Gravity Explained: Why Objects Fall

[eep]

My limited understanding of relativity tells me that "gravitational forces" are really just due to the curvature of spacetime. That is, the reason why the Earth orbits the sun is because the Earth is following a "straight" line but the actual spacetime is curved so the "straight" line appears curved. Like an ant on a sphere, it walks straight but the sphere is curved so the path is curved. Anyways, when you hold an object in the air and then let go, it falls. Why? What's giving it the "push" to fall?

 

[Garth]

My limited understanding of relativity tells me that "gravitational forces" are really just due to the curvature of spacetime. That is, the reason why the Earth orbits the sun is because the Earth is following a "straight" line but the actual spacetime is curved so the "straight" line appears curved. Like an ant on a sphere, it walks straight but the sphere is curved so the path is curved. Anyways, when you hold an object in the air and then let go, it falls. Why? What's giving it the "push" to fall?

Nothing. It just follows its "straight" line through space-time curved by the Earth's mass. Before you let go you were giving it the "push" for it not to fall.

The question is what is giving you the "push" to stand up? The answer is the strength of your legs and the floor you are standing on!

That "push" you feel as your weight.

Garth

 

[eep]

So you're saying that objects in gravitational fields are already on a straight line path? I'm having a hard time wrapping my head around that. In my head I'm picturing the object that I'm holding as being at rest. When I let it go, there is no "force" acting on it so it should stay there. If it were already moving, then it would follow it's straight-line path in the curved spacetime, but if I've put it at rest then what gets it moving?

 

[Flatland]

One thing you got to remember, there's no such thing as "rest"

 

[chaah]

"Push" is a Newtonian concept, eep. In Einstein's way of thinking, talk of "curved spacetime" is meant to replace talk of "pushes" (or "forces"). So the reason why your object moves is because the spacetime around it is curved. If you like, the curved spacetime is what "pushes" it to move, but that's not the best way to put it.

You can also see this by considering what Newton's first law looks like in Einstein's terms. Part of Newton's first law is this:
A body at rest remains at rest unless acted upon by an external force.

In Einstein's terms, however, this law would be restated this way:
A body at rest remains at rest unless the spacetime around it is curved.

Maybe you now want to ask, but why should an object move simply because the spacetime around it is curved? Perhaps this is your true question? Because asking for what "pushes" the object just confuses Newton's way of thinking with Einstein's way of thinking. An object that is initially "at rest" in a curved spacetime does not require an extra "push" to make it move. It starts to move simply because the spacetime around it is curved.

 

[eep]

Ah, okay. It's hard to decouple my mind from the idea of forces. It seems strange that an object in a curved spacetime would simply move because of the curvature. My mind still wants to say that there is something pulling/pushing it down the curve!

 

[chaah]

Ya, know what you mean, your question is quite understandable actually. In fact, those popular diagrams that illustrate the effect of curved spacetime can be misleading. They depict something like a trampoline, in the center of which rests a heavy ball ("the sun"), which causes a depression in the middle of the trampoline. And then they show a smaller ball at the edge ("a planet"), and if you release the smaller ball, it falls towards the "sun". (Or goes around in an orbit if it has a suitable initial momentum.) And the caption is, "Curved spacetime attracts planet to sun". But of course the only reason why a ball at the edge of a depressed trampoline would fall towards the depression is because of gravity as we know it. So that's confusing - it suggests that the curved trampoline isn't enough, you also need a force in addition. A little misleading.

 

[eep]

Yeah, that's sort of the picture I had in my head. Misleading, indeed! So relativity gets rid of the concept of gravitational forces altogether then? And Einstein was trying to get a theory which does the same for the other fundamental forces as well?

 

[chaah]

It's hard to know the best way to put it. It might be better to say that Einstein showed a way of understanding what a gravitational "force" actually is and how it comes about. (Newton had virtually no analysis of a "force", you see.) The main thing here is that a gravitational "force" is not to be regarded as something over and above the effect of curved spacetime. As for what Einstein was trying to do with the other fundamental forces, I don't really know. From my limited understanding, the other forces of nature (e.g., the electromagnetic force) were analysed via a "quantum" approach, unlike Einstein's "spacetime" approach to gravity. Mathematically, these approaches take a different form and everyone has been trying since then to produce just one mathematical framework to handle all the four forces, rather than keep working with two different ones. I don't know if Einstein was trying to quantimize gravity, or whether he was trying to spacetimeize the other forces, or indeed whether I am any longer talking any sense at this point.

