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Gravitational Redshift, Inertia, and the Role of Charge

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Gravitational Redshift, Inertia, and the Role of Charge 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 uni- formly 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-Nordstr¨ommetric . .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 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 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 j = kaξ j . 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 (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 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 = p 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 = p 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 approx- imations mogeneous only to a first approxi- mation.' [Einstein, 1911, p. 900] p 1 1 + x ≈ 1 + x 2 The principle, thus, only holds in a `small neigh- 1 1 p ≈ 1 − x: (2.6) bourhood' of a point-like observer. Nonethe- 1 + x 2 less, a treatment of the redshift effect in a uni- form 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.
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