I Gravity: Force or Acceleration?

[PeterDonis]

It also works with multiple bodies.

There are no exact solutions for multiple bodies, so I'm not sure what you are basing this on.

You just add up the Newtonian potentials for each body and plug that in.

In the weak field regime for an isolated system, you can get away with this because the non-linearities in the EFE can be assumed to be negligible. But there are still limitations with this approach, and you're still doing numerical solutions since no exact solutions are known.

However, this is still limited to the weak field regime and an isolated system. You can't do it in the strong field regime, e.g., for a neutron star. And you can't do it for, e.g., FRW spacetime.

You can do a whole galaxy (minus the strong-field parts) this way.

Please give a reference.

If you're more comfortable with the phrase "time-only term" than "time-curvature term", I don't have any great objection.

Neither one is correct; note that I didn't say "time-only term". I said $g_{tt}$, and I explicitly noted that it is for a particular system of coordinates.

"flat" implies no force-like effects

It implies no such thing. There are "force-like effects" in an accelerating rocket in flat spacetime, or inside a rotating chamber in flat spacetime, for that matter.

"curved" implies force-like effects

It implies no such thing. Spacetime curvature is geodesic deviation, which can be shown entirely from the behavior of freely falling objects that feel zero force.

this clearly has force-like effects (curved geodesics). The metric is not flat, but the Minkowski part is flat. So all the non-flatness comes from the time dilation.

All this is personal theory and you are getting very close to a warning at this point.

 

[PeterDonis]

Do you mean an infinitesimal volume of spacetime or just a infinitesimal volume of a "space" slice (i.e. a spacelike hypersurface) ?

An infinitesimal volume of spacetime, since that is what is relevant for the differential conservation law.

 

[Dale]

In the Newtonian (weak-field, low-speed) limit of GR, gravity is 100% caused by time dilation.

This is 100% completely wrong, even in the Newtonian limit. Causes always preceed effects. Time dilation and gravity occur at the same time, so they do not have a cause-effect relationship.

 

[pervect]

Of course I know there is no acceptable quantization. That does not have to mean that GR is not a field theory.

It's true that GR is a field theory. For instance, the Lagrangian approach derives GR from a Lagrangian, though the Lagrangian involves the curvature scalar R of space-time.

While it is true that GR is a field theory, it is rather questionable if one can regard it as a field theory in the Minkowskii space-time of special relativity. If one ignores the global topological aspects, approaches such as Straumann in "Reflections on Gravity", https://arxiv.org/abs/astro-ph/0006423, can work, and they may feel more comfortable to those who aren't familiar with differential geometry. The "fields" in Straumann's approach modify the lengths of rulers, and the ticking of clocks. Straumann does not address the global topological issues, unfortunately.

Some quotes from Straumann's abstract.

Although this field theoretic approach, which has been advocated repeatedly by a num-
ber of authors, starts with a spin-2 theory on Minkowski spacetime,
it turns out in the end that the flat metric is actually unobservable,
and that the physical metric is curved and dynamical

So in the weak-field case, one can reproduce GR with Straumann's "fields" on a Minkowskii space-time, but it's probably not true that one could explain general strong-field space-times, for instance the space-time of a black hole, via this approach.

Furthermore, Straumann himself notes that the underlying Minkowskii space-time in his approach is not observable, one is inevitably led to curvature as soon as one settles on physical measurements of distance or time, such as the SI meter or the SI second.

In the end, one probably needs to learn differential geometry at some point to deal with GR. And that is certainly the "standard" approach to teaching the theory.

 

[PeterDonis]

it's probably not true that one could explain general strong-field space-times, for instance the space-time of a black hole, via this approach.

Also, as I've noted, one can't explain spacetimes like FRW, which do not have the same conformal structure at infinity as Minkowski spacetime (i.e., are not asymptotically flat), with this approach.

 

[PeterDonis]

Moderator's note: An advanced discussion of the gauge theory method of deriving the Einstein Field Equation has been moved to a separate thread here:

https://www.physicsforums.com/threads/gauge-theory-of-gravity.1008720/

 

[valenumr]

Please do give an example, yes.

Sorry for the delayed response, just a bit busy the past few days. And apologies for some of the liberties below, please correct any misguided assumptions.

Anyhow, I think of the elevator analogy, i.e. experiments conducted in a frame "at rest" on a massive body (in an accelerated gravitational field?) should be indistinguishable from those conducted in an properly accelerated (equivalent) frame. So that is fine *inside* the elevator, but not so much when taking into account everything *outside*. The former can pop the hatch of the elevator and the measured net kinetic energy of the universe is essentially the same, but this won't be the case for the latter (they would likely have a bad dose of gamma radiation). At least I think that's the case; either way, their respective measurements will be vastly different after a significant amount of time.

