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In Bryce DeWitt's Lectures on Gravitation, in eq. 2.7 on page 25 when he describes the rigid motion of a continuum he states $$x^\mu(\xi,\tau)=x^\mu(0,\sigma)+\xi^in^\mu_i(\sigma)\,\,\,(i=1,2,3) \, .$$

In this equation $\xi$ are the material coordinates and $x^\mu(0,\tau)$ is the origin of the labels $\xi^i$, $\sigma$ is the proper time for the particle taken as origin for material coordinates and $\tau$ is the proper time for the particle with labels $\xi$. I don't understand why he assumes that as motion takes place the coordinates $\xi$ do not change assuming that the three bases $n^\mu_i(\sigma)$ vary properly. I know this assumption is correct in Newtonian mechanics because you can imagine a rigid triad moving with the object and respect to it the object is static but this seem to be impossible according to relativity.

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4 Answers 4

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I don't have the book in front of me so I can't say for sure, but I can guess: The formula for $x^\mu$ is an assumption because it is a reasonable definition of the phrase rigid body. I would guess the author starts out with the assumption and then moves on to deduce what is really physically happening to achieve this.

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  • $\begingroup$ The problem is that the assumption does not seem reasonable to me because it implies all points of space does not change position with respect tot the moving frame and this is impossible according to SR $\endgroup$
    – facenian
    May 16, 2015 at 22:55
  • $\begingroup$ @facenian It's true that you'd have to take care to avoid letting particles travel faster than the speed of light. Other than that, there is no contradiction. $\endgroup$
    – user12029
    May 17, 2015 at 2:00
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There is no such thing as a rigid body, but there is such a thing as rigid motion. It is that kind of motion when all the parts of a given body have worldlines such that, on a sequence of spacelike hypersurfaces that can be chosen in some reasonably natural way, the relative layout and proportions of the body are the same from one hypersurface to the next. In special relativity these spacelike hypersurfaces could, for example, be the planes of simultaneity for a sequence of inertial frames which are also instantaneous rest frames of some part of the body such as its centroid.

This kind of motion is highly constrained and therefore rare in practice, but merits study. It includes, for example, rigid rotation, and the Rindler metric (constantly accelerating frame).

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Page 25 is in the section on special relativity. In section 2.1 the $\xi$ is basically a label based on the initial position. If you think of it as a function of $x^\mu$ then on page 23 there are nonzero $\xi^i_{,\mu}$.

I'd you are referring to the equation after eq. 2.7 the author is tracking the motion if a particular label, and that is why there is no $\xi^i_{,\mu}$ in the equation after 2.7.

As for rigid bodies, there is no body that is rigid in all situations, no way to relativistically enforce rigidity in some situations in dynamics in a relativistic way.

But you brought up GR when that section of the book has no GR. And that section is also not talking about dynamics, there are no forces, just motion. Like if it rotated itself just because it always has been.

It's looking at the cases in SR that are most similar to Newtonian rigid motions.

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  • $\begingroup$ I decided to use the tag of GR according to book's title and yes the question is about SR's kinematics. Regarding the question it is precisely the fact that there is no rigid body in the sense of newtonian mechanics that causes my not understanding the equation 2.7 $\endgroup$
    – facenian
    May 16, 2015 at 23:17
  • $\begingroup$ @facenian based on your comments to the other answer, I'm not sure if you know what equation 2.7 represents. It is not a coordinate transformation, it is the coordinates now of a point original labelled $(\xi^1,\xi^2,\xi^3)$. It is entirely possible in SR that a small isolated body spins at a slow fixed unchanging rate always and forever (past present and future) and calling it rigid might make sense even though the real body would not be rigid in other situations that didn't happen and aren't happening and won't happen. $\endgroup$
    – Timaeus
    May 17, 2015 at 17:48
  • $\begingroup$ I don't see it as a coordinate transformation. The problem is that according to the equation the $\xi^i$ seem to have a direc geometric meaning as cartesian coordinates of any point as seen from the moving origen in it's hyperplane of simultaneity meaning that any material particle appears to be in the same spatial position as seen from the moving frame as if it were knewtonian mechanics $\endgroup$
    – facenian
    May 18, 2015 at 11:36
  • $\begingroup$ @facenian Aren't the $\xi^i$ the cartesian coordinates at some fixed reference time? Really they are just labels, like the markings you could make on a stick to make it look like a ruler. That's how $\xi$ was defined in section 2.1 and so they have zero relationship to coordinates in any frame at any other time, they are just labels. They are the things you want to track over time. The origin might not even be moving, maybe your labels rotate about the origin. And as for a moving frame we are still using inertial frames and we only need one, so just use it and call it stationary. $\endgroup$
    – Timaeus
    May 18, 2015 at 15:07
  • $\begingroup$ @facenian Long ago the parts of the body that had coordinates (then) of $\xi^i$ are now (potentially) somewhere else (in the stationary inertial frame), and eq 2.7 tells us the coordinates now based on the labels about where they were back then. I'm not sure why you think otherwise. And if thinking otherwise causes confusion then I am even more confused as to why you think otherwise. $\endgroup$
    – Timaeus
    May 18, 2015 at 15:11
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In equation 2.7 the author is assuming there exists a reference frame with respect to which the continuum is at rest as is the case of newtonian mechanics. Apparently this is not in contradiction with SR as along as the dimensions of the medium are properly bounded as stated by the condition $(1+\xi^ia_{0i})^2>\xi^i\Omega_{ik}\xi^j\Omega_{jk}$

However, the assumption is not explicitly stated by the author and I think this is not the general case of rigid motion if as such we accept the defintion $\dot{\gamma_{ik}}=0$

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