Gravitational Redshift and the Equivalence Principle

Many different forms of the “equivalence principle” for theories of gravitation have been discussed in the literature. Most discussions refer to weak and strong versions, differing in their scopes of applicability. Roughly speaking, the weak principle (associated with Galileo and Newton) is usually said to be applicable to the dynamical behavior of material particles, whereas the strong principle (associated with Einstein) is regarded as a generalization or extrapolation to include all physical phenomena. There are controversies about how best to state these principles, e.g., in terms of “sufficiently small” regions or hypothetical homogeneous fields, in order to avoid falsification while retaining physical meaning. However, even if we focus on just the dynamics of material objects, there are different ways of expressing the equivalence principle, each with subtly different heuristic tendencies. Gravitational redshift is sometimes said to be a feature of any theory that satisfies the equivalence principle, but it’s important to note that this is true only when combined with local Lorentz invariance, i.e., special relativity in the absence of gravitation. It’s also worth remembering that, according to general relativity, gravitational redshift applies to the dynamical behavior of material particles, not just to electromagnetic waves, so the redshift cannot be attributed to the supposed extrapolation of the equivalence principle. It can be derived entirely in the context of the behavior of material particles.

Consider the following three ways of expressing the “equivalence principle”:

Galileo’s Equivalence Principle (GEP): Inertial and gravitational mass are equal, so all particles of matter undergo the same acceleration in a gravitational field.

Newton’s Equivalence Principle (NEP):  The laws of mechanics inside a sufficiently small elevator car (for a sufficiently short time) in free-fall in a gravitational field are indistinguishable from the laws of mechanics inside such a car floating freely in empty space, far from any gravitating body.

Einstein’s Equivalence Principle (EEP):  The laws of mechanics inside a sufficiently small elevator car at rest in a stationary gravitational field are indistinguishable from the laws of mechanics inside such a car in empty space (far from any gravitating body) subjected to constant acceleration.

We call NEP Newton’s principle because Newton explicitly noted that GEP implies NEP in Corollary 6 of the Laws of Motion in his “Principia”:

If bodies, moved in any manner among themselves, are urged in the direction of parallel lines by equal accelerative forces, they will all continue to move among themselves, after the same manner as if they had not been urged by those forces.

Thus NEP does indeed follow from GEP, and is satisfied by Newton’s laws. We call EEP the Einstein equivalence principle because, even though many authors claim Einstein’s only innovation was to extrapolate Newton’s principle to all physical phenomena (including electromagnetics and gravity itself), this is not entirely consistent with how Einstein himself described the principle. Admittedly he wrote about how someone in free fall doesn’t feel his own weight (echoing Newton’s proposition), but he also gave different statements of the principle, such as in his 1923 Princeton lectures (based closely on his 1916 paper), where he said

Let K be an inertial system. Masses which are sufficiently far from each other and from other bodies are then, with respect to K, free from acceleration. We shall also refer these masses to a system of coordinates K’, uniformly accelerated with respect to K. Relatively to K’ all the masses have equal and parallel accelerations [and] behave just as if a gravitational field were present and K’ were unaccelerated… The assumption of the complete physical equivalence of the systems of coordinates, K and K’, we call the “principle of equivalence”.

This is what we have called the Einstein Equivalence Principle (EEP), which asserts the equivalence between accelerated coordinates in flat space and stationary coordinates in a homogeneous gravitational field. Notice that Einstein describes the principle here purely in terms of the motions of “masses”, and only in the recapitulation does he refer to “complete physical equivalence”. The EEP, as distinct from NEP, emphasizes the equivalence of acceleration and gravity. Although the EEP and NEP are closely related to each other, the EEP tends to be more suggestive of theory in which gravitation is identified with acceleration. We will show that Newton’s theory, although it satisfies the equivalence principles, does not predict any difference between the frequency of a sequence of particles shot from the top of a tower and the frequency at which they arrive at the bottom of the tower. In other words, none of the equivalence principles, by themselves, imply a frequency shift if we assume Galilean relativity applies in the absence of gravity.

As an aside, we note that the restriction to “sufficiently small” regions is slightly problematic, because for a sufficiently small region the gravitational redshift itself can be made imperceptibly small, as can other effects such as the deflection of light. However, we assume the principles apply to large enough regions so that they have physically meaningful implications. In other words, the equivalence applies to regions small enough so that we ignore high-order effects, but large enough to exhibit the low-order effects.

