Four-momentum & four-acceleration in GR - Physics Discussion

[Karl Coryat]

TL;DR Summary

Asking how four-acceleration and four-momentum play into general relativity

I had a discussion with an engineer about forces and accelerations in the context of general relativity. My contention (which I confirmed in these forums some time ago) was that in GR, the real force and real acceleration at the Earth’s surface is upward, not downward. He contended that in physics, the direction of a force is defined by which system is changing momentum, and that if something is in free fall, its momentum is changing but the surface of the Earth's momentum is not — and therefore the force must be considered downward.

To me that sounds like an engineering definition, not a general definition. I suggested that the surface of the Earth's momentum is changing in a falling object's inertial reference frame, but that it’s four-momentum. Is that correct? In the inertial frame of, say, the center of the Earth, is the four-momentum of the surface perpetually changing, all around the planet’s perimeter?

Also, if the acceleration is considered upward from the surface in GR — it's obviously not a linear acceleration, so is that considered a four-acceleration, with the vector pointed away from the Earth's center?

Thank you!

 

[PeroK]

Summary:: Asking how four-acceleration and four-momentum play into general relativity

I had a discussion with an engineer about forces and accelerations in the context of general relativity. My contention (which I confirmed in these forums some time ago) was that in GR, the real force and real acceleration at the Earth’s surface is upward, not downward. He contended that in physics, the direction of a force is defined by which system is changing momentum, and that if something is in free fall, its momentum is changing but the surface of the Earth's momentum is not — and therefore the force must be considered downward.

If you experience a real force (which you do standing on the Earth), then you have an accelerating reference frame. In that frame other objects may be changing their momentum without any real forces on them.

For example, if you are in a car or train that is accelerating, you feel a real force and objects outside the car are changing their momentum relative to you, without being subject to a real force.

In any case, if you are in free fall, you cannot measure the force of gravity on yourself. If you were in a sealed laboratory, you could not tell directly whether you were in free fall in a gravitational field or at rest (or moving inertially) away from any gravity.

 

[PeterDonis]

He contended that in physics, the direction of a force is defined by which system is changing momentum

Momentum is frame-dependent, and in relativity, physics is not contained in frame-dependent quantities but in invariants. So this definition, however useful it might be as a good heuristic in your engineer friend's work, is not correct for relativity.

I suggested that the surface of the Earth's momentum is changing in a falling object's inertial reference frame

Yes.

but that it’s four-momentum. Is that correct?

You can use either 3-momentum or 4-momentum; in either case, the "direction of the change" will be a direction in space (in the case of 4-momentum, the direction of its change is a spacelike 4-vector).

In the inertial frame of, say, the center of the Earth

There is no such thing. There are no inertial frames large enough to cover the entire Earth.

is the four-momentum of the surface perpetually changing, all around the planet’s perimeter?

Yes, in the sense that the surface is not in free fall. But that's not a sense of "changing" that requires any "movement" of the surface. The Earth is (at least to a good first approximation, leaving out things like earthquakes and tides) in hydrostatic equilibrium, so all of its parts are at rest relative to each other.

if the acceleration is considered upward from the surface in GR — it's obviously not a linear acceleration, so is that considered a four-acceleration, with the vector pointed away from the Earth's center?

Yes--more precisely, it's a spacelike 4-vector that points away from the Earth's center. But 4-acceleration can be linear as well--for example, a rocket firing its engine in deep space and moving in a straight line has 4-acceleration.

 

[Dale]

My contention (which I confirmed in these forums some time ago) was that in GR, the real force and real acceleration at the Earth’s surface is upward, not downward.

Yes, this is correct. An accelerometer detects the acceleration due to all real forces so the net real force is upward, not downward.

He contended that in physics, the direction of a force is defined by which system is changing momentum, and that if something is in free fall, its momentum is changing but the surface of the Earth's momentum is not — and therefore the force must be considered downward.

Which system changes momentum depends on the reference frame. In particular, a non-inertial reference frame has "inertial forces" or "fictitious forces" that will change which systems are changing momentum. The surface of the Earth's momentum does change in local inertial frames which are identified by frames where accelerometers read 0.

Also, if the acceleration is considered upward from the surface in GR — it's obviously not a linear acceleration, so is that considered a four-acceleration, with the vector pointed away from the Earth's center?

Yes, this is what is measured by accelerometers.

 

[Ibix]

In particular, a non-inertial reference frame has "inertial forces" or "fictitious forces" that will change which systems are changing momentum.

Just to add to this, what Newton would call "the force of gravity", Einstein would call a fictitious/inertial force that comes from using the upwards-accelerating frame of the surface of the Earth, in much the same way Coriolis and centrifugal forces in a car turning a corner come from using the rotating frame of your car.

 

[pervect]

Summary:: Asking how four-acceleration and four-momentum play into general relativity

I had a discussion with an engineer about forces and accelerations in the context of general relativity. My contention (which I confirmed in these forums some time ago) was that in GR, the real force and real acceleration at the Earth’s surface is upward, not downward. He contended that in physics, the direction of a force is defined by which system is changing momentum, and that if something is in free fall, its momentum is changing but the surface of the Earth's momentum is not — and therefore the force must be considered downward.

To me that sounds like an engineering definition, not a general definition. I suggested that the surface of the Earth's momentum is changing in a falling object's inertial reference frame, but that it’s four-momentum. Is that correct? In the inertial frame of, say, the center of the Earth, is the four-momentum of the surface perpetually changing, all around the planet’s perimeter?

Also, if the acceleration is considered upward from the surface in GR — it's obviously not a linear acceleration, so is that considered a four-acceleration, with the vector pointed away from the Earth's center?

Thank you!

What you can unambiguously say to him is that the 4-force and the 4-acceleration are tensors , because they transform as tensors (rank 1 tensors, or vectors) under the Lorentz transform. You can also say that the 3-force is not a tensor, because it doesn't have these properties.

Furthermore, you can say that the 4-acceleration of an object in free fall is zero, and the 4-acceleration of an object on the Earth's surface is not zero,

The question of what is "real" is a matter of philosophy. I happen to agree with your philosophical position, that it's best to regard the tensor quantity as "real". I'd even agree that that's more or less the modern position. But there's no experiment that can distinguish between "real" and "not real", it's a philosophical question and not a scientific one, because it can't be decided by experiment.

I could go on for a bit about why I regard tensors as real, but it would get rather long and I'm not sure if you're interested. And as I said, that's a matter of philosophy, and while I have some opinions about the matter, I don't have a lot of philosphical references, because I haven't done a lot of philosophical reading.

But I will mention one of the clear advantages of tensors, they are manifestly covariant. Because all tensors transform in a standardized way under a change of coordinates, a tensor statement that is true in one coordinate system is true in all coordinate systems. This is not necessarily true (and usually isn't true) for non-tensor systems. So with tensors, one has the freedom to view coordinates as a human convention, a choice of labels. Non-tensor formulations tend to restrict which coordinate systems are allowed, because they won't work in any coordinate systems, they have some pre-requisites on which coordinates one uses.

 

[Karl Coryat]

You folks never let me down! I just want to thank you all for taking the time to answer people's questions, authoritatively and politely, all of these years.