Understanding the Equivalence Principle and Einstein's Curved Spacetime

[PAllen]

But your explanation results in a required change of the laws of physics, in order to remain consistent with a synchronized experiment on the opposite side of the Earth, since the Earth cannot be moving in both directions simultaneously.

If the explanation does not work for the synchronized experiment, then it can't be valid for either individual half, else you'd be changing the laws of physics based on what? A change in half of the experiment? That doesn't make sense either.

If I have a twin on the opposite side of the planet simultaneously do the same thing as me, drop the ball, the Earth cannot be moving in both directions simultaneously.

We can enforce simultaneity by using equally long electrical cords for signaling, and assuming both of us have flawless reaction time (or by having a computerized arm drop the ball, if you like). Experiments involving distant clocks requiring simultaneity have already been done for determining the properties of Neutrinos (speed is the property in question,) for example, so I know the respected experimentalists in the scientific community already use the same concepts I'm talking about, otherwise the experiment they did wouldn't even make sense.

It is you who is creating a contradiction by perpetually ignoring the information you've been given that the equivalence principle is local. This in no way means it is impossible to set up consistent spacetime coordinates throughout the solar system and beyond. It just means that that the equivalence principle can only be applied in small regions of space and short times such that tidal gravity can be ignored.

The global generalization of the equivalence principle is that free fall is defined by geodesics of the overall geometry - as shown by AT's links, which you show no signs of having attempted to understand. So balls on opposite sides of the Earth are each following inertial, geodesic, maximally straight paths in spacetime. Global geometry has the feature that these geodesics converge. The surface of the Earth and you standing on it are following non-geodesic paths; proper acceleration is a measure of the deviation from geodesic paths. These non-geodesic paths (that exist as paths of objects only by virtue of EM forces of atoms and molecules) cause your foot to accelerate (there is no relativity here - acceleration is locally measurable and absolute = invariant).

 

[Wade888]

The surface is accelerating in spacetime. It is most emphatically not accelerating in space. With relativity, you need to start thinking in terms of the former, and forget about the latter.

All of the space dimensions are still a part of space-time, so either way it's a contradiction, as it would still be moving in opposite directions simultaneously.

The only thing significant about space-time is that it ties time together with the other dimensions, so that you can't treat time as being non-influenced by the events in the space dimensions. It does not suddenly change the logic of up or down, left or right, front or back, well, at least not except possibly in the presence of an event horizon of a black hole, but we weren't talking about black holes, so that doesn't matter.

Anyway, classical space plus time, or relativity "space-time" either way, left is left, and right is right, and the north pole is the north pole, and the prime meridian is the prime meridian. Someone could invent another coordinate system for the globe, but it would be a foolish waste of time for the most part, since the existing systems already work well enough, which the GPS system still uses. If one experimenter is at the north pole, and the other at the south, then they are definitely where we think they are, and they happen to be facing in opposite directions, regardless of what coordinate system you try to invent to get around that fact.


I have heard Relativity explanations, read books on the subject, as well as QM and string theory, and watched countless documentaries on it, and have a College text with a hefty chapter or ten on the subject, and I have never once heard of anyone at all giving the sort of explanation that was here given.



Also, the notion that the Earth moves, and so pushes me up, as the Ball falls, by an amount of 5 feet actually contradicts the Equivalence principle, because if I were to actually be accelerated upward for a distance of 5 feet, the amount the Earth would need to move, and consequently push me, I would experience a "jerk" and upward acceleration the moment I dropped the object, not to mention the sudden stop at exactly 5 feet, which I do not experience, but the ball does. More-over, if the object were dropped in a sand box, we could observe the energy dispersed and exploding the sand away, meanwhile I have experienced absolutely no stress or impact on my body in any way, which I should have if I were actually accelerated by the amount required to explain the motion of the ball, and then suddenly stopped and accelerated to halt, simulating the classical "instantaneous acceleration," which is normally attributed to the ball hitting the ground and bringing it to a halt.

I don't understand why this is hard for you guys to see this. The "Moving Target" explanation cannot explain both experiments at the same time, regardless of what you think of the first.

 

[PeterDonis]

The only way that "target moves to meet the object" scenario could happen is if the ground literally passed through my body...

