GR explanation of Newtonian Phenomena

[blumfeld0]

1. How would a general relativist explain why an object falls towards the earth?

2. Is it correct to say that it is not the apple that falls towards the Earth but it is the Earth that accelerates towards the apple?
Why is this ok to say?
Is it because, in GR, there are no preferred reference frames?

a qualitative and/or quantitative explanation would be great.

thanks!

 

[pervect]

In "pure" GR, one would generally say that the apple is following a geodesic path through space-time, and that there are no forces on it.

One would say that someone standing on the surface of the Earth is not following a geodesic path, and that the reason for this is that the ground is in the way.

Note that in GR the center of the Earth is essentially following a geodesic path, so it's not really right to think of the Earth as accelerating towards the apple. The Earth (or at least it's center) is following a geodesic path, the apple is following a geodesic path, and the geodesics are converging towards each other.

 

[blumfeld0]

thank you Pervect. I was remembering something Brain Greene wrote in "The Fabric of the Cosmos". He writes on page 67 paragraph three(a short excerpt):

"Einstein argued that only those observers who feel no force at all-including the force of gravity-are justified in declaring that they are accelerating...Such force-free observers provide the true reference points for discussing motion...When Barney jumps from his window into an evacuated shaft, we would ordinarily describe him as accelerating down toward the Earth's surface. But this is a description that Einstein would not agree with...According to Einstein, Barney is not accelerating, he feels no force, he is weightless...you and the Earth all all the other things...are accelerating upwards. Einstein wold argue that it was Newton's head that rushed up to meet the apple, not the other way around"
So I was a little confused about that statement and how it follows from gravity = acceleration principle.

 

[MeJennifer]

Greene is correct.

All matter surrounding the center of the Earth is constantly accelerating away from the center.

 

[Ich]

"Einstein argued that only those observers who feel no force at all-including the force of gravity-are justified in declaring that they are accelerating...

... that they are not accelerating?
The "force of gravity" is seen as a fictious force, the imagined counterpart of the true force that drives you away from your geodesic path.

 

[pmb_phy]

Greene is correct.

All matter surrounding the center of the Earth is constantly accelerating away from the center.

Don't you mean towards the center?

Pete

 

[pmb_phy]

... that they are not accelerating?
The "force of gravity" is seen as a fictious force, the imagined counterpart of the true force that drives you away from your geodesic path.

That is a matter of opinion. Some people claim that gravity isn't a force while others do claim its a force.

The question comes down to "What does it mean to be real?" Einstein said that one can produce a gravitational field merely by changing spacetime coordinates(I personally think he stated that poorly myself). In such case the gravitational force is real since it produces a time rate of change on the particle's momentum. The gravitational field can be "transformed away" by changing spacetime coordinates to that of particle with the origing located at the center of the particle. Those who oppose this do so because they believe that all "real" forces should have a frame independant existence.

However Einstein argued that gravity was a "real" force as have some physicists who know GR who worked during the 19th century.

Peter

 

[MeJennifer]

Don't you mean towards the center?

No, it is away from the center.
Anything that moves with respect to the metric field accelerates in general relativity.

 

[Garth]

Don't you mean towards the center?

No, it is away from the center.
Anything that moves with respect to the metric field accelerates in general relativity.

Anything supported that is.

I think the relevant phrase here is "relative to the freely falling frame"

Garth

 

[Ich]

Those who oppose this do so because they believe that all "real" forces should have a frame independant existence.

I'm one of those. That's the least a concept of "reality" should include to be of any use.

However Einstein argued that gravity was a "real" force as have some physicists who know GR who worked during the 19th century.

I certailnly would not argue with any physicist who knew GR in the 19th century.

 

[pmb_phy]

No, it is away from the center.
Anything that moves with respect to the metric field accelerates in general relativity.

To be precise; The gravitational force is towards the center while the force due to the supporting structure points away from the center of the earth.

