Spacetime curvature due to acceleration causing gravity?

[Dale]

Is it true that curved spacetime (such as caused by the Earth) is the cause that an object standing on the Earth feels a force otherwise known as proper acceleration?

No. You feel proper acceleration whenever you move non inertially, regardless of the curvature.

 

[Buckethead]

No, he didn't say that. Spacetime curvature is not a force. The apparent force that you feel when you stand on the floor is not due to spacetime curvature.

Whether you're in curved spacetime or in flat spacetime, you don't feel proper acceleration unless something is pushing up on you. On the Earth, you only feel proper acceleration because the floor is pushing up on you. The same is true in outer space: You feel proper acceleration because the floor of the rocket is pushing up on you. In both cases, if the floor gives way, then you will no longer feel proper acceleration. So absolutely no: spacetime curvature is not what causes you to feel proper acceleration.

The more accurate way to think about it is this: There is a certain type of motion called "inertial motion", which is the motion of an object that is not acted on by any forces. You only feel proper acceleration when something forces you to travel in a noninertial way.

The difference between curved spacetime and flat spacetime is that in flat spacetime, inertial motion has a very nice property: If you start off putting two objects a small distance apart and initially traveling in the same direction at the same speed, then under inertial motion, they will continue traveling in the same direction, and will remain the same distance apart. In contrast, curved spacetime has an effect called "geodesic deviation". If you start two objects off traveling in the same direction at the same speed a small distance apart, they will not continue to travel in the same direction and will not continue to stay the same distance apart. The most stark example is: You drop two objects from rest far above the surface of the Earth. Initially, they have the same velocity, namely 0. But as time goes on, they will converge toward the center of the Earth. The distance between them will get smaller, and their velocities will point in different directions.

That's a key indicator of spacetime curvature, is that geodesics (the paths of objects traveling inertially) do not remain parallel when they start out parallel. This fact has nothing directly to do with "feeling proper acceleration". You feel proper acceleration when you are forced to move noninertially.

Thank you for the clarification especially about proper acceleration being a deviation from an inertial path which is exactly what happens when you are standing on the surface of the Earth, you are being forced to deviate from the inertial path through spacetime and the result is a proper acceleration.

I suppose that is what I was trying to say all along, I just didn't say it as well as you. The point I was trying to make was that without spacetime curvature at the Earth, you would not feel a proper acceleration while standing on the Earth because you would not be forced to deviate from an inertial path through spacetime. So in a roundabout way you could say curved spacetime is what "causes" you to feel proper acceleration while standing on the Earth even though as you said, it's the being forced to deviate from the inertial state that is causing the proper acceleration.

That being said, doesn't it follow that if there is no curved spacetime at the surface of the Earth, one would not feel proper acceleration while standing on the surface of the Earth? In other words, there are only two ways to feel proper acceleration, either in an accelerating elevator, or standing on a non accelerating object while in a curved spacetime.

 

[Buckethead]

No. You feel proper acceleration whenever you move non inertially, regardless of the curvature.

This is very clear to me now. See my post just before this.

 

[PeterDonis]

without spacetime curvature at the Earth, you would not feel a proper acceleration while standing on the Earth because you would not be forced to deviate from an inertial path through spacetime.

You're making an implicit comparison here which isn't really meaningful. When you say "without spacetime curvature at the Earth", what are you comparing the actual situation with? And how are you going to pick out "the same place" in whatever you are comparing the actual situation with? There is no well-defined way to answer such questions. So there is no well-defined way to make the comparison you are trying to make here.

OTOH, if I say "you feel proper acceleration because the Earth is pushing on you", or you say "you feel proper acceleration because the rocket is pushing on you", both of those are well-defined situations and can be compared by simply comparing the proper accelerations, without having to make any ill-defined comparisons between different situations.

doesn't it follow that if there is no curved spacetime at the surface of the Earth, one would not feel proper acceleration while standing on the surface of the Earth?

No, because, again, the comparison you are trying to make here is not well-defined.

In other words, there are only two ways to feel proper acceleration, either in an accelerating elevator, or standing on a non accelerating object while in a curved spacetime.

No. You can feel proper acceleration in a rotating cylinder whose center of mass is moving inertially.

If you are looking for a general rule about when you feel proper acceleration, a better way to think about it is, what kinds of things can push on you? It has nothing to do with whether the spacetime is flat or curved. It has to do with what kinds of things can exert a force on you that you feel.

 

[stevendaryl]

That being said, doesn't it follow that if there is no curved spacetime at the surface of the Earth, one would not feel proper acceleration while standing on the surface of the Earth?

In other words, there are only two ways to feel proper acceleration, either in an accelerating elevator, or standing on a non accelerating object while in a curved spacetime.

