Understanding the Equivalency Principle: Elevator Example and Its Limitations

[ryan albery]

The classic example of the equivalency principle is being inside an elevator where an acceleration is in indiscernible from a gravitational field. I'm thinking the elevator is a bad example though, and that the equivalence principle only holds for a single point in spacetime. If you have any distance to make a measurement, at the top and bottom of the elevator for example, you'll measure a gradient if you're in a gravitational field, but not if you're simply accelerating. Our current technology can measure a change in gravity with less than a centimeter of elevation change on the surface of the earth... a gradient that wouldn't exist if you were out at space and accelerating.

Any thoughts, does this make sense?

 

[Nickelodeon]

... you'll measure a gradient if you're in a gravitational field, but not if you're simply accelerating.

I think this is a very good reason why not to base a theory of gravitation on The Equivalence Principle

 

[A.T.]

I'm thinking the elevator is a bad example though, and that the equivalence principle only holds for a single point in spacetime.

The idea of the Equivalence Principle is that gravity is an inertial force, that affects everything equally in that point.

The uniform inertial force in an accelerating elevator is just an example of an inertial force that is easy to understand. There are other inertial force examples, like the centrifugal force, Coriolis force, which are not uniform. But none of them is exactly like gravity around a spherical mass, so we use the simplest one as an analogy: linear acceleration.

 

[Al68]

The classic example of the equivalency principle is being inside an elevator where an acceleration is in indiscernible from a gravitational field. I'm thinking the elevator is a bad example though, and that the equivalence principle only holds for a single point in spacetime. If you have any distance to make a measurement, at the top and bottom of the elevator for example, you'll measure a gradient if you're in a gravitational field, but not if you're simply accelerating. Our current technology can measure a change in gravity with less than a centimeter of elevation change on the surface of the earth... a gradient that wouldn't exist if you were out at space and accelerating.

Any thoughts, does this make sense?

The equivalency is only valid for a theoretical uniform gravitational field, not a non-uniform one. The equivalency is only approximate locally when the field is non-uniform, like Earth's field.

The significance of the equivalence principle is that the concept of gravity is equivalent to an accelerated reference frame, not that a particular example of a gravitational field is equivalent to a particular accelerated frame.

 

[Nabeshin]

Intelligent textbook authors add the caveat,
"Being in an accelerated elevator is indistinguishable from being in a gravitational field for experiments performed over small enough distances and time scales"

 

[DrGreg]

If you have any distance to make a measurement, at the top and bottom of the elevator for example, you'll measure a gradient if you're in a gravitational field, but not if you're simply accelerating.

Actually, you will measure a gradient in the "simply accelerating" lift, but not quite the same gradient as under gravity.

To a first order approximation (i.e. approximated as proportional to distance) both gradients are the same, but to higher accuracy they do differ.

 

[ryan albery]

I'm not sure if I understand that correctly, as it seems that if an elevator was accelerating at 1g, the entire body would be accelerating at 1g (assuming a non rotational acceleration). Am I missing something?

 

[yuiop]

I'm not sure if I understand that correctly, as it seems that if an elevator was accelerating at 1g, the entire body would be accelerating at 1g (assuming a non rotational acceleration). Am I missing something?

The elevator would be length contracting as it accelerates and so the floor of the elevator will be traveling slightly faster than the ceiling of the elevator at any given instant, according to a non accelerating external observer. This would result in occupants of the elevator noticing slightly less acceleration at the top of the elevator than at the bottom. They would would also notice a slight redshift of light from a source on the floor of the elevator when looking at from the top of the elevator. So as DrGreg said, you would notice a gradiant inside the accelerating elevator, similar to a gravitational field to first order.

Also, a clock on the floor of the elevator would be ticking slightly slower than a clcok on the ceiling of the accelerating elevator for the same reason.

 

[ryan albery]

Is that correct? I understand that an external, non accelerating observer would note the length contraction (with a 3 meter tall elevator going from 0 to the speed of light in 1 second resulting in a roughly similar gradient as the same elevator stationary on the surface of the earth), but would it be noticed by someone inside? And if it was noticed, wouldn't it be a blue shift?

 

[bcrowell]

You could have a gravitational field without a gradient. For example, if you measure the field within a valley, it's possible for the gradient to be positive, negative, or zero.

You could have a varying acceleration that would simulate a gradient.

In general, GR has an interplay of local and global properties. Local means expanding things in a Taylor series up to a certain order. You're talking about trying to tell what order you can expand to. Obviously there's a limit to how big the radius of convergence can be. E.g., there's no way you can describe a cosmological spacetime as a perturbation of a flat spacetime. For one thing, there are topological properties you'd never get that way.

 

[ryan albery]

Here on earth, the lowest gravitational gradient I've measured was 1.2uGal/cm (1 Gal = 1cm/s^2)... with 4.4 atop a 4km tall mountain. It's somewhat a matter of debate cause the densities aren't exactly known, but somewhere deep in the Earth the gradient is zero, with gravity decreasing to zero at the center of the earth. For an infinite flat plane, there is no gradient due to gravity (which is a great approximation for monitoring ground water with gravity). Another cool integral to do is that gravity and the gradient are both zero anywhere inside a hollow sphere.

I can see how a changing acceleration could be identical to a particular gravitational field, if the conditions were set up just right. I'm definitely not questioning the equivalence principle (I actually participated in a cool experiment dropping heavy water in a free fall gravimeter that verified it), just suggesting that so long as you have a distance for measuring the gradient, such as inside an elevator, you can tell the difference between being stationary in a gravitational field, and in an accelerating reference frame.

 

[yuiop]

Is that correct? I understand that an external, non accelerating observer would note the length contraction (with a 3 meter tall elevator going from 0 to the speed of light in 1 second resulting in a roughly similar gradient as the same elevator stationary on the surface of the earth), but would it be noticed by someone inside? And if it was noticed, wouldn't it be a blue shift?

0 to (nearly) the speed of light in one second is extreme and much greater than the acceleration due to gravity on the surface of the Earth. You have not specified who measures one second, but I assume you mean the proper time as measured by someone inside the elevator, According to Baez in this link http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html it would take 8 years of proper time to reach 0.9999998c with constant acceleration of 1G (as measured by an accelerometer inside the elevator) and that would be even longer according to a non accelerating observer (1,840 years). If the elevator accelerated forever at a constant 1G, it still would not achieve exactly the speed of light.

The length contraction would not be noticed inside the accelerating elevator, but gradient effects would be noticed. It would be blue shift if the light source was on the ceiling of the elevator and the observer was on the floor and red shift if the source and observer switch positions as mentioned in my last post. Same in a gravitational field. Red shift for a signal going upwards and blue shift for a signal going downwards.

If I remember correctly, the acceleration measured inside the accelerating elevator drops off proportional to 1/d as you climb up inside the elevator while inside a stationary elevator on the surface of a massive body, the acceleration drops off according to 1/d^2. With sensitive enough instruments you could probably detect that difference.

 

[Ich]

just suggesting that so long as you have a distance for measuring the gradient, such as inside an elevator, you can tell the difference between being stationary in a gravitational field, and in an accelerating reference frame.

That's ok. It does not invalidate the equivalence principle, it's rather nitpicking the popular description. As Nabeshin said, one would have to include a disclaimer when stating the EP in plain English, but that doesn't make it more readable.