Rigid bodies in special & general relativity

[atyy]

It's often said there are no rigid bodies in relativity. Is the notion of "angle" primary or derived in relativity? Does "angle" require rigid bodies, at least "locally"?

For example, to use light rays to measure the metric, do we need mirrors which are "rigid"?

 

[Fredrik]

The angle of what? Do you have a specific situation in mind? The interior angles of a triangle are well-defined once you have specified three points in space. Also worth mentioning is the fact that an angle is a distance divided by another distance, and if one of them is Lorentz contracted...well, you get the idea.

 

[JesseM]

There are no bodies that remain rigid when accelerated, but objects moving inertially in SR behave just like rigid objects moving inertially in Newtonian physics (in their rest frame anyway, obviously rigid rulers moving in Newtonian physics aren't Lorentz-contracted). Can't you measure angles using an inertial device?

 

[atyy]

The angle of what? Do you have a specific situation in mind? The interior angles of a triangle are well-defined once you have specified three points in space. Also worth mentioning is the fact that an angle is a distance divided by another distance, and if one of them is Lorentz contracted...well, you get the idea.

There are no bodies that remain rigid when accelerated, but objects moving inertially in SR behave just like rigid objects moving inertially in Newtonian physics (in their rest frame anyway, obviously rigid rulers moving in Newtonian physics aren't Lorentz-contracted). Can't you measure angles using an inertial device?

Yes, once we have an inertial frame it's no problem. I guess my question is are inertially rigid bodies just allowed by the theory and nice to have, or is their existence fundamentally necessary for the theory? From memory, I think d'Inverno says we could use rigid bodies, but we'll use light instead, since we can't really define rigid bodies. But Rindler and Ludvigsen postulate angle as primary. If we take angle as primary, then I guess "locally" rigid bodies must be primary, and not just something nice to have (ie. even using light, we can't get rid of rigid bodies completely, contrary to d'Inverno's motivation for preferring the method)?

 

[Fredrik]

I guess my question is are inertially rigid bodies just allowed by the theory and nice to have, or is their existence fundamentally necessary for the theory? From memory, I think d'Inverno says we could use rigid bodies, but we'll use light instead, since we can't really define rigid bodies. But Rindler and Ludvigsen postulate angle as primary. If we take angle as primary, then I guess "locally" rigid bodies must be primary, and not just something nice to have (ie. even using light, we can't get rid of rigid bodies completely, contrary to d'Inverno's motivation for preferring the method)?

We can't really define what a "clock" is either, but we still postulate that what a clock measures is the proper time of the curve in Minkowski space that represents its motion. The word "clock" here has to be interpreted as "what we normally think of as a clock". We need a similar postulate to tell us what mathematical quantity to associate with length measurements. It's a bit tricky to get such a postulate right. How about this one:

Suppose that light is emitted at an event where a clock shows time -t (with t>0), then reflected by a mirror, and finally detected at the event where the same clock shows time t, the distance between the clock and the mirror at the event where the clock shows time 0 is approximately ct, and the approximation is exact in the limit t→0+.

A bit awkward, but it seems to do the job without rigid bodies. I'm not sure that's a good thing though. If we take this approach, we must also explain why what we measure with a non-accelerating meter stick agrees with the coordinates assigned by the inertial frames that have time axes that coincide with the world line of the meter stick (or never use solid objects to measure distances). That means we still have to include postulates about the properties of solids.

We could try a postulate that uses a ruler as the fundamental length measuring device instead, but that's complicated too, because we need to talk about acceleration and synchronized clocks. Maybe we need to define acceleration measurements before length mesurements, I don't know.

I'm probably going to sit down and reallly think this through some day, but not today. It bugs me that I've never seen a complete list of the postulates we need. Such a list would be a definition of special relativity, so the implication is that I've never really seen a definition of what special relativity is.

 

[atyy]

Suppose that light is emitted at an event where a clock shows time -t (with t>0), then reflected by a mirror, and finally detected at the event where the same clock shows time t, the distance between the clock and the mirror at the event where the clock shows time 0 is approximately ct, and the approximation is exact in the limit t→0+.

A bit awkward, but it seems to do the job without rigid bodies. I'm not sure that's a good thing though. If we take this approach, we must also explain why what we measure with a non-accelerating meter stick agrees with the coordinates assigned by the inertial frames that have time axes that coincide with the world line of the meter stick (or never use solid objects to measure distances). That means we still have to include postulates about the properties of solids.

I agree that does the job without needing meter sticks. But how about the mirror? Is that a "locally rigid body" (whatever that may mean)?

 

[Fredrik]

Does it matter? Why not think of the "mirror" as the point in space where the reflection event ocurred?

Regarding the notion of a "locally rigid body". I would interpret that as an object doing Born rigid motion, so a spinning wheel for example could never be a locally rigid body.

 

[atyy]

We can't really define what a "clock" is either, but we still postulate that what a clock measures is the proper time of the curve in Minkowski space that represents its motion. The word "clock" here has to be interpreted as "what we normally think of as a clock".

I guess an ideal clock will do everything, since it gives ds². And since atoms are ideal clocks, we can conveniently postulate that an ideal clock exists, without postulating rigid bodies. Then why is it always said the metric gives distance and angle, since from this point of view it only gives distance, and angle is secondary? Are there "geometries" with the notion of distance, but no notion of angle?

 

[atyy]

Does it matter? Why not think of the "mirror" as the point in space where the reflection event ocurred?

Regarding the notion of a "locally rigid body". I would interpret that as an object doing Born rigid motion, so a spinning wheel for example could never be a locally rigid body.

Well, yes, perhaps it shouldn't be called "locally rigid body", since that's already a "reserved" term. But we would need to postulate the existence of mirrors? Hmmm, I'm vacillating between clock only, or clock and mirror.

 

[Dale]

Regarding the notion of a "locally rigid body". I would interpret that as an object doing Born rigid motion, so a spinning wheel for example could never be a locally rigid body.

A spinning wheel can be Born rigid, it just cannot increase or decrease its rotation rate in a Born-rigid way. In contrast, you can have Born rigid linear acceleration.

@ atyy. I would say that angles are, in general, frame variant. Particularly angles of velocity. E.g. in the light clock experiment in the rest frame the angle of the light ray is perpendicular to the mirror, and not perpendicular in any other frame.

 

[atyy]

@ atyy. I would say that angles are, in general, frame variant. Particularly angles of velocity. E.g. in the light clock experiment in the rest frame the angle of the light ray is perpendicular to the mirror, and not perpendicular in any other frame.

Thanks, that's key. So we have ds² as primary (invariant), and angle as derived (variant).

I'm now wondering to set up ds²: clock only or laser and mirror, or whatever minimal combination. At the same time, I'm sensing this question is a bit silly, like whether there are one two postulates needed for SR, since in actuality to verify that a particular object is a clock, laser or mirror is presumably a very unminimal procedure.