In the context of Newtonian physics, consider an elevator car floating freely in empty space, far from any gravitating body, and suppose bullets are being fired from the bottom of the car once per second. The bullets will obviously arrive at the top once per second. In view of the NEP, it follows that the same must be true for an elevator car in freefall near a gravitating body. In other words, there will be no change in frequency between the bottom and top of a free-falling car. (We stipulate that the car is not falling far during our experiment, so the acceleration of gravity is essentially constant during this period.)

We can also infer from Newtonian theory that if an elevator car is stationary on the Earth’s surface, each bullet will take an equal amount of time to travel from the bottom to the top of the car, so again the frequency of arrival at the top will equal the frequency of firing at the bottom. In this case the bullets will be spatially closer together when they reach the top, and moving more slowly, but the frequency will be unchanged. Thus Newton’s theory does not predict any gravitational “redshift” for ballistic particles, whether the car is floating freely in empty space, in freefall near a gravitating body, or stationary near a gravitating body. (For a detailed proof of this, see below.)

Also, still within the context of Newtonian theory, if an elevator car in empty space, far from any gravitating body, is subjected to constant upward acceleration (equal to, say, the acceleration of gravity on the earth’s surface), each successive bullet takes the same amount of time to travel from the bottom to the top of the car, as can be seen by noting the congruence of each bullet’s journey described in terms of the inertial coordinate system co-moving with the bottom of the car at the moment of emission – with the Newtonian assumption that these coordinate systems are related by Galilean transformations. To verify this explicitly, consider the figure below, which shows the parabolic trajectories of the top and bottom of the car, and of a projectile fired upward from the bottom to the top.

Letting “a” denote the acceleration and “L” the height of the car, the bottom and top of the car follow the paths

respectively. (We assume Galilean-rigid acceleration.) Letting U denote the speed of the bullet relative to the floor at the moment of firing, and noting that the bottom of the elevator at time t is 2at, we have the relation

Making the substitutions x₁ = at₁ and x₂ = at₂ + L and simplifying, we get

Solving this for the time interval, the time required for the bullet to reach the top of the car is

(Naturally this interval approaches L/v₀ in the limit as a goes to 0.) This confirms that the interval is independent of the point in the elevator’s trajectory when the bullet is fired, so the interval between two consecutive bullets arriving at the top is equal to the interval between firing at the bottom. Hence there is no redshift, just as there is none for a stationary car in a gravitational field. Thus Newton’s theory satisfies EEP, even though it does not imply gravitational redshift. We get gravitational redshift only when we combine the equivalence principle with local Lorentz invariance (LLI), which signifies that local relatively-moving inertial coordinate systems are related by Lorentz transformations rather than Galilean transformations. In an accelerating elevator car in flat spacetime according to special relativity, the time required for each successive bullet to reach the top of the car is not invariant, because the composition of the speed of the bullet and the speed of the car is not simply additive, and we have relativistic time dilation due to the change in speed of the car during the time of transit. This special relativistic time dilation associated with acceleration in flat spacetime, combined with the equivalence principle, implies gravitational time dilation in a car that is stationary in a gravitational field. Without the time dilation effects of special relativity in flat spacetime, no version of the equivalence principle would imply gravitational redshift.

Oddly enough, the literature contains many claims to the contrary. For example, Bernard Schutz (in his book “Gravity from the Ground Up”) gives the following “proof” that Newton’s theory predicts gravitational redshift. He expresses this in terms of light instead of bullets, but he argues that the reasoning is perfectly general, claiming that “any theory of gravity predicts the effect if it respects the [weak] principle of equivalence”. And of course, the redshift in general relativity doesn’t apply only to light, it applies to bullets fired from a gun as well, because it corresponds to gravitational time dilation, which applies to all physical processes. Here is generic Schutz’s “explanation” for gravitational redshift:

Imagine a beam of light shining upwards from a source on the ground. An experimenter stands on a tower directly over the source and measures the frequency of the light when it reaches him.  [He] has a companion who falls off the tower at the moment that the beam of light leaves the ground. As a freely falling experimenter, the companion finds that the light moves as if it were in outer space, in particular, the frequency that he measures does not change with time. At the instant he leaps off, he is still at rest with respect to the ground, so he measures the same frequency as an experimenter on the ground would measure. By the equivalence principle, this is also the frequency he (the companion) measures when the light reaches the top of the tower a moment later. But in this brief moment, the companion has begun to fall. Relative to the companion, the experimenter at the top is moving away from the light source at the time of reception of the light at the top, and so the fixed experiment's frequency will be redshifted with respect to the companion's frequency.

