Gravitational Redshift, Inertia, and the Role of Charge

Gravitational redshift, inertia, and the role of charge

Johannes Fankhauser,∗ University of Oxford.

(October 5, 2018)

I argue that the gravitational redshift effect cannot be explained purely by way of uniformly accelerated frames, as sometimes suggested in the literature. This is due to the fact that in terms of physical effects a uniformly accelerated frame is not exactly equivalent to a homogeneous gravitational field let alone to a gravitational field of a point mass. In other words, the equivalence principle only holds in the regime of certain approximations (even in the case of uniform acceleration). 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.

Contents [Einstein, 1911]. His thought experiment ini- tiated the revolutionary idea that mass warps space and time. There exist some misconcep- 1 Introduction 1 tions, as evidenced in the literature, regarding the nature of the gravitational field in Einstein’s 2 Gravitational redshift 2 General Theory of Relativity (GR) and how it 3 Uniformly accelerated frames and relates to redshift effects. My aim is to give the equivalence principle 3 a consistent analysis of the gravitational redshift effect, in the hope of thereby advancing in 4 Equivalence and gravitational red- some small measure our understanding of GR. shift 6 Moreover, I will show that when charge is taken into account, gravitational redshift is subject to 5 Redshift due to charge 7 further corrections. 5.1 The weight of photons . . . . .7 5.1.1 Einstein’s thought experiment ...... 7 5.1.2 Inertia of energy . . . .8 In the first part of this paper I shall de- 5.2 Reissner-Nordströmmetric . . .9 rive and discuss the gravitational redshift in the framework of GR, from the equivalence princi- 6 Conclusion 9 ple, and from energy conservation principles to then compare and relate the different results. 1 Introduction In the second part of this paper I shall examine effects on the redshift due to charge with some In 1911 Einstein foresaw a phenomenon remarks on the relationship between GR and thereafter known as ‘gravitational redshift’ electromagnetism. ∗[email protected]

a 2 Gravitational redshift in the direction of ξa, we have ua = √ ξ 1 b −ξ ξb P1 It is a straightforward task to derive the rela- a a ξ p b and u2 = √ . The lengths −ξ ξb = tive shift in coordinate time of two clocks in a −ξbξ b P given gravitational field with metric g . Since 2 µν p−g ξbξc are obtained by contraction with the we will employ some alternative approximate bc metric. We let the observers O and O , whose approaches to derive the gravitational redshift 1 2 clock rates we wish to compare, describe their in the following sections, we shall choose to world-lines. The difference in the world-lines’ present the exact and most general derivation lengths in spacetime consequently determines from GR first, variants of which are standard the amount of gravitational redshift. Figure 1 fare (see for example, [Wald, 2010, p. 136]). 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 a a k to a receiver O2, placed at point P2, at height 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 |P = kaξ |P . a a 1 2 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 An emitter O1 on the surface of the Earth r ds2 = g dxµdxν = − 1 − S c2dt2 sends a train of electromagnetic pulses from µν r point P with energy momentum 4-vector ka r −1 1 + 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 a ∂ a where Killing field ξ = ∂t . Since the 4-velocities 2GM of the two observers are unit vectors pointing rS = (2.3) c2 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 is the so-called Schwarzschild radius, r the dis- source of gravitational field but moving in uni- tance from the Earth’s centre, G the gravita- form accelerated motion with respect to an in- tional constant, c the speed of light, and M the ertial frame [Norton, 1985].3 mass of the Earth. This yields However, it is clear that Einstein was well aware of the mere linearly approximate validity of the equivalence principle when he wrote: q p b 2GM −ξ ξb 1 − 2 ν c r2 1 P2 ‘...we arrive at a principle [the = = q ν2 p b 2GM equivalence principle] which, if it is −ξ ξb 1 − 2 c r1 P1 really true, has great heuristic im- GM 1 1 gh portance. For by theoretical consid- ≈ 1 + 2 − ≈ 1 + 2 , (2.4) c r1 r2 c eration of processes which take place relative to a system of reference with or ∆ν GM 1 1 uniform acceleration, we obtain in- ≈ − , (2.5) formation as to the behaviour of ν c2 r r 1 2 processes in a homogeneous gravi- GM with g := 2 2 the gravitational constant at r1, tational field. ... It will be shown c r1 ν = ν1, ∆ν = ν1 − ν2, and r2 − r1 = h. For in a subsequent paper that the grav- the last approximation in the second last line itational field considered here is ho- 1 1 r2−r1 h mogeneous only to a first approxi- we have used r − r = r r ≈ r2 if r1 ≈ r2 1 2 2 1 1 mation.’ [Einstein, 1911, p. 900] and r1, r2 h. Moreover, we used the approximations The principle, thus, only holds in a ‘small neigh- √ 1 bourhood’ of a point-like observer. Nonethe- 1 + x ≈ 1 + x 2 less, a treatment of the redshift effect in a uni- 1 1 form static gravitational field proves instruc- √ ≈ 1 − x. (2.6) 1 + x 2 tive, insofar as it shows that certain conse- quences of GR can be explained by means of ge- Experimental tests of the gravitational ometry without resorting to gravitational fields. redshift were first conducted by Cranshaw, Dealing with uniform accelerations to derive Schiffer and Whitehead in the UK in 1960 the gravitational redshift, however, is a deli- [Cranshaw et al., 1960]. It was not clear cate business, and we shall see that the field, whether significant conclusions could be drawn resulting from uniform (proper) acceleration, is from their results. In the same year, the ex- not uniform if we demand a constant (proper) periments by Pound and Rebka in Harvard suc- distance between emitter and observer! cessfully verified the gravitational redshift effect We consider a spaceship that is uniformly [Pound and Rebka Jr, 1960]. accelerated. An emitter E and receiver R inside the spaceship, separated by a height h, compare frequencies of signals ascending the spaceship. 3 Uniformly accelerated frames For an illustration, see Figure 2. and the equivalence principle As in the derivation of the gravitational redshift from the Schwarzschild metric, we let the Einstein’s equivalence principle (also called the observers describe their world-lines. It suffices weak equivalence principle) assumes that any to consider only one spatial dimension x. Accel- experiment in a uniform gravitational field eration a is measured in an inertial frame S with yields the same results as the analogous exper- momentary velocity v relative to the inertial iment performed in a frame removed from any 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’.