Motion is relative. In a local inertial frame, the falling object is at rest and the ground is accelerating upward. If you want to apply the EP, that is the frame you use. That frame doesn't extend far enough to cover the other side of the Earth, so it can't be used to compare what happens there with what happens locally.

In a global frame in which the Earth is at rest, the falling object is accelerated downward and the ground is at rest, as is the ground on the other side of the Earth, where down is the opposite direction. But this frame isn't local so it can't be used to apply the EP.

There is no inconsistency in any of this.

 

[PAllen]

All of the space dimensions are still a part of space-time, so either way it's a contradiction, as it would still be moving in opposite directions simultaneously.

No. On one side of the earth, the Earth surface is accelerating towards star S1 relative to a local free fall frame (the type that follows the laws of SR). On the other side of the earth, the surface is accelerating toward star S2 relative to a local free fall frame. There is no contradiction at all.

The only thing significant about space-time is that it ties time together with the other dimensions, so that you can't treat time as being non-influenced by the events in the space dimensions. It does not suddenly change the logic of up or down, left or right, front or back, well, at least not except possibly in the presence of an event horizon of a black hole, but we weren't talking about black holes, so that doesn't matter.

No, it changes the picture universally. Directions are relative to something. A fundamental feature of curved spacetime is that the straightest possible paths (geodesics) can converge or diverge. This means, in fact, that 'up' can be two different directions in different places in spacetime. It also means that the static ground on opposite sides of the Earth have proper acceleration in opposite direction.

I have heard Relativity explanations, read books on the subject, as well as QM and string theory, and watched countless documentaries on it, and have a College text with a hefty chapter or ten on the subject, and I have never once heard of anyone at all giving the sort of explanation that was here given.

I'm sorry that you failed to understand this material. The explanations given here are consistent with such material.

Also, the notion that the Earth moves, and so pushes me up, as the Ball falls, by an amount of 5 feet actually contradicts the Equivalence principle, because if I were to actually be accelerated upward for a distance of 5 feet, the amount the Earth would need to move, and consequently push me, I would experience a "jerk" and upward acceleration the moment I dropped the object, not to mention the sudden stop at exactly 5 feet, which I do not experience, but the ball does.

I'm sorry you read all that material and don't understand anything about the equivalence principle, which has been presented clearly enough for a hundred years.

When you stand on the Earth you are being continuously accelerated relative (get that word - it seems to be missing from your understanding) a local free fall frame. You feel this acceleration on your feet. There is no jerk expected - that is idiocy. The ball is accelerating also by virtue of your hand holding - until you drop it. When you drop it, it ceases accelerating in its local frame, while you and the Earth continue accelerating.

More-over, if the object were dropped in a sand box, we could observe the energy dispersed and exploding the sand away, meanwhile I have experienced absolutely no stress or impact on my body in any way, which I should have if I were actually accelerated by the amount required to explain the motion of the ball, and then suddenly stopped and accelerated to halt, simulating the classical "instantaneous acceleration," which is normally attributed to the ball hitting the ground and bringing it to a halt.

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I can hardly count the number of elementary errors here. I ask you again to consider that you are in a uniformly accelerating rocket with a sandbox, and drop a ball. Everything would be indistinguishable from your experience standing on the ground and dropping the ball.

I don't understand why this is hard for you guys to see this. The "Moving Target" explanation cannot explain both experiments at the same time, regardless of what you think of the first.

And I don't understand how you could have possibly read what you claim and understand almost nothing of it.

 

[PeterDonis]

it would still be moving in opposite directions simultaneously.

As I said in my previous post, motion is relative; it depends on the frame you are using.

I have heard Relativity explanations, read books on the subject, as well as QM and string theory, and watched countless documentaries on it, and have a College text with a hefty chapter or ten on the subject, and I have never once heard of anyone at all giving the sort of explanation that was here given.

This makes me confused about the position you are taking. Are you claiming that relativity is wrong, or just that we are explaining it wrong? If the latter, what is your explanation of the phenomena under discussion, based on the understanding of relativity that you have from all those explanations, books, documentaries, and a college text?

Also, the notion that the Earth moves, and so pushes me up, as the Ball falls, by an amount of 5 feet actually contradicts the Equivalence principle, because if I were to actually be accelerated upward for a distance of 5 feet, the amount the Earth would need to move, and consequently push me, I would experience a "jerk" and upward acceleration the moment I dropped the object, not to mention the sudden stop at exactly 5 feet, which I do not experience, but the ball does.