As written, this is absurd, because it says “At the instant he leaps off, he is still at rest with respect to the ground, so he measures the same frequency as an experimenter on the ground would measure”. Needless to say, this is the frequency that both experimenters on the tower would measure at this instant, since they are in identical locations and states of motion. Schutz invokes the equivalence principle only to argue that the frequency for the falling observer “does not change with time”, but surely he’s not suggesting that the frequency for the stationary observer changes with time, and that it suddenly begins to change simply because his companion has slipped off the tower. The fact is that, based on Newtonian physics with the equivalence principle (but local Galilean relativity), both of the experimenters (the stationary one at the top of the tower and the free-falling one) will encounter the same frequency, not redshifted.

To prove this, consider the figure below, in which the two vertical lines represent the world lines of the bottom and top of a tower of height L standing on the Earth’s surface, and the parabolas represent tandem free-falling worldlines. These could be thought of as the bottom and top of an elevator car in free fall. The right-hand parabola can also be regarded as the extended world line of the free-falling companion in Schutz’s thought experiment.

The gun at the bottom of the tower is fired at event E₁, and companion slips off the tower at that same time. (Alternatively, the companion could have been thrown upward along the parabolic world line, and been momentarily at rest at the top.) The first bullet passes the free-falling experimenter at event A, and the stationary experimenter at event B. A small amount of time ∆t later, another bullet is fired, but now we must decide if this takes place at event E₂ at the stationary position of the bottom of the tower, or if it takes place at event E₃ at the bottom of the imaginary elevator car that is in free fall. For the stationary situation, the bullet is fired with the same upward speed from event E₂ and arrives at event D at the top of the tower, and the elapsed time between B and D is simply ∆t, so there is no frequency shift or time dilation. For the free-falling elevator car, the second bullet is fired at event E₃, with an upward speed that is diminished by the fact that the bottom of the elevator is moving downward. This bullet arrives at the top of the car at event C, and the elapsed time between A and C is (again) simply ∆t, so there is no frequency shift or time dilation.

To verify these statements, note that the stationary positions of the bottom and top of the tower are x=0 and x=L, and the equations of the bottom and top of the free-falling elevator car are

The equation of motion of the first bullet, fired from event E₁, is

The equation of motion of the second bullet, if it is fired from event E₂ (the bottom of the stationary tower) is

On the other hand, if the second bullet is fired from event E₃ (the bottom of the free-falling elevator), we must take into account not only the fact that the position is slightly lower but that the upward speed is diminished (since the bullet has constant speed v₀ relative to the gun), so the equation of motion is

Solving for the intersections of these worldlines, we get

Therefore, we have

so the time intervals between arrivals of the bullets are identical to the time interval between shots. This applies for any value of ∆t, no matter how small. This might seem surprising, since the bottom of the tower and the bottom of the free-falling elevator are almost coincident near the “apogee”. However, the upward speed of the second bullet is being reduced linearly, and this causes the arrival time to be delayed by just the amount to ensure that there is no frequency shift.

One might object that, for light, the speed is independent of the source, but this relies on special relativity (local Lorentz invariance), which entails time dilation related to relative velocities, but special relativity is not part of Newtonian physics. According to Newtonian theory, local inertial coordinate systems (in the full sense) are related by Galilean transformations, and there is no time dilation of any kind. This is perfectly compatible with the equivalence principles, including EEP.

Before continuing, we should mention a common source of confusion on this subject. Consider firing bullets downward from the top of the tower. Clearly the bullets fired downward in the stationary car will undergo acceleration during their transit, so they will have a higher speed when they reach the bottom than they had when they were emitted from the top. One University of Illinois professor “explains” gravitational redshift by analogy with falling mechanical objects as follows:

Consider a falling object. Its speed increases as it is falling. Hence, if we were to associate a frequency with that object the frequency should increase accordingly as it falls to earth.

