Understanding the Equivalence Principle and Einstein's Curved Spacetime

[Quarlep]

Hi
I am reading Stephan Hawking's Universe in the Nutshell and there I didnt understand this sentence
"This equivalence didn't work for a spherical Earth because people on opposite sides of the world would be getting farther away from each other.Einstein overcome this idea to make spacetime curved"
If you explain it more basic I will be glad
Thanks

 

[Nugatory]

He's saying that if we apply the equivalence principle to two people on opposite sides of the earth, they'd have to be accelerating in opposite directions to explain the appearance of gravity pulling them both towards the center of the earth. If they're accelerating in opposite directions, the distance between them should be changing, but it's not - it's constant and equal to the diameter of the earth.

 

[A.T.]

"This equivalence didn't work for a spherical Earth because people on opposite sides of the world would be getting farther away from each other.Einstein overcome this idea to make spacetime curved"

Locally, the gravity we observe on the surface doesn't require intrinsic curvature, just curve-linear coordinates (equivalent to an accelerating frame of reference) like shown here:

https://www.youtube.com/watch?v=DdC0QN6f3G4

But the above video shows just a small local patch of space-time. To make all those small patches fit together globally, space-time must be curved like shown here:

http://www.adamtoons.de/physics/gravitation.swf

Unlike the local cone-like patches, this global space-time surface cannot be rolled out flat, without distortion. This is called "intrinsic curvature".

 

[Wade888]

He's saying that if we apply the equivalence principle to two people on opposite sides of the earth, they'd have to be accelerating in opposite directions to explain the appearance of gravity pulling them both towards the center of the earth. If they're accelerating in opposite directions, the distance between them should be changing, but it's not - it's constant and equal to the diameter of the earth.

But electromagnetism from the molecules of the Earth are pushing each of them away from one another, thereby offsetting the gravity, so there is no contradiction in any regards.

Same principle as an automobile pushing against something like a rocket sled. If either is more powerful, they move in one direction or another, but if they are balanced they don't accelerate at all.

 

[Wade888]

Locally, the gravity we observe on the surface doesn't require intrinsic curvature, just curve-linear coordinates (equivalent to an accelerating frame of reference) like shown here:

https://www.youtube.com/watch?v=DdC0QN6f3G4

But the above video shows just a small local patch of space-time. To make all those small patches fit together globally, space-time must be curved like shown here:

http://www.adamtoons.de/physics/gravitation.swf

Unlike the local cone-like patches, this global space-time surface cannot be rolled out flat, without distortion. This is called "intrinsic curvature".

If no forces are acting during the fall, why does a dropped object smash through it's target at the bottom of a fall? No force should have equaled no kinetic energy, no momentum. Why doesn't it bounce off harmlessly, or just "stick" for example?

I don't know about you, but I've accidentally dropped a thing or two on a finger or toe, and always feel just as much, usually a lot more, force, pressure, whatever you want to call it, than simply lifting the object. I realize issues like "Instantaneous Acceleration" modify the amount of damage a given amount of momenta can cause, but the explanation of that video appears to contradict both that notion and common experience as well.

Can you explain this discrepancy?

And for example, in the rubber sheet model of space-time distortion due to mass, the objects don't actually fall toward one another due to the warping of the rubber sheet in the model, but rather the "real" gravity of the Earth. The only reason the model works is because "real" gravity is actually "faking" the modeled gravity, which makes the model invalidated.

 

[WannabeNewton]

A.T.'s construction in his video is not even remotely close to the "rubber sheet analogy". The latter is inherently flawed and non-descriptive of general relativity whereas the former is perfectly accurate. This is a question we get very often on this forum.

The short answer is that you are forgetting to include the time dimension; when one models gravity using space-time geometry and space-time curvature, there is curvature in the time dimension that causes an object initially at rest to start freely falling under the influence of gravity.

See this thread for more details: https://www.physicsforums.com/showthread.php?t=726837
and this post for a more mathematical version of what I just said above: https://www.physicsforums.com/showpost.php?p=4597436&postcount=19

 

[Nugatory]

The short answer is that you are forgetting to include the time dimension; when one models gravity using space-time geometry and space-time curvature, there is curvature in the time dimension that causes an object initially at rest to start freely falling under the influence of gravity.

Which is of course is the key to the second sentence in the original post.

 

[Nugatory]

But electromagnetism from the molecules of the Earth are pushing each of them away from one another, thereby offsetting the gravity, so there is no contradiction in any regards.

You are misunderstanding the equivalence principle. The EP basically says that there is no gravity to offset; the force you feel from the molecules of the Earth pushing against the soles of your shoes is accelerating you away from your natural inertial trajectory and it is the only force acting on you.