Pete

 

[Hurkyl]

Centrifugal force (and Coriolis force, and the like) are of the same kind of force as GR asserts gravity is; they are artifacts of your choice of coordinate system.

Here's a cute comic on the topic: http://xkcd.com/c123.html

 

[blumfeld0]

sorry my fault. It says in "The Fabric of the Cosmos" that
"Einstein argued that only those observers who feel no force at all-including the force of gravity-are justified in declaring that they are NOT accelerating...

ok but sorry I still do not understand when Greene writes
According to Einstein, Barney is not accelerating, he feels no force, he is weightless...you and the Earth all all the other things...are accelerating upwards. Einstein wold argue that it was Newton's head that rushed up to meet the apple, not the other way around".

I understand that centrifugal forces, coriolis forces are pseudo-forces
and gravity can also be interpreted as a kind of pseudo-force (depending on the coordinate system one chooses right?) but I don't understand
1. why the statement (which i get i think)
"The gravitational force is towards the center while the force due to the supporting structure points away from the center of the earth."
leads to the conclusion that things are accelerating upward as Greene implies.

2. i don't understand how/why GR specifically asserts that things are acclerating upward?
I mean do we really need GR to tell us "The gravitational force is towards the center while the force due to the supporting structure points away from the center of the earth."
is this a consequence of the equivalence principle? thank you

 

[jambaugh]

Here is a way to see the phenomenon via analogy. Take a sheet of paper and draw a straight line on it. Roll the paper into a cylinder with the line on the outside. Visualize the axis of the cylinder as "space" and the angle around the cylinder as "time" curled up on itself. The line represents an object at rest as time passes.

Curl the paper into a cylinder from the corner instead and you see the line as a spiral representing an object uniformly moving through space and time.

Now show "curvature" by expanding the cylinder on one side so it is more of a cone. You'll see the line start almost "stationary" and begin to "accelerate" toward the big end of the cone. The line is following a geodesic path but due to the "curvature" it is being accelerated in one direction.

The analogy isn't exact but it is a good tool to help you visualize gravity. Acceleration is the curving of the path of an object through space-time. Curvature of space-time itself means the minimally curving path (geodesic) is still curving. "Motion" through time gets turned into motion through space and time.

Regards,
James Baugh

 

[MeJennifer]

Now show "curvature" by expanding the cylinder on one side so it is more of a cone. You'll see the line start almost "stationary" and begin to "accelerate" toward the big end of the cone. The line is following a geodesic path but due to the "curvature" it is being accelerated in one direction.

The analogy isn't exact but it is a good tool to help you visualize gravity.

Sorry James, but I think this analogy is very likely to mislead.

You can fold a sheet of paper any way you want but you won't be able to demonstrate spacetime curvature with it. Folding a sheet of paper only induces extrinsic curvature but not intrinsic curvature. Spacetime curvature is intrinsic curvature.

Take a square sheet of rubber with the two axes representing time and space. If you draw a curved line on this sheet you have a curved path on a flat spacetime. It does not matter how you fold the rubber sheet or how you draw the path the spacetime remains flat.
But if you deform the rubber sheet you induce spacetime curvature.

 

[jambaugh]

Sorry James, but I think this analogy is very likely to mislead.

You can fold a sheet of paper any way you want but you won't be able to demonstrate spacetime curvature with it. Folding a sheet of paper only induces extrinsic curvature but not intrinsic curvature.

Yes I well understand this point but note I'm not using a "distance on the paper" for the t-coordinate but rather the angle about the axis of the cone. Thus there is an intrinsic curvature in the coordinate manifold (which is not the sheet of paper).

There is a difficulty with any demo using paper or rubber sheets et al because you will still always have a locally Euclidean manifold and can't easily express the intrinsic Minkowski signature.

But you will note that my analogy demo does incorporate one of the main features of the gravitational potential, time "slows" in a sense as you fall into the potential. Look at my demo again more carefully.

