# Equivalence principle - are all accelerations actually equivalent to gravity?

I have read the basis of the the equivalence principle, that inertial mass = gravitational mass, and that it leads to the conclusion that any acceleration = equivalent gravitational field intensity. (I believe 'gravitational field intensity' is the same as 'gravitational acceleration'.)

Einstein used an elevator example for this - one in a gravitational field and another accelerating in deep space.

Modifying that slightly, let us consider the following scenarios:
- A person in a rocket accelerating using its engines at something like 20g will probably get crushed against the floor and die soon, because the human body cannot take that kind of acceleration for any extended period
- A person in a rocket that is being accelerated at 20g by a gravitational force would be happily alive, and wouldn't feel the acceleration at all, because he is in 'free fall'

My question is, provided the above thinking is correct, how strictly true is the equivalence principle, and what are its limits? (And, if the above thinking is not correct, what am I getting wrong?)

Nugatory
Mentor
Modifying that slightly, let us consider the following scenarios:
- A person in a rocket accelerating using its engines at something like 20g will probably get crushed against the floor and die soon, because the human body cannot take that kind of acceleration for any extended period
- A person in a rocket that is being accelerated at 20g by a gravitational force would be happily alive, and wouldn't feel the acceleration at all, because he is in 'free fall'

Add a third scenario to that list: the rocket has landed on the surface of a planet. It's a super-massive planet with a surface gravity 20 times that of earth.

As in your first scenario, the person is crushed against the floor and dies. The equivalence principle asserts that this case is equivalent to the first case and that both are different from the second scenario, the free-fall one.

Your second scenario is analogous to what would happen in Einstein's elevator example if we were to cut the cable and allow the elevator to fall: the person inside the elevator would be in free fall and wouldn't feel the acceleration at all (until the free fall ends with a collision with the bottom of the elevator).

Add a third scenario to that list: the rocket has landed on the surface of a planet. It's a super-massive planet with a surface gravity 20 times that of earth.

As in your first scenario, the person is crushed against the floor and dies. The equivalence principle asserts that this case is equivalent to the first case and that both are different from the second scenario, the free-fall one.

Your second scenario is analogous to what would happen in Einstein's elevator example if we were to cut the cable and allow the elevator to fall: the person inside the elevator would be in free fall and wouldn't feel the acceleration at all (until the free fall ends with a collision with the bottom of the elevator).

I am talking about a sustained and gradually building acceleration, not an impulse (shock) acceleration, so the scenario you mention does not hold.

Look at it this way - the rocket does not go from 1g to 20 g in a split second, but gradually climbs in acceleration by 1g/hour, reaching 20g in 20 hours. I believe the person would still die by crushing some time within those 20 hours.

If the rocket's acceleration was climbing in the same way though gravitation, going from 1g to 20g over 20 hours, I believe the person would not feel a thing, and would be alive and well after 20 hours.

Are you saying this is incorrect?

Nugatory
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I am talking about a sustained and gradually building acceleration, not an impulse (shock) acceleration, so the scenario you mention does not hold.

There's no impulse or shock involved in the third scenario, the one that is equivalent to the first in the way that the elevator suspended at rest in a gravitational field is equivalent to the elevator accelerating in empty space. The ship is just sitting quietly on the surface of the planet.

Nugatory
Mentor
If the rocket's acceleration was climbing in the same way though gravitation, going from 1g to 20g over 20 hours, I believe the person would not feel a thing, and would be alive and well after 20 hours.

Are you saying this is incorrect?

Yes. If we're accelerating at 20g at a particular moment, then at that moment we feel 20 times heavier than on earth. It doesn't matter what we were doing a moment before or will be doing a moment later.

[EDIT - I wrote the above, but it's completely wrong - I misread the question. My feelings won't be hurt if a moderator just deletes this post for me].

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PeterDonis
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2020 Award
- A person in a rocket that is being accelerated at 20g by a gravitational force would be happily alive, and wouldn't feel the acceleration at all, because he is in 'free fall'

This person is not being accelerated; he feels no weight. He is in free fall. The equivalence principle does not say that free fall is the same as feeling weight.

"Acceleration" in the equivalence principle means proper acceleration; the "acceleration" of the person in the quote above is coordinate acceleration only.