Those are NOT two different ways. Those are the SAME way. In both cases, you feel the floor pushing up. The qualifier "non accelerating" doesn't by itself mean anything. There are two different common meanings of the word "accelerating". One is PROPER acceleration and the second is COORDINATE acceleration. In both cases (a floor in a rocket, or a floor on the surface of the Earth), there is nonzero proper acceleration. In both cases, whether there is coordinate acceleration or not depends on what coordinate system you are using. There is a noninertial coordinate system according to which the floor is at rest, and there is an inertial coordinate system according to which the floor is accelerating.

What is interesting about spacetime curvature is that you can choose a coordinate system according to which

1. Everywhere on the surface of the Earth is at "rest" relative to this coordinate system,
2. But all those points have nonzero proper acceleration in DIFFERENT directions. If you're standing up at the North pole, your proper acceleration is pointing straight up. If you're standing at the South pole, your proper acceleration is pointing in the opposite direction. This is very different from being on board a rocket, where everything that is at "rest" relative to the rocket has proper acceleration pointing in the same direction (the direction the rocket is going).

The fact that the direction of the proper acceleration is in different directions has a major consequence.

If you try to maintain a rocket in constant proper acceleration, you have to expend energy. To make the rocket accelerate one way, you have to throw mass (burnt fuel) in the opposite direction. Eventually, you're going to run out of fuel. So this kind of proper acceleration cannot be maintained forever.

In contrast, consider drilling a hole through the center of the Earth from the North Pole to the South Pole. Put one rocket on the North Pole. Put another rocket on the South Pole. Connect them by a long rod. The rocket at the north pole is accelerating (proper acceleration) in one direction, and the rocket at the south pole is accelerating in the opposite direction. But they are never getting farther apart! So the force of the rod connecting them is enough to keep them accelerating forever.

 

[Dale]

The point I was trying to make was that without spacetime curvature at the Earth, you would not feel a proper acceleration while standing on the Earth because you would not be forced to deviate from an inertial path through spacetime.

Again, no. It is the unbalanced real contact force with the ground that causes the proper acceleration.

Any scenario in which there is an unbalanced real force will lead to the sensation of proper acceleration (with or without curvature). Any scenario in which there is no unbalanced real force will lead to the absence of proper acceleration (with or without curvature). The presence or absence of curvature is irrelevant.

Remember that spacetime curvature is tidal gravity, the sensation of proper acceleration is not. So what does spacetime curvature do in this scenario? It allows you to have proper acceleration in the opposite direction of a person on the other side of the world without the distance between you increasing.

That being said, doesn't it follow that if there is no curved spacetime at the surface of the Earth, one would not feel proper acceleration while standing on the surface of the Earth?

This is getting annoying. No matter how many times you repeat it the answer will still be no.

 

[1977ub]

"Early versions of Einstein's theory did not include the curvature of the geometry of ordinary space around the sun."
https://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/general_relativity_massive/

Is this the same as "tidal gravity" ?

 

[Buckethead]

You're making an implicit comparison here which isn't really meaningful. When you say "without spacetime curvature at the Earth", what are you comparing the actual situation with? And how are you going to pick out "the same place" in whatever you are comparing the actual situation with?

Am I not comparing it to empty space where there is no curvature? I visualize spacetime curvature as something that shapes the geodesics through spacetime and as a result, when you stand on the Earth, you and the Earth are being forced to deviate from an inertial path through spacetime. This causes the "push" between you and the Earth and the result is the sensation of proper acceleration. Is this not a good way to look at this?

 

[PeterDonis]

Am I not comparing it to empty space where there is no curvature?

How are you going to do the comparison? Empty space has no distinguishable places; how are you going to pick out the place where the "Earth" is and compare the proper accelerations at the two places?

Is this not a good way to look at this?

No. As has already been said a number of times in this thread. And it has been repeatedly explained why.

 

[PeterDonis]

The OP question has been answered. Thread closed.

 

[Dale]

Sorry @PeterDonis, I wanted to post one last explanation of what curvature actually does

I visualize spacetime curvature as something that shapes the geodesics through spacetime

This is true, but spacetime curvature is not just any shape. The specific shape that spacetime curvature gives to geodesics is that spatially separated geodesics diverge. Around Earth the shape of the curvature means that geodesics at your head and foot are pulling apart while geodesics on your left and right are squeezing together. Hopefully you recognize this as tidal gravity.

In a given region with a fixed spacetime curvature you can get rid of the proper acceleration simply by removing the floor. The proper acceleration disappears but the curvature doesn’t. However, the tidal force is present whether you are in free fall or not, it does not disappear under any condition as long as there is curvature.

 

[PeterDonis]

The specific shape that spacetime curvature gives to geodesics is that spatially separated geodesics diverge.

Or converge.