To make the analogy even better allow the arc-length along the curve to be coordinate time and the angle around the cone to be the proper time of the falling body.
This isn't good for comparing two paths but it does use the Eucleanization trick:
$$d\tau^2 = dt^2 - dx^2 \to dt^2=d\tau^2+dx^2$$

I didn't want to get into this much detail for a simple analogy but if you want to get technical...
Regards,
James Baugh

 

[blumfeld0]

Thanks guys! I was reading this
http://arxiv.org/PS_cache/gr-qc/pdf/9312/9312027v2.pdf
specifically pages 8-11
I think I get it in conjunction with your explanations.

Can anyone think of any other everyday phenomena that GR explains in a different way than Newtonian physics would?
What about why is it harder to walk up the hill than come down the hill?
I guess that's just a consequence of local energy conservation?


thanks

 

[jambaugh]

Thanks guys! I was reading this
http://arxiv.org/PS_cache/gr-qc/pdf/9312/9312027v2.pdf
specifically pages 8-11
I think I get it in conjunction with your explanations.

Can anyone think of any other everyday phenomena that GR explains in a different way than Newtonian physics would?
What about why is it harder to walk up the hill than come down the hill?
I guess that's just a consequence of local energy conservation?

thanks

The GR corrections to Newton are so small that they are outside the domain of "everyday phenomena". This means also it is quite tough to test. The closest you will find are GR and SR corrections used in the GPS system.

 

[mendocino]

Sorry James, but I think this analogy is very likely to mislead.

You can fold a sheet of paper any way you want but you won't be able to demonstrate spacetime curvature with it. Folding a sheet of paper only induces extrinsic curvature but not intrinsic curvature. Spacetime curvature is intrinsic curvature.

Take a square sheet of rubber with the two axes representing time and space. If you draw a curved line on this sheet you have a curved path on a flat spacetime. It does not matter how you fold the rubber sheet or how you draw the path the spacetime remains flat.
But if you deform the rubber sheet you induce spacetime curvature.

Can you tell me if we can describe Newtonian gravity as curvature of time?
If yes, then it can exist even spacetime remains (intrinsic) flat,
Since it only curves in one dimension like a cone or cylinder

 

[pervect]

Can you tell me if we can describe Newtonian gravity as curvature of time?
If yes, then it can exist even spacetime remains (intrinsic) flat,
Since it only curves in one dimension like a cone or cylinder

It really all depends on what you mean by "curved". As you point out, a cone has no intrinsic curvature. So in what sense is a cone "curved"?

Formally, one might say that it's curved because the metric coefficients are not constant. Equally formally, this implies that the Christoffel symbols are not zero.

Equally formally, one might insist that a cone is a flat geometry. Since it's an argument about defintions and semantics, the argument could go on for a long time.

For a very rough analogy, consider y = ax + b. Is this a "curved" line?

answer #1 - a line can't have any intrinsic curvature, so there's no such thing as a curved line.

answer #2 - yes, it's curved because y isn't constant.

answer #3 - no, it's a straight line, not a curved one. you need y = a nonlinear function of x for it to be curved. The first derivative of y with respect to x is nonzero, so y is not constant, but because the second derivative of y with respect to x is zero, it's "straight".


So the analogy with space-time would be

answer #1 - doesn't apply as long as we have at least 1 space + 1 time

answer #2 - a space-time is "currved" whenever the metric coefficients aren't constant.

answer #3 - a space-time is curved only if the Riemann tensor (roughly analogous to the second derivative of a function) is non-zero.

Answer #3 looks the best in this light, but on the one hand you have the famous elevator experiment saying that the force you feel in an accelerating elevator is gravity, and on the other hand you have the fact that the space-time in an accelerating elevator is the same flat space-time of an inertial coordinate system with different labels, a lot light using polar coordinates on a flat sheet of paper. And you also have people saying that gravity is curved space-time (or possibly curved time). All three answers can'be be correct, but they're all in common use. The culprit behind the incompatibility of the satements is basically the overloading of the word "curved", which is used in multiple senses.