 

[Nugatory]

If no forces are acting during the fall, why does a dropped object smash through it's target at the bottom of a fall? No force should have equaled no kinetic energy, no momentum. Why doesn't it bounce off harmlessly, or just "stick" for example?

"No force" does not mean "no kinetic energy, no momentum". This isn't a relativity thing; if you look at the classical definition of momentum ($p=mv$) and kinetic energy ($E_k=\frac{1}{2}mv^2$) you'll see that they depend on having a non-zero velocity relative to the observer, not a non-zero force. The relativistic definitions are just a bit more complicated, but there's still no force involved in them.

The reason a dropped object smashes through its target is that it's on a collision course with its target and when they collide they hit hard enough to break the target... Imagine what happens to the windshield of a car if it drives into a weight hanging from an highway overpass, for example. There's no force between the car and the weight, no gravity involved anywhere, but if the car is moving fast enough its windshield will be smashed.

The General Relativity picture of a dropped object is that the object is at rest while the ground underneath it is accelerating upwards towards the object. If the ground is moving fast enough when they meet, something will smash.
We know it's the ground accelerating upwards while the object is just floating free in space because when we stand on the ground we can feel it pushing up against the soles of our shoes; and if we were falling alongside the object we'd be weightless until we and the ground collided.

 

[Wade888]

I can press my hand against a table, either from above or from below, and feel the same pressure as I would were I standing on my hands instead of my feet. The table doesn't magically change the direction of it's natural motion (or non-motion). The experience is the same either way, and I must be in the same reference frame either way. Therefore the explanation does not work, because it is not consistent with observation.

 

[PeterDonis]

If no forces are acting during the fall, why does a dropped object smash through it's target at the bottom of a fall? No force should have equaled no kinetic energy, no momentum.

No force is acting on the free-falling object; but a force *is* acting on the target. In the GR viewpoint, the free-falling object sits at rest and the target is accelerated upward until they smash together. The further away the target is at the start, the more time it has to accelerate, so the harder the smash will be.

 

[PeterDonis]

"No force" does not mean "no kinetic energy, no momentum".

This is true, but I don't think it's the answer to Wade888's question. There *is* a force present; but it acts on the *target*, not on the falling object. See my response to him, just posted.

 

[Wade888]

No force is acting on the free-falling object; but a force *is* acting on the target. In the GR viewpoint, the free-falling object sits at rest and the target is accelerated upward until they smash together. The further away the target is at the start, the more time it has to accelerate, so the harder the smash will be.

Okay, so synchronize two experimenters on exactly opposite points of the Earth. Each drops their ball on their target. Your explanation requires the Earth to expand in both directions simultaneously, in order to "accelerate" the target "upward." Of course, this scenario (not the theory) happens all over the world all the time in asynchronous fashion, but it helps to do the synchronized thought experiment to make the point.

So now I am to understand that the theory of Relativity says that if we lift objects, the Earth shrinks, but if we drop them, the Earth expands, but I drop the object from shoulder height. The Earth cannot be expanding, because I did not sink up to my arm pits in the soil or rock. The explanation is still inconsistent with easily, readily observed reality.

 

[PeterDonis]

Okay, so synchronize two experimenters on exactly opposite points of the Earth.

In which case you're no longer talking about the equivalence principle, since that only applies locally, where "locally" means "over a small enough region of space and time that tidal gravity is negligible". The exact size of such a region depends on how accurate your measurements are, but it's safe to say that experimenters on opposite sides of the Earth are *not* in each other's local regions of spacetime.

So now I am to understand that the theory of Relativity says that if we lift objects, the Earth shrinks, but if we drop them, the Earth expands, but I drop the object from shoulder height.

No, relativity says no such thing. See above.

 

[PeterDonis]

You are misunderstanding the equivalence principle.

No, he's trying to apply it in a situation where it doesn't apply. See my previous post. (The same comment applies to the OP; what Hawking was really saying was that you can't apply the EP over a region large enough to include experiments on both sides of the Earth simultaneously, because tidal gravity is not negligible. Hawking just chose a more colorful way of saying it, as he usually does.)

 

[Nugatory]

No, he's trying to apply it in a situation where it doesn't apply. See my previous post. (The same comment applies to the OP; what Hawking was really saying was that you can't apply the EP over a region large enough to include experiments on both sides of the Earth simultaneously, because tidal gravity is not negligible. Hawking just chose a more colorful way of saying it, as he usually does.)

You sure? When he talks about the electromagnetic forces from the surface of the Earth "offsetting" gravity, I'm hearing a deeper misunderstanding than the one that Hawking was addressing - that description is inconsistent with the equivalence principle even locally, and it's more akin to the Newtonian description that the GR one.

