How accurate is the metaphor of trampoline

[victorvmotti]

In popular perception of curved spacetime the metaphor of trampoline and a dimple on it is often used to explain general relativity.

I see that this may explain the spatial curvature and show how the only straight line in curved space around Sun for planets like Earth would be a timelike geodesic.

But can we explain Precession of the perihelion of Mercury which concerns the curvature through spacetime using the same metaphor of trampoline?

 

[WannabeNewton]

No. The metaphor is entirely flawed so don't expend any of your effort on it. We've already had a thread asking the exact same question before by the way: https://www.physicsforums.com/showthread.php?t=726837&page=2

 

[A.T.]

In popular perception of curved spacetime the metaphor of trampoline and a dimple on it is often used to explain general relativity.

I see that this may explain the spatial curvature

Yes, not explain but visualize it:
http://en.wikipedia.org/wiki/Schwarzschild_metric#Flamm.27s_paraboloid
But it has high misinterpretation potential:
http://en.wikipedia.org/wiki/Gravity_well#Gravity_wells_and_general_relativity

and show how the only straight line in curved space around Sun for planets like Earth would be a timelike geodesic.

Not sure what you mean here. Every geodesic is a (locally) straight line.

But can we explain Precession of the perihelion of Mercury which concerns the curvature through spacetime using the same metaphor of trampoline?

Yes, the orbit precession is related to the spatial geometry. See bottom picture here:
http://www.physics.ucla.edu/demoweb..._and_general_relativity/curved_spacetime.html
And this thread:
https://www.physicsforums.com/showthread.php?t=626729

However, you cannot correctly explain the orbit itself with that purelly spatial curvature. To explain the coordinate acceleration towards the big mass, you have include the time dimension:

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

 

[victorvmotti]

Thanks for the feedback.

Well on paper and in formal math everything works perfectly well.

But trying to explain or correctly visualize it for people who don't like the math seems to be kind of a mission impossible or waste of effort.

Only adding to confusion and misconception.

 

[A.T.]

But trying to explain or correctly visualize it for people who don't like the math seems to be kind of a mission impossible or waste of effort.

You can visualize the concepts and give the correct ideas to a layman who does have some basic geometric skill.

 

[Happy Recluse]

I have a similar question. If gravity were a field generated by the mass of an object, we see why objects are drawn to a large object, i.e., why the moon is drawn to Earth. But if gravity is curved space, then why assume that space is curved to draw smaller objects toward larger objects? We can’t say that large mass bends space so that smaller objects move toward it because mass is no longer a factor.

Following the analogy, a heavier ball in the middle of the trampoline causes the trampoline to droop and smaller balls roll toward the larger ball. But if this concept of weight is replaced with curved space, why does space always curve such that smaller objects are drawn to larger objects where weight and gravitational force play no role?

 

[A.T.]

But if gravity is curved space,

Space-time not just space. See the very last part of post #3.

Following the analogy, a heavier ball in the middle of the trampoline causes the trampoline to droop and smaller balls roll toward the larger ball.

This is a flawed analogy that has nothing to do with how "attraction" works in GR. See the link I already posted:
http://en.wikipedia.org/wiki/Gravity_well#Gravity_wells_and_general_relativity

why does space always curve such that smaller objects are drawn to larger objects where weight and gravitational force play no role?

Mass/energy still plays a role. The big object has more mass/energy.

 

[Happy Recluse]

Mass/energy still plays a role. The big object has more mass/energy.

If that role is not gravitational force, then what is it?

 

[Nugatory]

But if gravity is curved space, then why assume that space is curved to draw smaller objects toward larger objects?

We don't assume just that, it's a consequence of the Einstein field equations of general relativity. The more massive an object is, the greater the curvature it causes in the space-time (not just space!) around it, so the effect of a larger mass on the trajectory of a nearby smaller mass is greater than the effect of a smaller mass on the trajectory of a larger one.

This is little more than a mathematical restatement of the classical picture, which does not say gravity will "draw smaller objects towards larger objects" - instead, they're both drawn towards their mutual center of gravity.

 

[Nugatory]

If that role is not gravitational force, then what is it?

In general relativity there is no gravitational force. There's just inertia, which says that objects want to move in a straight line at a constant speed unless a force is applied to make them do something different - and ostensibly parallel straight lines (in spacetime, not just space) happen to intersect because of curvature.

Imagine, for a moment, that you and I are both standing on the Earth's equator, one kilometer apart. We both start walking due north... and even though we're just moving straight ahead and there's no attractive force between us, somehow we'll draw closer and closer until we bump into one another at the north pole.

Now, imagine that before we start our trek from equator to north pole, we each grab opposite ends of a rod one kilometer long. As long as we hang onto our respective ends, we won't collide at the north pole - but we'll feel the rod is pushing us off our natural straight-line paths.

