Forces and gravitational time dilation

[pervect]

The following is more of an interesting example and observation than a question that I am presenting for public comment. It's somewhat related to a recently thread, which was closed for moderation, but I think it's different enough not to fall under the ban of reposting threads that have been closed, and that it's educational and worthwhile to post.

The problem relates to some observations as to how forces act in "Einstein's elevator", a case which can be considered to be modeled by special relativity even though many methods commonly used to address the problem are introduced in General relativity. It's also case which involves both the concept of forces and (gravitational) time dilation, that illustrate how they interact. Thus it serves as a vehicle for discussing the laws of physics in cases where time dilation exists.

Consider a weight hanging from a cable of constant cross section in an accelerating elevator. The cable is considered to be "massless" though perhaps it would be better to say that the density of the cable is zero, thus the cable doesn't have to support its own weight. This is not realistic, but does not violate any fundamental laws of physics. It does violate the weak energy condition, but that's not a fundamental law. IT's also noteworthy that this idea of a zero-density cable is frame dependent, so this part of the problem specification is a bit coordiante dependent.

We can ask the question "Is the tension in the cable constant along it's length". And the answer to that question is no - the tension in the cable is not constant.

At the I level, we can draw this conclusion from considerations of the conservation of energy. The work done in lifting the weight by a certain distance through the rope, which is assumed for simplicity to be well approximated by Born rigidity, must conserve energy, and this requires that the tension not be constant. The issue involves Bell's spaceship paradox, whereby the proper acceleration at the top of the cable is dfferent than the proper acceleration at the bottom of the cable where the weight is attached.

The A-level answer is more interesting and I personally find it more convincing than the I-level argument, but it does require graduate level concepts.

For the A-level approach, we use the Rindler metric, and we note that the relativistic replacement for F=ma, an ordinary differential equation, is ##\nabla_a T^{ab} = 0##, a partial differential equation, where the concept of force is replaced by an entity known as the stress-energy tensor ##T##, in which tension in the rope is just one component of the tensor. The components are basically density and tension, and we set the density to zero, as the rope is (in the frame of the elevator) specified as being massless. It is worth noting that the concept of "massless" here is frame dependent.

Solving this partial differential equation in previously mentioned Rindler coordinates (which is convenient, though of course other coordinate choices could be made, as long as some attention is paid to sepcifying the idea that the density of the rope is zero in a coordinate independent form), and converting the tension in the rope from a coordinate basis to an orthonormal basis, we observe that the stress in the cable is not constant, and that the total force exerted by the rope on it's attachment point at the top of the elevator is not equal to the total force exerted by the rope on the weight at the bottom of the elevator.

I'm pretty sure I posted the details of this calculaltion elsewhere in the PF forums once upon a time - if it becomes necessary I believe I can dig it up.

 

[vanhees71]

This is not about gravitation but about the description of relativistic motion in an accelerated reference frame. It is utmost important to keep in mind that gravitation is only locally equivalent to describing physics in an accelerated reference frame. True gravitation cannot be transformed away by a coordinate transformation since if there's true gravitation, the Riemann curvature tensor is non-vanishing, and that's an invariant property, which can't "transformed away". In physical language: Gravitation is equivalent to an accelerated frame of reference only at such scales, over which tidal forces can be neglected.

 

[A.T.]

We can ask the question "Is the tension in the cable constant along it's length". And the answer to that question is no - the tension in the cable is not constant.

At the I level, we can draw this conclusion from considerations of the conservation of energy. The work done in lifting the weight by a certain distance through the rope, which is assumed for simplicity to be well approximated by Born rigidity, must conserve energy, and this requires that the tension not be constant. The issue involves Bell's spaceship paradox, whereby the proper acceleration at the top of the cable is dfferent than the proper acceleration at the bottom of the cable where the weight is attached.

Do you have an I level explanation using the inertial frame of reference? The cable is contracting here, so the weight accelerates more than the cable attachment. But how does this lead to varying tension in a cable with zero density?

 

[Dale]

Gravitation is equivalent to an accelerated frame of reference only at such scales, over which tidal forces can be neglected.

But this example is in flat spacetime, so there are no tidal effects.

 

[vanhees71]

Exactly! So there is no gravitation.

 

[PeterDonis]

The issue involves Bell's spaceship paradox, whereby the proper acceleration at the top of the cable is dfferent than the proper acceleration at the bottom of the cable where the weight is attached.

That's not the Bell spaceship paradox. In the Bell spaceship paradox, the "top" and "bottom" proper accelerations are the same--and that, in relativistic kinematics, means the ends will separate (i.e., the motion will not be Born rigid).

