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. (October 5, 2018) I argue that the gravitational redshift effect cannot be explained purely by way of uni- formly 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 red- shift 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 exper- iment . .7 5.1.2 Inertia of energy . .8 In the first part of this paper I shall de- 5.2 Reissner-Nordstr¨ommetric . .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] 1 a 2 Gravitational redshift in the direction of ξa, we have ua = p ξ 1 b −ξ ξb P1 It is a straightforward task to derive the rela- a a ξ p b and u2 = p . 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 uaj 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 ξaj = g j 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 jP = kaξ jP . 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 (d# + sin #d' ); 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 approx- imations The principle, thus, only holds in a `small neigh- p 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- p ≈ 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 red- shift 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.