But from another point of view, consider two (I think) inertial frames: one in circular orbit around a massive body, and one falling directly on a normal vector towards the same massive body. The former is inertial, not accelerating, not changing energy, etc. But the latter is also inertial (right? Does that imply not accelerating?), but changing energy. In both cases, the best I can say is energy is "with respect to the rest of the universe", which to be fair is ill defined, but say, take the CMB background as a standard rest frame.

Finally, considering an observer "at rest" on the same massive body. This is non-inertial? So accelerating, but also not changing energy. So essentially the elevator "at rest" in a gravitational frame. It feels like this is essentially more like our orbiting observer than our free falling observer, but I'm pretty sure that is not right.

So all these various cases just are hard to reconcile in my personal head. I'm not saying there is anything wrong, proposing alternative physics, or any such thing, or even that any of my assumptions are correct. I am just saying that these concepts are pretty hard to wrap my brain around.

 

[PeterDonis]

that is fine *inside* the elevator, but not so much when taking into account everything *outside*.

Of course not; the equivalence principle, which is what you are referring to, is local. It doesn't say everything everywhere is the same in both cases. It just says things inside the elevator, i.e., inside a small enough patch of spacetime that the effects of spacetime curvature are negligible, is the same in both cases. We do not expect that things will look the same once we look outside the elevator; obviously they will generally not be.

consider two (I think) inertial frames: one in circular orbit around a massive body, and one falling directly on a normal vector towards the same massive body. The former is inertial, not accelerating, not changing energy, etc. But the latter is also inertial (right?

Yes, both are inertial, but they have different 4-velocities.

Does that imply not accelerating?)

"Inertial" means zero proper acceleration, yes.

but changing energy.

Depends on what concept of "energy" you use. Both of them are on geodesic trajectories, meaning both have constant energy at infinity.

In both cases, the best I can say is energy is "with respect to the rest of the universe", which to be fair is ill defined, but say, take the CMB background as a standard rest frame.

Using that, every object in the solar system has huge kinetic energy since the solar system barycenter is moving at about 600 km/s relative to the CMB rest frame, which is a far larger speed than any internal speed in the solar system.

considering an observer "at rest" on the same massive body. This is non-inertial?

Yes.

So accelerating

Yes.

but also not changing energy

Again, it depends on what concept of "energy" you use. "Energy" is not a single unique absolute quantity; there are different concepts of energy you could use. For example, relative to a free-falling observer, the object at rest on the Earth's surface is changing energy.

So essentially the elevator "at rest" in a gravitational frame.

There is a non-inertial frame in which this elevator is at rest, yes.

It feels like this is essentially more like our orbiting observer than our free falling observer

Why? "At rest" is frame-dependent; so is "energy". But nonzero proper acceleration is not; it's an invariant. So proper acceleration is a much better way to judge which observers are "more like" each other.

all these various cases just are hard to reconcile in my personal head

So far nothing you have pointed out is any kind of problem. So I'm not sure what problem you think you are perceiving. The only issue I can see with anything you've said is that you need to be more careful to distinguish frame-dependent concepts (like "energy" and "at rest") from invariants (like proper acceleration). All of the actual physics is contained in the latter.

 

[PeroK]

So all these various cases just are hard to reconcile in my personal head. I'm not saying there is anything wrong, proposing alternative physics, or any such thing, or even that any of my assumptions are correct. I am just saying that these concepts are pretty hard to wrap my brain around.

This seems to me something of a mental block. Many of the ideas that you consider hard to reconcile appear in non-relativistic physics (e.g. that energy is not absolute but frame dependent; and, if we consider non-inertial frames, then even the change in energy is frame dependent). Take a look at this homework problem from elementary (Newtonian) kinematics and you'll see how many of the ideas you find difficult arise quite naturally in non-relativistic physics:

https://www.physicsforums.com/threads/mechanics-calculating-launch-angle-of-projectile.1008645/

The "clever" solution is post #24. And, see my comment in post #25.

 

[valenumr]

This seems to me something of a mental block. Many of the ideas that you consider hard to reconcile appear in non-relativistic physics (e.g. that energy is not absolute but frame dependent; and, if we consider non-inertial frames, then even the change in energy is frame dependent). Take a look at this homework problem from elementary (Newtonian) kinematics and you'll see how many of the ideas you find difficult arise quite naturally in non-relativistic physics:

https://www.physicsforums.com/threads/mechanics-calculating-launch-angle-of-projectile.1008645/

The "clever" solution is post #24. And, see my comment in post #25.

Not exactly a mental block, but I suppose you might feel like your trying to explain that the Earth is a sphere to someone who never heard of such "craziness", having experienced their entire life perceiving it as flat.

Anyhow, I am working on learning the actual math, albeit slowly. I am still learning about covectors, tensor inner product, tensors as metrics and such things. Any other suggested topics would be welcome.