. Godfrey. Harvard.

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