Implications of the statement "Acceleration is not relative"

Mentz114

(my bold)

Me neither. It looks completely consistent. Using the fictitious force to keep the 'rocket' twin stationary does not change the physics - viz. the travelling twin is non-inertial some of the time but the other one is always in free-fall. Therefore the travelling twin ages less as she should according to the other frames.

PAllen

Consistency is not the same a uniqueness. There are different, reasonable, choices for simultaneity for the accelerating twin. Each produces a different metric (though they converge near the 'time axis' represented by the accelerating twin), with different statements as to how much of the aging (of the inertial twin) occurs during turnaround (assuming e.g. coast, turn, coast). Further, the most common way this is presented will not work at all for a W shaped traveler trajectory (the simultaneity surfaces will intersect and multiply map the inertial twin world line, preventing you from having any coordinate chart in which to integrate proper time). So then you must use a different set of simultaneity surfaces to handle this case.

Mentz114

OK.

If both worldlines are specified, is it possible to find a simultaneity choice that enables the WLs to be integrated ?

If the worldlines are specified, is the amount of ageing at turnarounds not uniquely defined ? I have to say that I'm not much interested in where the ageing occurs.

OK, but I was talking about the simplest scenario.

I understand you are advocating caution, but I was addressing the OP's question about a consistent treament of the twins in which the travelling twin remains stationary ( ie has a vertical worldline).

PAllen

Of course.
It is definitely not uniquely defined. Only the observables are uniquely defined. Simultaneity defined by the Einstein convention (two way light signal), and by a simultaneity based on spacelike geodesics 4-orthogonal to the traveling world line tangent, produce quite different answers. The former will work fine for the W trajectory. The latter is the one most commonly used, and will not work at all for the W trajectory.
My point, having seen religious subservience to a convention that is only locally favored, is to stress the non-unuiqueness. Consistency is not a problem. But the non-uniqueness means there isn't one answer to how much ageing of the distant twin occur during turnaround. It really is just as silly as a short line on a piece of paper supposedly having a unique point of view about where the extra length of a longer line is.

Mentz114

Thanks for the responses. I guess that finishes off the CADO nonsense.

I have now realised that the OP was bothered because there is no treatment of the twins case with the travelling twin being inertial. As you and others have already pointed out, that is impossible.

GregAshmore

No, that is not at all what bothers me. I am fully aware that the traveling twin is not inertial; of course a non-inertial frame cannot be treated as inertial. I am bothered that a theory which is only suited for treating inertial frames is used to deal with a problem involving a non-inertial frame.

Mentz114

Thanks for the clarification. I've never had a problem with that. We can get some useful results by including acceleration in SR. For instance, the Rindler frame, the Langevin frame, Born coordinates and probably others.

The Lorentz transformation works even if the β parameter depends on time, so we have a transformation from inertial to non-inertial coordinates.

PAllen

Well, the correct statements are:

- The mathematics of SR is simplest in inertial frames, but all phenomena may be analyzed in such frames, including non-inertial motion.
- There is no such thing as a global non-inertial frame; non-inertial frames are local.
- It is possible, in many ways, to set up coordinates in which a non-inertial world line has constant spatial coordinates of 0. For any such coordinates, you have to transform the Minkowski metric. This transformed metric leads to different formulas for time dilation, light paths, and geodesics. Different choices for such coordinates will produce different answers for coordinate dependent properties, but will produce the same answers as inertial frames for any observations or measurements.

GregAshmore

No. That is the limited principle of relativity, for the special case of inertial frames. The principle of relativity states that the laws of physics are the same for all frames of reference.

If you deal with non-inertial frames within the confines of special relativity, then you have the same problem that Newton had: There is an absolute quality to acceleration; there is a preferred frame.

I maintain my position that this does damage to the principle of relativity.

PAllen

Using Lorentz transform with varying β picks out a special class of coordinates with a specific simultaneity convention. If you want to treat more general coordinates, you use a more general transform. In particular, going from inertial coordinates to coordinates based on Einstein (or radar) simultaneity, will not use a Lorentz transform.

PAllen

There is the special principle of relativity and 'general principle of relativity'. They are different principles, with different physical content. The special principle of relativity, as physical principle, says you cannot detect inertial motion except in reference to other things. The general principle of relativity does not say you cannot detect non-inertial motion. It says you cannot locally distinguish whether your non-inertial motion comes from holding position relative to a gravitational source versus accelerating far from any source.