 

[George Jones]

At first this, it's very difficult warp one's mind around this stuff! Like anyting, though, intuition is built up through experience.

The geodesic ("straight line") along which an object moves when it falls freely is a straight line in spacetime, not a straight line in space. The usual version of Newton's first law refers to a straight line in space.

Whether or not an object is freely falling, it "moves" along a line spacetime. For example, suppose I hold a ball around which a watch is strapped, so that it is "at rest in space". The ball is still "moving" along its worldline in spacetime, i.e., at every reading on the watch, the ball is at a different event on its worldline.

If I hold the ball for a while and then release it so that it falls freely, part of the ball's worldline corresponds to the time when its "at rest" in space, and part of the ball's worldline corresponds to the time when its freely falling. So, the question is, "Which part of the worldline, if any, is straight in spacetime?"

A mathematical model, general relativity, that is backed up by loads of empirical evidence answers "When the ball is freely falling." In this model, objects "fall at the same rate", because they move along (almost) the same grooves in spacetime. Ignoring the spacetime curvature caused by a test object, these "straight line" gooves are intrinsic properties of spacetime, so "falling" at the same rate makes sense.

Now, in special realtivity, consider an accelerometer in a spaceship located deep in in interstellar space.

The accelerometer consists of two main parts - a hollow sphere like a basketball (go Suns!) inside of which is a slightly smaller sphere. Initially, the centres of the spheres coincide, so that there is a small, uniform gap between the spheres.

If the ship is accelerating, the gap will be closed, and contact between the spheres will be made. An alarm that indicates "curved" motion will sound. If the ship is not accelerating, no alarm will sound, and "straight line" motion is indicated.

Now move the accelerometer to a place near the surface of the Earth, and assume that the accelerometer is small enough that tidal forces can be neglected. When the accelerometer is held at rest (in space), the alarm sounds, but if the accelerometer falls freely, no alarm sounds.

Regards,
George

 

[eep]

So your "accelerometer" indicates whether one is traveling in a straight line through spacetime?

 

[George Jones]

So your "accelerometer" indicates whether one is traveling in a straight line through spacetime?

Yes. The accelerometer sounds the alarm when the 4-acceleration is non-zero. This may seem strange, but when I hold a ball at rest, its 4-acceleration is non-zero, and after I release the ball, its 4-acceleration is zero.

Regards,
George

 

[DaveC426913]

A very clever thought experiment.

(Though I guess it's a convulated way of demonstrating the Equivalence Principle)

 

[Ich]

The best thing about this thought experiment is that some people work with accelerometers all the time. You buy them and measure what George says.
When Newton says that they are at rest, they show acceleration. When he says tha they accelerate, they show no acceleration.
And when I design a machine and want to calculate the forces due to its weight, I simply calculate it as being accelerated upwards all the time. The whole concept is very straightforward.

 

[masudr]

When Newton says that they are at rest, they show acceleration. When he says tha they accelerate, they show no acceleration.

Be careful. This only concerning what Newton might have called gravitational acceleration.

 

[George Jones]

The setup I gave was inspired by Moore's book A Traveler's Guide to Spacetime, which I don't have at hand, but which I recently reread. As DaveC426913 pointed out, this is a version of the principle of equivalence.

Regards,
George

 

[Ich]

Newton´s view:
The falling accelerometer is in fact accelerated, but due to the equivalence principle all parts of it accelerate the same way, so you get the wrong result (no acceleration). The same with the accelerometer at rest: You measure the wrong result. So, even if Newton´s predictions are correct, Einsteins view is much cooler:
Spacetime is curved, so that the local inertial system is "falling down" all the time, and everything "at rest" has to be constantly pushed. Especially as you have the readings of accelerometers as evidence, it´s an easy introduction to GR.