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.