The Bornd rigid motion you are using is described by the Rindler congruence.

 

[PeterDonis]

the total force exerted by the rope on it's attachment point at the top of the elevator is not equal to the total force exerted by the rope on the weight at the bottom of the elevator

As long as by "total force" you mean the locally measured force, yes, this is true. The more general idea here is that "forces redshift", and this idea carries over to curved spacetime scenarios such as an object suspended in the gravitational field of a massive object. All you need is the correct redshift factor for the given scenario.

 

[pervect]

Do you have an I level explanation using the inertial frame of reference? The cable is contracting here, so the weight accelerates more than the cable attachment. But how does this lead to varying tension in a cable with zero density?

Not currently - I think the biggest obstacle is an I-level definition of what we mean by "energy". The conservation of energy argument requires a notion of energy to be conserved, as well as a way to easily deal with the possibility that energy might be stored in the cable. While "##\nabla_a T^{ab} = 0##" could be argued to be a notion of conservation of energy, and sufficient to illustrate the example, it's not I-level :(.

I don't think I spelt out the energy conservation argument in enough detail, the basic argument is that the cable can be used to transport power from the low level to the high level. We also need the idea that by making the process sufficiently slow, we don't have to deal with energy storage in the cable (due to elastic stretching in the cable, for instance). We also probably need some notion of rigidity of the cable to be able to argue that if we move the bottom of the cable by some amount, the top moves by the same amount.

Then transfering energy up - and down - the cable with idealized 100 percent efficiency must match the idea of transfering energy via a perfectly efficient conversion of the energy to radiation (light, for example), transfering radition up or down, then converting it back to energy with 100 percent efficiency. If we could transfer energy up the cable with no loss via the cable, then get that energy and more back by converting it to radiation, letting it blue-shift, then converting it back to radiation, we'd have a perpetual motion machine, which illustrates that the energy input at the bottom of the cable in the inertial frame of the bottom of the cable isn't equal to the energy extracted at the top of the cable in the inertial frame of the top of the cable.

The rigidity of the cable is what makes the distance the top of the cable moves equal to the distance that the bottom of the cable moves, and the notion that work = force * distance implies the force in the inertial frame at the top of the cable can't be the same as the force at the inertial frame at the bottom of the cable.

 

[pervect]

Exactly! So there is no gravitation.

While I agree that the notion of "gravitational time dilation" may be unfortunate in cases where there is aguably no gravitation, it's commonly understood what the term means. There is a lot of language that is in common use that can be misleading. It's always worth while to consider if there's a better, less misleading way of wording things that is also understandable at the desired level, but I can't think of one in this case.

 

[A.T.]

The rigidity of the cable is what makes the distance the top of the cable moves equal to the distance that the bottom of the cable moves, and the notion that work = force * distance implies the force in the inertial frame at the top of the cable can't be the same as the force at the inertial frame at the bottom of the cable.

In the inertial frame the ends do not move the same distance, because the accelerating cable is contracting.

 

[vanhees71]

While I agree that the notion of "gravitational time dilation" may be unfortunate in cases where there is aguably no gravitation, it's commonly understood what the term means. There is a lot of language that is in common use that can be misleading. It's always worth while to consider if there's a better, less misleading way of wording things that is also understandable at the desired level, but I can't think of one in this case.

That's easy. An ideal ("point-like") clock moving along a time-like trajectory in spacetime measures it's proper time (in German it's called "Eigenzeit", which means "its own time"), which is an invariant,
$$\mathrm{d} \tau=\mathrm{d} \lambda \sqrt{g_{\mu \nu} \dot{q}^{\mu} \dot{q}^{\mu}},$$
where ##q^{\mu}(\lambda)## describes the time-like world line of the clock.

 

[pervect]

In the inertial frame the ends do not move the same distance, because the accelerating cable is contracting.

If you analyze the problem in one single inertial frame, the analysis will need some extensive reworking. The idea that the cable is "pure stress" with zero density will hold locally in any co-moving inertial frame but it won't hold in a global inertial frame. The cable also won't be static in the global inertial frame either - parts of it will be moving relative to other parts. It'll be quite confusing, and the assumption that there's no energy stored in the cable will also fall by the wayside.

The easiest way to get any answers would be to compute the various tensor quantities in the accelerating frame where the analysis is easy, and then covert to the global inertial frame, I think.

I would argue that even if you did work the problem in the global inertial frame, to get some physical sense of the significance of the result, you'd want to convert the results from the global inertial frame to a frame that's instantaneously at rest with respect to the cable at whatever point one was interested in anyways. This isn't one single frame, it's a different frame for every point on the cable. But that's just what doing the analysis in the "accelerated frame" involves.