# Is the lay explanation of the equivalence principle wrong?

The common explanation/trope for the equivalence principle always has something to do with you being inside an elevator or spaceship, and your supposed inability to differentiate, say, gravity from a rocket, or free-fall from being motionless in empty space. For example, Wikipedia says "being at rest on the surface of the Earth is equivalent to being inside a spaceship (far from any sources of gravity) that is being accelerated by its engines." Many other sources say similar things, often making broad, sweeping generalizations like "there is no measurement you can make to distinguish the two scenarios."

Isn't that just plainly wrong? Gravitational fields aren't homogeneous. Here on the Earth, a clock on the floor runs more slowly than a clock on the table, and we have clocks precise enough to measure such small differences due to the gravitational gradient. But doesn't a clock in an accelerating spaceship run at the same rate no matter where in the ship you put it?

I searched the top 50 or so questions about the equivalence principle and found this answer that talks about tidal effects in gravitational fields, but the explanation is very confusing. As far as I can tell, it seems to be saying the "attraction" between the particles that arises as a result of slightly different gravity vectors is somehow equivalent to the actual mutual gravitational attraction the particles should have absent the external gravitation field. I don't see how that could possibly be the case.

Also in that answer is a link to this explanation of tidal forces and the equivalence principle, which seems to be saying the opposite: that tidal forces are, in fact, a distinguishing characteristic between gravitational acceleration and rocket acceleration, and that the equivalence principle only exactly applies to small enough points in space over a small enough duration, where tidal effects are negligible.

I know what seems most correct to me, but as I'm not an expert in this field, would an actual expert please shed some light on these varying views of the equivalence principle?

• More sophisticated arguments at that level include some language equivalent to "in a sufficiently small laboratory", which is strictly correct as the equivalence principle is a local (in the technical sense) statement. – dmckee Mar 27 '15 at 23:11
• Also, I remember reading that a clock on an accelerating spaceship will run at different speeds when placed at different heights, which is consistent with the equivalence principle. – Javier Mar 27 '15 at 23:19
• A reading assignment: mathpages.com/home/kmath622/kmath622.htm – Alfred Centauri Mar 27 '15 at 23:28

But doesn't a clock in an accelerating spaceship run at the same rate no matter where in the ship you put it?

Remarkably, the answer is, even in the context of SR, no.

It turns out that acceleration of an extended object is quite subtle.

That is to say, we can't meaningfully speak of the acceleration of an extended object.

Essentially, the 'front' (top?) of the spacecraft accelerates less than the 'back' (bottom?) of the spacecraft if the spacecraft is not to stretch and eventually fail structurally.

Thus, the clocks at the front (top) run faster than the clocks at the back (bottom) as would be the case for clocks at rest at different heights in a gravitational potential.

This is actually well known and best understood in the context of Rindler observers.

Note that Rindler observers with smaller constant x coordinate are accelerating harder to keep up! This may seem surprising because in Newtonian physics, observers who maintain constant relative distance must share the same acceleration. But in relativistic physics, we see that the trailing endpoint of a rod which is accelerated by some external force (parallel to its symmetry axis) must accelerate a bit harder than the leading endpoint, or else it must ultimately break.

Now, this isn't meant to answer your general question but, rather, to address the particular question quoted at the top.

• I hadn't considered that Lorentz contraction meant there was an acceleration gradient along the length of an accelerating spacecraft. Other tidal effects of gravity observable on Earth, I believe, arise as a result of Earth's particular configuration of matter, which doesn't have anything to do with gravity per se. – Nickolas Mar 29 '15 at 3:24
• Does the acceleration gradient follow an inverse square law? – Asad Saeeduddin Feb 9 '16 at 10:24

Why does an apple fall from a tree? Why do all objects accelerate towards earth at $9.8$ $m/s^2$? The 'out-of-the-box answer' is that the objects themselves don't move. It's the ground that rushes up! Regardless whether attached to the tree or not, Newton's apple is suspended motionless: it's earth's surface that accelerates up and meets the apple. This simple insight immediately explains why all objects regardless their mass accelerate at the same pace of $9.8$ $m/s^2$.

Right?