 

[Happy Recluse]

In general relativity there is no gravitational force. There's just inertia, which says that objects want to move in a straight line at a constant speed unless a force is applied to make them do something different - and ostensibly parallel straight lines (in spacetime, not just space) happen to intersect because of curvature.

Right. My conditional contains the premise: "there is no gravitational force." So we agree there is no gravitational force, but that doesn't explain the cause of the curvature.

In the olden days we said that large objects exerted more force over smaller objects because the increased mass produced more force. Although we no longer recognize the existence of gravitational force, the results of a new theory of space-time curvature has the same results. That is, the new theory also concludes smaller objects are drawn to larger objects, but not because of force. We think the curvature of space-time causes smaller objects (with their inertia) to be drawn to larger objects.

Why does space-time curve in a way that produces the same effects found in the (mistaken) concept of gravitational force? Why can't a small stone in outer space might bend space-time so as to draw large planets toward it? Again, if we dispose of gravitational force, what causes space-time to curve as it does?

 

[Dale]

Again, if we dispose of gravitational force, what causes space-time to curve as it does?

In GR the source of curvature is the stress energy tensor. This was indirectly mentioned above in the reference to the Enstein field equations.

In the appropriate limit the EFE reduce to Newtons law of gravitation.

 

[A.T.]

Why does space-time curve in a way that produces the same effects found in the (mistaken) concept of gravitational force?

It doesn't produce exactly the same results. But it obviously cannot produce completely different results in cases where Newtonian gravity matches observation quite well, because it has to match the same observation.

Again, if we dispose of gravitational force, what causes space-time to curve as it does?

The relationship between mass/energy distribution and the geometry of space-time doesn't involve the concept of a "force". (This is the most misleading / circular part of the rubber-sheet analogy, where the weight of the balls pulls it down). The theory also doesn't explain any underlying mechanism, it just states the relationship between mass/energy distribution and the geometry of space-time quantitatively.

 

[victorvmotti]

The theory also doesn't explain any underlying mechanism, it just states the relationship between mass/energy distribution and the geometry of space-time quantitatively.

When you have the perfect fluid at rest in a comoving coordinates then the trace of energy-momentum tensor is simply three times pressure minus density.

Doesn't this explain the mechanism involved?

Pressure and density curve spacetime so that freely falling objects move along a timelike geodesic which it "appears as a force" in Newtonian gravity.

 

[Nugatory]

When you have the perfect fluid at rest in a comoving coordinates then the trace of energy-momentum tensor is simply three times pressure minus density.

Doesn't this explain the mechanism involved?

Is that an explanation of why the universe acts as it does, or is it just a precise mathematical description of how it acts? The trace of the tensor is a powerful mathematical tool indeed, but I'm not sure why it has anything to do with the curvature of space-time.

Of course, similar issues arise with ordinary Newtonian gravity. Someone asks why more massive objects exert greater gravitation, I tell them it's because of Newton's law $F=Gm_1m_2/r^2$, they leave satisfied with my explanation... But that "explanation" is nothing except the original observation about mass and gravity, formalized mathematically. It's not an explanation, it's a restatement, and as an answer to the why question, I might as well have said "just because it is".

Bottom line: Physics doesn't answer "why" questions, although it gives us a powerful and predictive understanding of how the universe behaves.

 

[Happy Recluse]

Fascinating. Is there any mathematical objection to action-at-a-distance?

 

[pervect]

Fascinating. Is there any mathematical objection to action-at-a-distance?

I would say no, that Newton's law of gravity is an example of a theory of gravity that's perfectly consistent mathematically, and has action-at-a-distance. The problem is that it (Newton's laws of gravity) just doesn't match experiment.

Some specific solar system experiments that are consistent with GR and not Newtonian gravity: the bending of light, propagation delays of light (Shapiro delay), the precession of Mercury's orbit, and most recently, the GP-B frame dragging experiments.

 

[Happy Recluse]

I would say no, that Newton's law of gravity is an example of a theory of gravity that's perfectly consistent mathematically, and has action-at-a-distance. The problem is that it (Newton's laws of gravity) just doesn't match experiment.

Nugatory says, that “The trace of the tensor is a powerful mathematical tool indeed, but I'm not sure why it has anything to do with the curvature of space-time.” Since your understanding of physics includes explanations beyond mathematical models, we can return to the original question: What is the physical connection between objects and space-time such that objects cause curvature in space-time?

 

[Dale]

What is the physical connection between objects and space-time such that objects cause curvature in space-time?

The connection between objects and spacetime is that objects have stress energy and stress energy is the source of curvature, per the EFE.

 

[Happy Recluse]

The connection between objects and spacetime is that objects have stress energy and stress energy is the source of curvature, per the EFE.

Okay, so how does the stress energy of an object cause the curvature of space-time?

 

[Dale]

Thread closed for silence, per request of Happy Recluse