3 the acceleration is measured to be a0.4 Rela- We want the (proper) acceleration a of the tivistic transformation of 3-acceleration gives spaceship to be constant. The right hand side d of Equation 3.1 is equal to dt (γv). Since a is a = γ3a0, (3.1) constant we integrate Equation 3.1 twice to ﬁnd the trajectory — so-called Rindler hyperbola — where γ = 1 is the Lorentz factor.5 q v2 of a uniformly accelerated spacetime point as 1− 2 c observed by the inertial frame S0: s c2 at2 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 Figure 2: The gravitational redshift experiment expected. in a uniformly accelerated spaceship. The red- We recover uniform acceleration in the New- shift effect can be explained by the equivalence tonian sense for t 1. That is, principle (to first order). at2 x(t) = x + , (3.4) 0 2 Note that the acceleration of the spaceship 2 needs to be measured in the (momentary) in- with x0 = c /a the position at t = 0. ertial frame with instantaneous velocity v such For an exact derivation, it would lead 0 dv that a = dt (proper acceleration). With re- to inconsistencies to assume emitter and respect to the accelerated frame, sure enough, the ceiver be subject to the same Rindler hy- ship’s acceleration is zero. However, the princi- perbola with only an additional spatial dis- ple of relativity — the requirement according to tance h in the coordinate x. For if we main- which the laws of physics take the same form in tained a constant height between E and R rel- any inertial frame — no longer holds in acceler- ative to the inertial observing frame S0, length ated, hence non-inertial, frames. Therefore, as contraction, as predicted by special relativ- expected, the two observers in the spaceship are ity, would stretch the spaceship and eventually going to feel a (pseudo)force F = m0a, where tear it apart (cf. also Bell’s spaceship paradox m0 is the rest mass (invariant mass) of an ob- in [Dewan and Beran, 1959] and [Bell, 1987, ject in the spaceship. Chapter 9]). This is key. As was also pointed 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]). 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 out by [Alberici, 2006], assuming the gravita- region of the spaceship. That is, however, not tional acceleration to be the same for the top a surprise, for we would not expect the equiva- and bottom observers leads to all kinds of para- lence principle to hold globally in the first place. doxes. Most notably, it is not possible in this Further, it follows that proper time intervals case to define a globally freely falling inertial along two different Rindler hyperbolae between frame because the corresponding metric would two events having the same coordinate velocity lead to a non-vanishing Riemann tensor, and are in a fixed proportion, hence curvature! The receiver R in the bow lying higher by height h with respect to the τR gE gEh = = 1 + 2 , (3.7) emitter E must follow the hyperbola τE gR c yielding the exact gravitational redshift formula 2 c2 for uniform acceleration. Alternatively, we can x2 − c2t2 = + h , (3.5) a write for the proper height (relative to S) to be con- νE gRh νR = = νE 1 − (3.8) gE h c2 stant. These are the two desiderata to sim- 1 + c2 ulate reasonably the gravitational redshift by uniform acceleration: First, the ship must have for the corresponding observed frequencies, to a constant acceleration; and second, the ship highlight the dependence on the two differ- must have a constant proper height. The world- ent proper accelerations of emitter and receiver lines of emitter and receiver are denoted in Fig- (cf. also the results in [Alberici, 2006]). From ure 3. 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