Of course there is always the possibility that I am misunderstanding OP's misunderstanding

 

[PeterDonis]

When he talks about the electromagnetic forces from the surface of the Earth "offsetting" gravity

Just to clarify, by "the OP" I meant Quarlep. The electromagnetic force thing was Wade888. Of course, I may be misunderstanding the respective misunderstandings as well.

 

[A.T.]

Your explanation requires the Earth to expand in both directions simultaneously, in order to "accelerate" the target "upward."

No, as Hawking explains, this is exactly the issue that space-time curvature solves. In curved-space time you can have two objects experiencing proper acceleration away from each other, without increasing their distance.

proper acceleration = what an accelerometer measures = deviation from free fall = deviation from straight (geodesic) path in space time

 

[PeterDonis]

as Hawking explains, this is exactly the issue that space-time curvature solves. In curved-space time you can have two objects experiencing proper acceleration away from each other, without increasing their distance.

Yes, but it should be noted that, as I said in my previous response to Wade888, if you're talking about spacetime curvature, you're no longer talking about the EP, because spacetime curvature is tidal gravity, and the EP only applies in a small enough patch of spacetime that tidal gravity is negligible.

 

[Dale]

So now I am to understand that the theory of Relativity says that if we lift objects, the Earth shrinks, but if we drop them, the Earth expands, but I drop the object from shoulder height. The Earth cannot be expanding, because I did not sink up to my arm pits in the soil or rock. The explanation is still inconsistent with easily, readily observed reality.

Expansion is different from acceleration. The surface of the Earth is accelerating upwards at 1 g proper acceleration. The Earth is not expanding. Those two statements are compatible because the spacetime around the Earth is curved.

 

[A.T.]

if you're talking about spacetime curvature, you're no longer talking about the EP,

Global curvature allows the EP to work locally everywhere.

 

[Dale]

Well said!

 

[Nugatory]

Just to clarify, by "the OP" I meant Quarlep. The electromagnetic force thing was Wade888. Of course, I may be misunderstanding the respective misunderstandings as well.

OK, all is now clear to me Or something

 

[TrickyDicky]

Global curvature allows the EP to work locally everywhere.

Global curvature is a somewhat ill-defined concept for GR's Lorentzian 4-manifolds. The EFE is about the local curvature, and different for each solution, so I fail to see what global curvature you allude to here.

 

[WannabeNewton]

Global curvature is a somewhat ill-defined concept for GR's Lorentzian 4-manifolds. The EFE is about the local curvature, and different for each solution, so I fail to see what global curvature you allude to here.

Indeed when the term "global" is interpreted strictly from a mathematical standpoint, A.T.'s statement is inaccurate because Riemann curvature is obviously local (it's a tensor field). However I think A.T. is using the term "global" in a loose non-mathematical manner to mean that there exist length scales and time scales within the physical system for which space-time curvature is negligible up to second order in a perturbation series expansion and that these length and time scales are small compared to the length and time scales of the physical system itself.

 

[TrickyDicky]

I think A.T. is using the term "global" in a loose non-mathematical manner to mean that there exist length scales and time scales within the physical system for which space-time curvature is negligible up to second order in a perturbation series expansion and that these length and time scales are small compared to the length and time scales of the physical system itself.

Could be, thus my asking what global curvature he was exactly alluding to.

 

[Wade888]

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

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?

Really? That's the consequence of what his explanation to me was.


God help you if you think I'm the one who's out of line.


I've never once seen the Earth expand upward to a ball dropped from a person's hand, and neither have you, because if it had the person would have been encased in 5 feet of dirt.

You've lost your senses if you really believe that.

 

[Wade888]

In which case you're no longer talking about the equivalence principle, since that only applies locally, where "locally" means "over a small enough region of space and time that tidal gravity is negligible". The exact size of such a region depends on how accurate your measurements are, but it's safe to say that experimenters on opposite sides of the Earth are *not* in each other's local regions of spacetime.



No, relativity says no such thing. See above.

No, you said this:

No force is acting on the free-falling object; but a force *is* acting on the target. In the GR viewpoint, the free-falling object sits at rest and the target is accelerated upward until they smash together. The further away the target is at the start, the more time it has to accelerate, so the harder the smash will be.

That is a logical inconsistency, because in this case the "target" is the surface of the Earth, upon which I am also standing. That was the point.

The only way that "target moves to meet the object" scenario could happen is if the ground literally passed through my body, which it never does. The object always falls from my shoulder height to my feet height, and the Earth most certainly never moves with respect to my body.

