Space Curvature & Potential Energy

[PaulCam]

So I'm an Software Engineer, not a physicist, nor a mathematician. So I like to work in the qualitative, not the quantitative.

Today I hit on a problem. I've been trying to remove the concept of "down" or "inward" from my thinking of gravity and GR.

When people show the concept of space/time curvature they show the trampoline or similar curved sheet with smaller balls curving around the larger mass. But of course this still requires the concept of "down" to resolve in your head. The balls want to fall to the centre but their angular momentum around the curved surface means they follow a parabola.

So my thoughts that maybe the way space is actually curved means that there is actually no acceleration acting on the Earth orbiting the sun, it is traveling on a straight un-accelerated path, it's just that spacetime is curved by the Sun in such a way that the straight path "wraps" around in classical space to form a circle. Wall of death motorcycle rider style.

But it all fell (oops pun) apart when I considered if the Earth was to rendered void of it's angular momentum. If it isn't traveling relative to the sun at all, what would happen?

Well that damn pesky concept of "down", or even "inward" raises it's head, as of course it will "fall" towards the sun.

So, either the premise that it is purely the curvature of space causing what we perceive as gravity is understood incorrectly by me, or something is missing from my train of thought.

What causes an object to move "down" a gravity well / space curvature from rest? In effect, how is gravitational potential energy explained in GR, or rather what is it that enacts on an object a will to travel to a lower potential energy on a gravity well? It is simply that objects want to return to lower energy states?

 

[Nugatory]

When people show the concept of space/time curvature they show the trampoline or similar curved sheet with smaller balls curving around the larger mass.

Those pictures are terribly misleading, and it's not surprising that they've left you more confused than when you started. If you search this forum for "rubber sheet" you'll find some reasons why it's a bad explanation, and also find some better ones. Pay particular attention to the video by member A.T comparing the worldlines (path through spacetime) of a falling apple and an apple still on the tree.

 

[A.T.]

When people show the concept of space/time curvature they show the trampoline or similar curved sheet with smaller balls curving around the larger mass.

A very misleading analogy, as explained here:
https://www.physicsforums.com/threa...the-force-of-gravitation.760793/#post-4791624

See this for a more relevant analogy:
https://www.physicsforums.com/threads/gravity-and-curved-space.917934/#post-5786330

And this:
https://www.physicsforums.com/threa...in-a-gravitational-field.673304/#post-4281670

 

[PeroK]

So I'm an Software Engineer, not a physicist, nor a mathematician. So I like to work in the qualitative, not the quantitative.

Today I hit on a problem. I've been trying to remove the concept of "down" or "inward" from my thinking of gravity and GR.

When people show the concept of space/time curvature they show the trampoline or similar curved sheet with smaller balls curving around the larger mass. But of course this still requires the concept of "down" to resolve in your head. The balls want to fall to the centre but their angular momentum around the curved surface means they follow a parabola.

So my thoughts that maybe the way space is actually curved means that there is actually no acceleration acting on the Earth orbiting the sun, it is traveling on a straight un-accelerated path, it's just that spacetime is curved by the Sun in such a way that the straight path "wraps" around in classical space to form a circle. Wall of death motorcycle rider style.

But it all fell (oops pun) apart when I considered if the Earth was to rendered void of it's angular momentum. If it isn't traveling relative to the sun at all, what would happen?

Well that damn pesky concept of "down", or even "inward" raises it's head, as of course it will "fall" towards the sun.

So, either the premise that it is purely the curvature of space causing what we perceive as gravity is understood incorrectly by me, or something is missing from my train of thought.

What causes an object to move "down" a gravity well / space curvature from rest? In effect, how is gravitational potential energy explained in GR, or rather what is it that enacts on an object a will to travel to a lower potential energy on a gravity well? It is simply that objects want to return to lower energy states?

One problem with (almost) all popular science explanations of GR is that they avoid the key concept of the Lagrangian. Newton's laws are generally known using the concept of a force: a particle moves in response to a force. But, Lagrange reformulated Newtonian mechanics (into something entirely equivalent) where the particle moves in order to minimise the Lagrangian. Or, more precisely, a certain integral involving the Lagrangian.

In GR there is no concept of a force, so that does not generalise. But, the Lagrangian principle does generalise, and it is in fact an axiom of GR that a particle moves in order to maximise a certain quantity - which turns out to be the amount of "proper" time that the particle experiences. Note that in "flat" spacetime, where there is no gravity, the paths that maximise the proper time are paths of constant velocity: "at rest or uniform motion in a straight line", in other words.

In the curved spacetime around a star or a planet, the paths that maximise the proper time are close to the Newtonian elliptical orbits etc.

The thing to note is that these paths do not represent any predefined paths in space (like a rollercoaster or a rubber sheet), but are a response to the relationship between time and space around a large spherical mass.

On a final point. When you do the mathematics of GR - based on the defined spacetime around the Earth or the Sun, say, and the Lagrangian principle - you generate an energy equation that is the same as the Newtonian energy equation, except that it has an additional term (that is generally negligible in the case of the solar system). This energy equation has an "effective potential" that is very close the the Newtonian potential. For that reason, to a good approximation, you can continue to use the concept of Newtonian potential energy to study the paths of planets in the solar system, and the motion under the Earth's gravity, for example.

In some cases, of course, the additional term becomes non-negligible and in those cases GR diverges significantly from Newtonian gravity.

 

[Mister T]

Today I hit on a problem. I've been trying to remove the concept of "down" or "inward" from my thinking of gravity and GR.

Without gravity there is no way to define the down direction.

 

[PeterDonis]

When people show the concept of space/time curvature they show the trampoline or similar curved sheet with smaller balls curving around the larger mass.

This doesn't show spacetime curvature; it only shows space curvature. (And that, as you will see if you look at the links others have posted, is coordinate-dependent, because how spacetime is split into "space" and "time" is coordinate-dependent.)

What causes an object to move "down" a gravity well / space curvature from rest?

The same thing that causes the Earth to go around the Sun in its orbit. You say, correctly, that the Earth is following an unaccelerated path through spacetime, and that no force is required to make the Earth follow such a path; it's the natural motion. The same is true of an object that is momentarily at rest relative to the Sun: its natural motion, the unaccelerated path, is to fall towards the Sun. To be held at rest relative to the Sun, a force would need to be exerted on it; just as a force needs to be exerted on you to stay at rest relative to the Earth (this force is exerted by the Earth's surface, pushing up on you).