PeterDonis
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2020 Award
If the rocket's acceleration was climbing in the same way though gravitation, going from 1g to 20g over 20 hours, I believe the person would not feel a thing, and would be alive and well after 20 hours.

Since he would be in free fall the whole time, yes, he would be alive and well. He would also never have been accelerated at all, according to the EP. See my previous post.

PeterDonis
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2020 Award
If we're accelerating at 20g at a particular moment, then at that moment we feel 20 times heavier than on earth.

Only if it's proper acceleration. If you are being "accelerated at 20g" by gravity, you are actually not being accelerated at all; you are in free fall. See my previous posts.

haushofer
In Newtonian gravity only the linear (time dependent) accelerations are "equivalent" to gravity. In GR all accelerations are locally equivalent to gravity.

Nugatory
Mentor
Only if it's proper acceleration. If you are being "accelerated at 20g" by gravity, you are actually not being accelerated at all; you are in free fall. See my previous posts.

I completely agree. I misread OP's question. Will fix my post if the edit window is still open.

Dale
Mentor
2020 Award
I have read the basis of the the equivalence principle, that inertial mass = gravitational mass, and that it leads to the conclusion that any acceleration = equivalent gravitational field intensity. (I believe 'gravitational field intensity' is the same as 'gravitational acceleration'.)

Einstein used an elevator example for this - one in a gravitational field and another accelerating in deep space.

Modifying that slightly, let us consider the following scenarios:
- A person in a rocket accelerating using its engines at something like 20g will probably get crushed against the floor and die soon, because the human body cannot take that kind of acceleration for any extended period
- A person in a rocket that is being accelerated at 20g by a gravitational force would be happily alive, and wouldn't feel the acceleration at all, because he is in 'free fall'

My question is, provided the above thinking is correct, how strictly true is the equivalence principle, and what are its limits? (And, if the above thinking is not correct, what am I getting wrong?)
You are completely missing the point of the equivalence principle. Accelerating in empty space at 20 g is equivalent to being at rest in a 20 g flat gravitational field (both accelerometers read 20 g). Being in free fall in a flat gravitational field is equivalent to travelling inertially in the absence of gravity (both accelerometers read 0 g).

There's no impulse or shock involved in the third scenario, the one that is equivalent to the first in the way that the elevator suspended at rest in a gravitational field is equivalent to the elevator accelerating in empty space. The ship is just sitting quietly on the surface of the planet.

Yes, now I see what you meant.

This person is not being accelerated; he feels no weight. He is in free fall. The equivalence principle does not say that free fall is the same as feeling weight

"Acceleration" in the equivalence principle means proper acceleration; the "acceleration" of the person in the quote above is coordinate acceleration only.

I see the difference clearly now. Thanks.

You are completely missing the point of the equivalence principle. Accelerating in empty space at 20 g is equivalent to being at rest in a 20 g flat gravitational field (both accelerometers read 20 g). Being in free fall in a flat gravitational field is equivalent to travelling inertially in the absence of gravity (both accelerometers read 0 g).

You've hit the nail on the head. I was missing the point completely. Very stupid of me. Don't know why I didn't realize it in the first place.

stevendaryl
Staff Emeritus
Modifying that slightly, let us consider the following scenarios:
- A person in a rocket accelerating using its engines at something like 20g will probably get crushed against the floor and die soon, because the human body cannot take that kind of acceleration for any extended period
- A person in a rocket that is being accelerated at 20g by a gravitational force would be happily alive, and wouldn't feel the acceleration at all, because he is in 'free fall'

The equivalence principle relates (1) accelerating at 20g through gravity-free space, and (2) being at rest in a gravitational field of strength 20g.

Alternatively, it relates (3) traveling at constant velocity in gravity-free space, and (4) falling in a gravitational field.

Of course, comparing (1) and (4) or comparing (2) and (3) you'll see a huge difference.

Does not anybody cares about the locality which is inherent in equivalence principle

In free fall all your molecules feel the same acceleration, that's why you don't perceive it and experience weightlessness. When you are standing still your feet are crushed by all the mass above them. Imagine 'standing' on your palms for a change. Your palms now feel the pressure of your feet.

In another scenario when you fall towards a black hole you experience spaghettification. So free fall on a black hole is radically different that free fall on earth. It's just that gravity is very weak so you experience almost exactly the same weightlessness as you would in deep space.