You're explanation is clearly flawed at best, and completely wrong otherwise.

 

[WannabeNewton]

Let me grab my popcorn because it's been a long time since there's been a confused troll in these regions.

 

[PAllen]

No, you said this:
That is a logical inconsistency, because in this case the "target" is the surface of the Earth, upon which I am also standing. That was the point.

The only way that "target moves to meet the object" scenario could happen is if the ground literally passed through my body, which it never does. The object always falls from my shoulder height to my feet height, and the Earth most certainly never moves with respect to my body.

You're explanation is clearly flawed at best, and completely wrong otherwise.

You are attached to the earth, the Earth is accelerating you upwards relative to the free fall ball. You are part of the 'target' as long as you are stationary relative to the earth. So is the ball until you drop it.

Think of the scenario of you standing in a rocket accelerating, and you drop a ball, and it hurts your toe. When you drop the ball, it is no longer accelerating, while you, the rocket and your toe, in particular, are. So you toe accelerates toward the ball, and the collision hurts your toe. If the ball is replaced by a cat, it also hurts the cat.

 

[Bandersnatch]

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?

You misunderstood.
When you drop a ball, it starts following "a straight line" in spacetime(a geodesic). The geodesic is curved towards the centre of the mass bending the spacetime(Earth), but for all intents and purposes it is the natural path for it, with no acceleration experienced by the falling item.
The surface of the Earth, and everything standing on it(including you), however, cannot follow the geodesic, as the material beneath pushes it away from it("upward"; away from the bending mass).
As the ball moves along the geodesic in free fall(again, no acceleration), it sees Earth accelerating upwards. The upwards acceleration of Earth is real, and once the ball lands on the ground, it begins to feel it to.

Have a second look at the video provided by A.T. in post #3. Once it clicks it becomes very obvious.

 

[Wade888]

If synchronizing observations on the opposite sides of the planet is not valid, then the whole of the field of astronomy with regards to things like Parallax measurements for stellar cartography is invalidated, since it requires synchronized measurements from opposite sides of the planet to be valid, OR requires time-stamped measures which can be re-constructed geometrically to acquire orientation for triangulation, either way, the concept is the same.Knowing this fact which has been implemented to within human capabilities for quite literally centuries, I can then safely say that the concept of a synchronized experiment on opposite sides of the planet must be valid for every observer, and the results that a theory about such events produces must also be logically consistent within itself, and logically consistent with the actual observations in reality.

Since the Earth cannot be expanding in every direction at once (to satisfy your explanation,) a synchronized ball drop from opposite sides of the planet easily disproves your position that says the target moves to the object.

 

[Bandersnatch]

Since the Earth cannot be expanding in every direction at once (to satisfy your explanation,)

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.

 

[Wade888]

You are attached to the earth, the Earth is accelerating you upwards relative to the free fall ball. You are part of the 'target' as long as you are stationary relative to the earth. So is the ball until you drop it.

Think of the scenario of you standing in a rocket accelerating, and you drop a ball, and it hurts your toe. When you drop the ball, it is no longer accelerating, while you, the rocket and your toe, in particular, are. So you toe accelerates toward the ball, and the collision hurts your toe. If the ball is replaced by a cat, it also hurts the cat.

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.

 

[PAllen]

If synchronizing observations on the opposite sides of the planet is not valid, then the whole of the field of astronomy with regards to things like Parallax measurements for stellar cartography is invalidated, since it requires synchronized measurements from opposite sides of the planet to be valid, OR requires time-stamped measures which can be re-constructed geometrically to acquire orientation for triangulation, either way, the concept is the same.


Knowing this fact which has been implemented to within human capabilities for quite literally centuries, I can then safely say that the concept of a synchronized experiment on opposite sides of the planet must be valid for every observer, and the results that a theory about such events produces must also be logically consistent within itself, and logically consistent with the actual observations in reality.

Since the Earth cannot be expanding in every direction at once (to satisfy your explanation,) a synchronized ball drop from opposite sides of the planet easily disproves your position that says the target moves to the object.

Who said synchronizing clocks on opposite sides of a planet is impossible? GPS system does it with great precision, and relies on general relativity to achieve its results.

What is true is that the equivalence principle applies only to the extent that you can ignore curvature (in GR terms) or tidal gravity (in terms of measured changes of direction of free fall and also changes in proper acceleration of static bodies with position). Thus, the equivalence principle cannot apply to opposite sides of the Earth at once.

The geometry in the second link that AT gave, applies globally. It shows that geodesics on all sides of the Earth converge toward the earth. This is not a surprising geometric fact - a hyperbolic geometry has converging geodesics.

 

[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.

.
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.