Motion is relative; the correct way to state all this is that the Earth (including you) and the ball are in relative motion. Viewed from one frame, the ball is moving and the Earth (including you) is at rest; viewed from another frame, the ball is at rest and the Earth is moving, along with you.

Now, what about when the ball stops? It is true that the ball feels a sudden jerk, and you don't. From the viewpoint of the frame in which the ball is at rest while it is freely falling, the same is true: when the ball hits the ground, it is suddenly no longer at rest--it is now being pushed upward by the ground. So the ball's state of motion changes suddenly in both frames, and the state of motion of the ground is constant in both frames (at rest in one, accelerating upward with a constant acceleration in the other). Similar remarks apply to your sandbox scenario.

The "Moving Target" explanation cannot explain both experiments at the same time, regardless of what you think of the first.

What does "at the same time" mean? The "moving target" explanation works perfectly well in any local inertial frame, but you need to pick different local inertial frames for different locations: the local inertial frames in your vicinity are different from the ones on the other side of the Earth. That's why they're called "local". Yes, you can't find a single frame in which the "moving target" explanation works everywhere on Earth, but the EP doesn't require that; again, that's why it's called "local".

 

[Dale]

So you really believe the Earth expands by 9.8m/s^2 in eveyr direction simultaneously?

No, I already told you that expansion was different from acceleration. The surface of the Earth has an upwards proper acceleration of 9.8 m/s² and a 0 expansion tensor.

If X is the unit generator of the congruence for the surface of the Earth then the proper acceleration is given by ${X^{\mu}}_{;\nu}X^{\nu}$

and the expansion is given by ${h^{\lambda}}_{\mu}{h^{\eta}}_{\nu}X_{(\lambda;\eta)}$ where $h_{\mu\nu}=g_{\mu\nu}+X_{\mu}X_{\nu}$

These are clearly different quantities, so acceleration does not imply expansion.

You really believe if I drop a ball from shoulder height, that the Earth accelerates up to the ball, without passing through my body, and the ball somehow gets "hit" by the Earth,e ven though I didn't move, and the Earth didn't move through me?

The Earth doesn't move through you because you are accelerating upwards along with the surface of the earth. If you have an iPhone you can get an app to see the reading on the built in accelerometer. It will show that you are accelerating just as much as the ground is.

I've never once seen the Earth expand upward to a ball dropped from a person's hand, and neither have you

Again, it accelerates, it doesn't expand. You cannot use flat reasoning to make correct conclusions in a curved spacetime.

 

[Dale]

In case Wade888 is still watching, or for other people's benefit, I thought that it might be useful to show how curvature relates to acceleration and expansion. I apologize in advance for the length of the post.

Consider a flat sheet with time in the vertical direction and position in the horizontal direction. An object which is unaccelerated will form a straight line on the sheet, while an object which is accelerated will have a curved line. So, an accelerometer measures how much a worldline curves. On a flat sheet of paper if two objects accelerate away from each other then they curve away from each other and the distance between them expands.

Now, on curved surfaces the idea of a straight line is generalized to what is known as a geodesic. A straight line is a geodesic in a flat space. In both curved and flat spaces the shortest distance between two points is a geodesic. So, just like straight lines, geodesics have no proper acceleration.

So, let's consider geometry on a sphere. We will start at the equator with two nearby longitude lines. Longitude lines are great circles, which are geodesics. So if time is north-south and space is east-west then an object going along a longitude line will have 0 proper acceleration (longitude lines are great circles, great circles are geodesics, geodesics are like straight lines so they have no proper acceleration). So, even though the two nearby longitude lines do not accelerate, the distance between them contracts, and eventually they will run into each other.

If they wish to keep the distance between them constant, then they must deviate from the path of the longitude line. This curving away from the geodesic means that such a path will have proper acceleration, locally they will be turning away from the other line. So, even though they are accelerating away from each other, the distance between them is not expanding. The acceleration is in fact required in order to counteract the convergence that would otherwise occur.

Hopefully that makes sense and explains how a curved space can require two objects to accelerate away from each other to maintain their distance from each other. Expansion and acceleration are different concepts, and flat-space intuition won't always work, so it is important to think about simple curved spaces as well.