This is a misconception. The fact that a sequence of material objects, pulses of light, or wavecrests accelerate during their fall does not imply that they will arrive more frequently. Each object (pulse, wavecrest) in a stationary car undergoes the same acceleration and travels the same distance, so the elapsed time to fall is the same for each of them, and hence the frequency of arrivals equals the frequency of departures. (The objects will be spatially further apart, and moving faster, such that the frequency is unchanged.)

Our Illinois professor may counter this by saying that he isn’t referring to a sequence of entities, each undergoing the same pattern of acceleration, but rather to an individual entity possessing an inherent frequency. This is incoherent, both figuratively and literally. (It seems to arise from misconceptions about quantum theory, and the mistaken idea that individual photons have frequencies.) If the entity in question is a wavecrest (for example), it obviously makes no sense to talk about the frequency of an individual wavecrest, or of any other specific phase. By definition the frequency of a wave is proportional to the frequency of arrival of successive phases (e.g., wavecrests). At this point, students sometimes imagine a ‘javelin theory’, according to which each object is conceived as having a certain length (in the direction of travel) with markings at fixed intervals along its length. If the javelin passes us more rapidly, the frequency of markings passing us will increase. However, this again is incoherent, as can readily be seen by noting that some radio waves have wavelengths of many miles, far longer than the height of the car or tower, and it clearly won’t do to have a javelin equal to an entire wavelength (let alone multiple wavelengths), because then the rigidity of the javelin would imply equal speeds at the top and bottom. Hence successive wavecrests must be on separate javelins, and we know there is no frequency shift in the arrival times of the javelins. One might posit a “rain” of many javelins in parallel, but each falls separately, and the average frequency of markings reaching the floor of the car is just equal to the frequency of arrival of the javelins multiplied by the number of markings per javelin. Hence (again) the average arrival frequency of the markings equals the departure frequency.

Many scientists seem to be confused about the origin and explanation of gravitational redshift. For example, Banesh Hoffmann (in his book Relativity and its Roots) begins by correctly noting that

By the principle of equivalence... the experimenter next to the ceiling clock must see the floor clock going at a slower rate [for a stationary room in a gravitational field]. But here there is no acceleration to account for the result. Instead there is gravitation. So Einstein concluded that if two clocks going at the same rate are placed with one closer to the earth than the other, the one closer to the earth will be seen in this way to go at a slower rate than the other.

This is true enough, but then he says

The gravitational red shift does not arise from changes in the intrinsic rates of clocks. It arises from what befalls light signals as they traverse space and time in the presence of gravitation.

At best the first sentence might be accepted as a tautology, since the “intrinsic rate” of a clock (or any regular physical process) is always the same by definition: Compared with itself, every clock advances one minute per minute. However, the second sentence is difficult to defend, because whatever “befalls” a sequence of particles, light pulses, or wavecrests as they travel from emitter to receiver befalls each of them equally, so it cannot account for any difference in frequency between emission and reception at two stationary points. A difference in frequency can only be due to a difference in the extrinsic rates of the clocks at those two points. This effect is cumulative, as can be confirmed after the clocks have been at different gravitational potentials for an arbitrarily long time, by bringing them together and making a direct comparison.

One often reads that the weak equivalence principle (meaning GEP and/or NEP) is sufficient to imply gravitational time dilation. For example, an article on gravitational redshift for Einstein Online provided by the University of California, Irvine, says

You do not need general relativity to derive the correct prediction for the gravitational redshift. A combination of Newtonian gravity, a particle theory of light, and the weak equivalence principle (gravitating mass equals inertial mass) suffices.

But this, as we’ve seen above, is not true. A stream of particles in Newtonian gravity, obeying the equivalence principle will not exhibit any frequency shift between two stationary points in a gravitational field, since every particle undergoes the same acceleration. Other resources, such as the article on gravitational redshift in Living Reviews on Relativity, note (correctly) that that the weak principle suffices when combined with special relativity in a metrical theory of spacetime, i.e., with the presumption that (local) inertial coordinate systems are related by Lorentz transformations with an invariant spacetime interval. Special relativity already entails time dilation effects along different worldlines, so it is already inconsistent with Newtonian physics.

