# Gravity, work, geodesics

1. Feb 6, 2010

From what I understand, Einstein basically scrapped the concept of gravity being a force and instead said that energy (and thereby mass) and momentum causes spacetime to curve. Objects still travel on geodesics in spacetime (Newton's first law), but since it is curved, the geodesics near massive objects are such that bodies following them look like they're attracted to the object.

Since gravity is no longer a force, it cannot by definition do any work. Instead it seems as though work would be required to keep an object from following its spacetime geodesic - which would correspond to acceleration in the newtonian formulation.

Thoughts?

2. Feb 6, 2010

### Staff: Mentor

Locally gravity can be considered a fictitious force. Fictitious forces can indeed do work in the appropriate non-inertial frame.

3. Feb 6, 2010

I started thinking about this when studying the equivalence principle. Basically all of GR comes from the assumption that (locally) gravity and an accelating reference frame is the same thing. Right? Hence there is no way to tell whether you're at rest in a (locally homogeneous) gravitational field or in deep space inside an accelerating rocket.

But the two situations are not the same, since one requires a continuous supply of power (that would eventually tend to infinity as the speed increases), while the other is an equilibrium state in which no energy consumption is needed.

But, if the equilibrium state on, say, the surface of the earth is viewed as a state of constant "acceleration from the geodesic", then the two situations become symmetric.

There is, however, the slight inconvenience about why there is no observed power consumption while resting on the ground... ;)

Am I totally wrong?

4. Feb 6, 2010

### HallsofIvy

The equivalence principal is only "locally" true. The fact that people on opposite sides of the earth feel gravitational pull in opposite directions shows that we are not accelerating. Your argument shows that it is "local" in time as well as in space.

5. Feb 6, 2010

Could you expand on how my argument shows this?

Also, I think the fact that the equivalence principle is only locally true doesn't mean that gravity cannot be seen as accelerated reference frames. The way I understand it is that in order to everywhere cancel the effect of gravity we need reference frames that have different velocities at every point in space - this is what gives rise to the curving of spacetime?

6. Feb 6, 2010

### Staff: Mentor

That is correct, the two situations are not the same, they are only locally the same. So there are no worries about anything that "would eventually tend to infinity". "Local" in the context of GR means both an infinitessimally small neighborhood of space and an infinitessimally brief amount of time.

7. Feb 6, 2010

### A.T.

Sure it can, just like the centrifugal-force can do work in a rotating frame.

It is not that gravity is not a force in GR, but rather that it is an inertial force. A frame of reference that spans a big area around a massive body is not an inertial-frame, and depending on the position you have inertial forces, similar to a rotating frame.

Power consumption is frame dependent. If something is resting in a frame, then there is no work done on it in that frame.

8. Feb 7, 2010

### Naty1

Hi Fred....lots of this is tricky...very subtle...and not easy to explain clearly....

I don't believe that geodesics played any part in Newton's absolute space and time. Sure a straight line happens to be a special case of a geodesic, but thinking that Newton viewed gravity as having anything to do with them attributes to him an insight I do not believe he had....Space [distance] for Newton was the absolute, unchanging measure of separation between bodies...but relative under Einstein...not clearly defined

Yes it does, but that is not the defining criteria I believe.
You can tell something is amiss with this when you consider two distant bodies....say initially at rest.....that system has some energy initially, and later when the bodies are closer together and in more rapid relative motion the system contains a different energy....some work has been done....

another way to say this, I believe, is that fictious forces can do work....gravity by any perspective can do work...along geodesics....

from Newton's view you are stationary on the surface of the earth; from Einstein's view you are accelerating...

With help from others here I have found two ideas helpful:

"an inertial frame is any frame where an ideal accelerometer at rest anywhere in the frame always measures 0 proper acceleration. Relative to a non inertial frame the force of an acceleration is feltâ€¦.either linear or rotational." and

"In both SR and GR, an inertial frame is one which is moving along a geodesic in space-time; these are free falling frames, ones where no forces act...."

I'm unsure exactly how to comment unambiguously here....Dalespam has helped steer me in the past and I trust his logic....but here, just suppose you "accelerate" enough to remain stationary, say to remain one foot above the earth's surface, or just outside the horizon of a black hole....your speed/distance remains fixed relative to those surfaces..

yes as is energy....KE, PE, even heat (as in the Unruh effect)....

all in all a lot to consider...

9. Feb 7, 2010

### pervect

Staff Emeritus
Energy and work in GR can probably best be associated with some sort of time translation symmetry, using Noether's theorem. Finding the symmetry can occasionally be tricky, though. This avoids all the Newtonian hand-me-downs involved in thinking of gravity as a force.

In simple situations, though, one can get away with thinking of gravity as a force. This is an offshoot of Newton-Cartan theory.