In general relativity as well, acceleration is distinguishable, and there is a precise mathematical difference between a local inertial frame and a local non-inertial frame in GR: in the former, the connection coefficients vanish, in the latter they do not.

As for laws taking the same form, this is just a matter of the mathematical way you write them (stevendaryl has explained this before on this thread, I believe). If, in SR, you write laws explicitly using the metric and vector/tensor quantities, as you do in GR, then the laws will take the same form in non-intertial coordinates as they do in inertial coordinates. This is still not GR, because there is no gravity involved, nor is the EFE (the equation defining GR) used.

PeterDonis

You've made this claim several times now. Can you give a reference? You talk as though this is "the" principle of relativity, but that doesn't match what I (and suspect others) know of the history and usage of the term.

Also, arguing about definitions is not the same as arguing about physics. Can you state a *physical* objection that doesn't depend on a particular definition for what "the principle of relativity" says?

DaleSpam

No. The principle of relativity as I stated it is the correct one for special relativity (SR). That is the form that it appears as a postulate of SR. The twins paradox is a SR problem, not a GR problem, since it does not use the Einstein Field Equations or curved spacetime.

However, the discussion about inertial vs non-inertial frames is not relevant to the statement "acceleration is not relative". The statement "acceleration is not relative", as we have mentioned, refers to proper acceleration. Proper acceleration is a property of a worldline, not a property of a reference frame.

It doesn't matter what reference frame you use, inertial or not, the proper acceleration is the same in all of them. So, the statement "acceleration is not relative" is about worldlines, not reference frames. I think that you are getting distracted by irrelevancies. The travelling twin has non-zero proper acceleration regardless of what reference frame is used.

DaleSpam

This is, IMO, a reasonable objection to make (I have made the same objection previously). The postulates of SR refer only to inertial frames, so how can you use them to make any claim about the physics in non-inertial frames?

Once you know how the physics works in inertial frames, then figuring out the physics in any other frame is simply a matter of performing a change of variables to the coordinates (aka coordinate transform). All of the usual math for doing a chang of variables still applies. Thus, even though the postulates only describe physics in inertial frames, you can use them indirectly to derive the physics in non-inertial frames.

DrGreg

To repeat what everyone else has said the "principle of relativity" is usually stated in terms of inertial frames only.

So let's consider an example, Newton's second law of motion. The relativistic 4D version of this, for a particle of constant mass, is[tex]
F^\lambda = m \frac{d^2x^\lambda}{d\tau^2}
[/tex]when measured in any inertial (Minkowski) coordinate system. This is pretty simple and almost the same as the non-relativistic version.

However in non-inertial coordinates, the equation becomes[tex]
F_\lambda = m \sum_{\mu=0}^3 g_{\lambda \mu} \frac{d^2x^\mu}{d\tau^2}
+ \frac{m}{2} \sum_{\mu=0}^3 \sum_{\nu=0}^3
\left(
\frac{\partial g_{\lambda \mu}}{\partial x^\nu} +
\frac{\partial g_{\lambda \nu}}{\partial x^\mu} -
\frac{\partial g_{\mu \nu}}{\partial x^\lambda}
\right) \frac{dx^\mu}{d\tau} \frac{dx^\nu}{d\tau}
[/tex]You don't need to understand the meaning of this, just observe that it's very complicated.

So, yes the laws of physics can be expressed in a form that is the same in all frames, inertial or non-inertial, but such expression is much more complicated than the inertial-frame-versions of the laws.

harrylin

Note that such reasoning does not generally hold, as I mentioned before: in the original variant by Langevin both are in free fall. Still it is the traveler who ages less (he didn't in 1911 account for gravitational time dilation but that isn't pertinent and gravitation at the turn-around only enhances the effect).

harrylin

It's just the same with Newton's mechanics. Its laws refer to inertial frames, but nothing prevents from deriving from those laws the corresponding ones for accelerating frames (e.g. coordinate accelerations such as Coriolis).
Likely you mean absolute frame. That was also Langevin's argument although neither Newton nor he saw that as a problem (note: he was one of the most prominent relativists in France). However, for some time Einstein considered that to be a problem. Historically that appears to be the central issue of the twin paradox.
Einstein tried to get rid of that issue with GR, but didn't really succeed. Perhaps you refer here to the introduction in his 1916 paper(in particular §2)?
- http://web.archive.org/web/200608290...ry/gtext3.html

It would be good if physics textbooks discussed this topic, but I don't know any that does.