Unless you are a 'flat-earther' the above probably comes across like total nonsense. But the point here is that there is absolutely nothing you can do that could convince me to revisit my position, unless you bring into the picture large-scale features such as earth's curvature. That is, you have to introduce non-local effects to force me into acceptance of the concept of gravity. Non-local effects such as the difference in direction between the acceleration of apples falling from trees located away from each other. Such differences in gravitational acceleration represent the true signature of gravity, and are known under the generic label 'tidal effects'.

There is no local measurement you can do on the falling apple that would prove the 'earth rushing up' model wrong. This is, in essence, Einstein's equivalence principle: in any small region of spacetime gravity is equivalent to uniform acceleration.

Gravitational fields aren't homogeneous. Here on the Earth, a clock on the floor runs more slowly than a clock on the table, and we have clocks precise enough to measure such small differences due to the gravitational gradient. But doesn't a clock in an accelerating spaceship run at the same rate no matter where in the ship you put it?

See page 962 lines 20-29 (Einstein's paper from 1911): https://www.relativitycalculator.com/pdfs/On_the_influence_of_Gravitation_on_the_Propagation_of_Light_English.pdf

1. Gravitational time dilation does not depend on whether or not the gravitational field is homogeneous. It depends on the gravitational potential between the locations of the clocks.
2. Gravitational time dilation is actually deduced by Einstein based on your (and his) argument that a clock in an accelerating spaceship runs at the same rate no matter where in the ship you put it. In his own words in the paper of above: "if we measure the velocity of light at different locations in the accelerated, gravitation free system K0, employing clocks U of identical properties we obtain the same magnitude at all these locations". Based on this argument and on the fact that the frequency of a wave must shift between two locations on the axis of acceleration of an accelerated spaceship due to the change in velocity during the propagation of the wave from one location to another, the supposed equivalence enforces the existence of gravitational time dilation in a system at rest in a gravitational field, for there is no change of velocity in such a stationary system, yet we expect a shift in the frequency of a wave between two locations in that system (equivalent to the frequency shift in the accelerated spaceship).
3. As you will appreciate from this paper, Einstein deduces gravitational frequency shift, gravitational bending of light and gravitational time dilation based on thought experiments in which a wave travels from a location S1 to a location S2 inside the "spaceship" (K and K' reference frames in his language), hence there is no basis for claiming that the equivalence is relevant only to regions of zero extent. The importance of the principle is no other than allowing to predict the different influences of a gravitational field by thought experiments that involve motion from one location to another.
4. The principle should be taken as an idealization. indeed

Gravitational fields aren't homogeneous

but suppose we somehow magically produced a homogeneous gravitational field (Einstein is great in thought experiments), could an observer in a closed room recognize its presence? Einstein answered this to the negative, and this answer has led him to his profound predictions.

1. In an accelerated spaceship remote clocks situated on the axis of acceleration are drawn out of synchronization (without a change in their tick rate). Clocks resting in a gravitational field and located in different gravitational potentials are drawn out of synchronization by ticking at different rates. If the equivalence principle is true, an observer in a closed room will fail to recognize whether the clocks are drawn out of synchronization because they are situated in different gravitational potentials or because the velocity of the room is changing with time.

Pete Brown wrote an essay which referred to this, see http://arxiv.org/abs/physics/0204044 and note the quote by Synge on page 20:

"I have never been able to understand this principle. Does it mean that the effects of a gravitational field are indistinguishable from the effects of an observer’s acceleration? If so, it is false. In Einstein’s theory, either there is a gravitational field or there is none, according as the Riemann tensor does not or does vanish. This is an absolute property; it has nothing to do with any observers world line … The Principle of Equivalence performed the essential office of midwife at the birth of general relativity, but, as Einstein remarked, the infant would never have gone beyond its long clothes had it not been for Minkowski’s concept [of space-time geometry]. I suggest that the midwife be buried with appropriate honours and the facts of absolute space-time faced."

In my view as a guy who "roots for relativity", the equivalence principle is only a principle, not some hard-and-fast golden rule. Being in an accelerating spaceship is like standing on the surface of the Earth, but it isn't exactly the same. The principle of equivalence applies to an infinitesimal region only, a region of zero extent, to no actual region. See Einstein's Leyden Address where he referred to a gravitational field as inhomogeneous space? Standing on a planet in inhomogeneous space is like accelerating through homogeneous space, but it isn't exactly the same.

## protected by Qmechanic♦Apr 19 '16 at 7:00

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