Of course, it’s well known that the equivalence principle is actually incompatible with global special relativity – which is precisely what led Einstein to the realization that a generalization was needed. According to global special relativity there exist global inertial coordinate systems in terms of which the speed of light is invariant, but the equivalence principles (weak or strong) combined with the inertia of energy implies that the energy of an electromagnetic wave (like every other form of energy) must be deflected in a gravitational field, and therefore the speed of the wavefront must vary with position in terms of a stationary coordinate system. This unambiguously implies that only local inertial coordinate systems can exist in the presence of gravity, and any global system of coordinates must possess intrinsic curvature and time dilation.

At the other extreme, one also finds claims in the literature (e.g., Florides) that not only is the weak principle insufficient to imply gravitational redshift, but that even the strong equivalence principle is not sufficient, and that only a full theory such as general relativity can unambiguously imply gravitational redshift.

For one last example of the confusion in the literature over this point, note that, in his “Lectures on Gravitation”, Richard Feynman gives two derivations of gravitational redshift. The first is in Section 5.2, where he says

The prediction of this frequency shift does not really need the machinery of our theory of gravitation, since it is implicit in the experimental result of Eotvos, that gravity forces (or potentials) are proportional to the energy content. Thus the frequency shift corresponds to the gravitational energy of the photon. According to Eotvos, the excited nucleus is heavier by (E₀/c²)g, if E₀ is the excitation energy, since we know from nuclear experiments its mass is M + E₀/c² if M is the mass in the ground state. When it is raised by a height h it contains the energy E₀ + (E₀/c²)gh + Mgh more than an unexcited nucleus at zero height. If we excite the lower nucleus we require only E₀. After the upper nucleus makes a transition, its total energy should exceed that of the lower nucleus only by Mgh. Since the photon frequency is [proportional to its energy according to the quantum relation] , the frequency of the photon emitted is ω = ω₀[1 + gh/c²]. It is thus obvious that the frequency shift is required by energy conservation.

This is valid reasoning, but it relies on not only the equivalence principle (as demonstrated by Eotvos) and the proportionality of the energy and frequency of a photon from quantum theory, but also crucially on the relation E = Mc² from special relativity. This confirms that the frequency shift follows not from the equivalence principle alone, but only from the equivalence principle combined with the postulate of special relativity (Lorentz invariance) in flat spacetime.

But later, in Section 7.2 (Some Consequences of the Principle of Equivalence), Feynman gives another derivation of the frequency shift, and claims that it does not rely on the relation E = mc². He says

The principle of equivalence also tells us that clock rates are affected by gravity. Light which is emitted from the top of the accelerating box will look violet-shifted as we look at it from the bottom… The time that light takes to travel down is to a first approximation c/h, where h is the height of the box. In this time, the bottom of the box has acquired a small additional velocity, v = gh/c. The net effect is that the receiver is moving relative to the emitter, so that the frequency is shifted [by the Doppler factor] 1 + v/c = 1 + gh/c². Note that this conclusion does not depend on E = mc² and on the existence of energy levels, which we had to postulate in the argument we have given previously. The conclusion is based on the expected behavior of classical objects; the calculation results from geometry and kinematics, and makes a direct physical prediction from the postulate of equivalence.

As explained previously, this argument is incorrect, because according to “the expected behavior of classical objects” (treating light as a stream of particles) in the context of Galilean relativity, the time for successive particles of light to traverse the box is invariant, provided they are emitted at a constant speed relative to the source, even though the speed of these particles relative to the receiver is different at the moment of reception. The confusion arises because people tend to assume that the Doppler effect (in a classical context) depends on the difference in relative speeds at emission and absorption, whereas it actually depends on a difference in travel times for successive particles (or wave crests). As we’ve seen, for classical particles in Galilean spacetime, the travel time for each successive particle is the same, so there is no frequency shift between emission and absorption, although the equivalence principle is perfectly satisfied. One might try to salvage the argument by postulating a classical wave theory of light in some stationary ether, but this not only conflicts with many experiments, it also restricts the argument to frequencies of electromagnetic waves, which would not account for the fact that time dilation entails changes in frequency for sequences of ballistic particles as well. Thus the only way that the equivalence principle leads to gravitational frequency shift is when it is combined with the postulate of local Lorentz invariance (equivalent to the postulate E = mc²).

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