νE ∆tx=h gEh = = 1 + 2 , (3.10) νR ∆tx=0 c which is consistent with the first order approx- Figure 3: The world-lines of emitter E and re- imation of the gravitational redshift from the ceiver R are Rindler hyperbolae when experi- Schwarzschild metric in Equation 2.4. Clocks encing constant proper acceleration. at E and R, whose rates one wishes to compare, are permitted to describe their world-lines, i.e. Rindler hyperbolae, with respect to the inertial Due to relativistic length contraction, the frame, and the value for the redshift is obtained receiver’s proper acceleration needs to be by comparing the lengths of their world-lines in slightly greater. By comparing the two hyper- spacetime. Therefore, the treatment here is ex- bolae it immediately follows that the accelera- act. The Rindler metric is, in fact, a solution tion gR of the receiver is related to the emitter’s to the vacuum Einstein field equations and has acceleration gE by vanishing curvature (Rµνρσ = 0). gE In the experiments of Pound-Rebka to con- gR = . (3.6) gE h firm gravitational redshift, the emitter sends a 1 + 2 c signal at equal intervals on a clock at the sur- Compare also the treatment and related para- face of the Earth. The receiver measures the doxes in [Fabri, 1994]. Therefore, the gravita- time interval between receipt of the signals on tional field is not constant over the extended 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 lo- cally, ...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 mas- sive 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. The Einstein equations for a charged point- like (non-rotating) mass with stress-energy tensor Tµν reads 6 Conclusion 1 8πG R − Rg = T . (5.3) µν 2 µν c4 µν There are several lessons to be drawn. Al- We neglect the cosmological constant Λ. From ternative derivations of gravitational redshift, this we obtain the Reissner-Nordströmmetric as well as Einstein’s historical ones, hold ap- (cf. [Reissner, 1916]) proximately only under special circumstances. Brown and Read used the equivalence principle, in conjunction with accelerated frames, and 2GM GQ2 2 2 2 claimed the gravitational redshift to derive from ds = − 1 − 2 + 4 2 c dt c r 4πε0c r this. Moreover, they supposed only multiple 2 −1 2GM GQ 2 Pound-Rebka experiments to reveal curvature. + 1 − 2 + 4 2 dr c r 4πε0c r If Brown and Read’s statements about the in- + r2(dϑ2 + sin2 ϑdϕ2), (5.4) ertial nature of gravitational redshift are valid in some sense, then only to first order and un- from which we recover the Schwarzschild met- der certain approximations. And if valid in this ric in the limit Q = 0. It is worth mentioning sense, then only in Rindler spacetime, for it that the charge term in the Reissner-Nordström is the only framework that consistently treats metric affects geodesics of particles even though uniform acceleration (and uniform gravitational they may be uncharged. For Q 6= 0, this met- fields, respectively), in which the Riemann cur- ric gives rise to an additional gravitational red- vature tensor vanishes globally. Arguments of shift. Similar to the derivation of gravitational equivalence can be used to infer a general re- redshift due to mass we obtain lationship between redshift and acceleration, but the details are subtle. Thus, the conclu- r sion that a single gravitational redshift effect 2GM GQ2 1 − c2r + 4πε c4r2 on the Earth’s surface can be derived from the ν1 2 0 2 = r equivalence principle and uniformly accelerated ν 2 2 2GM GQ frames should be taken with a grain of salt. In 1 − c2r + 4πε c4r2 1 0 1 experiments involving strong gravitational field gh g h ≈ 1 + − C2 , (5.5) variations only the exact Schwarzschild solution c2 c2 is accurate. GQ2 where g defined as before, and gC := 2 3 . If the field generating source is charged, 2 4πε0c r The approximations are as in the case without a blueshift effect arises, which is captured by charge (first order terms in h and large radii the Reissner-Nordströmmetric. It seems not r1, r2). difficult to convince oneself that any ‘source The effect is quadratic in the charge Q, and, of energy’ may likewise lead to further red- in fact, leads to a blueshift of the photon. Thus, shift effects. Instances of these are the strong it partly compensates the gravitational redshift and weak force, angular momentum, magnetic due to mass. Note that gravity is fully ‘ge- fields, and all other terms that contribute to ometrised’ by GR. That is, geodesics of the met- the stress-energy tensor in Einstein’s field equa- ric fully describe the motion of test particles. tions.

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