Extended Rigid Bodies in Special Relativity I was reading Landau & Lifshitz's Classical Theory of Fields and I noticed that they mention that an extended rigid body isn't "relativistically correct". 
For example, if you consider a rigid rod and apply a torque at one end, by definition of being rigid, the whole body must start rotating at the same instant. But the information about the force being applied cannot travel faster than the speed of light. 
The book hasn't mentioned how to resolve this paradox, and after thinking about it for a while, I'm wondering if one can even define a rigid body in a relativistically correct manner.
To summarize: How does one define a rigid body in special relativity?
 A: 
if you consider a rigid rod and apply a torque at one end, by definition of being rigid, the whole body must start rotating at the same instant. But [...]
I'm wondering if one can even define a rigid body in a relativistically correct manner

In the strict sense indicated: one can not.
An important clue in the argument has been stated by J. L. Synge [1]:

  For us time is the only basic measure. Length (or distance [or quasi-distance, or indeed any assertion of "spatial extension"]), in so far as it is necessary or desirable to introduce it, is strictly a derived concept 

If distinguishable "ends" are signalling between each other instantaneously (such that if a signal is "applied" at one end and the "reaction" or "echo" from the other end because of the signal is received back at the same time, a.k.a. in coincidence) then these two ends are not called "spatially extended" and "separated" from each other, but "co-located".
As far as "by definition of being [perfectly] rigid" is understood as instantaneous signalling between distinguishable parts, therefore the notion of an "perfectly rigid extended body" is an oxymoron.
However:
In recognition of this consequence of the basic notions of RT, the notion of "rigidity" is readily used instead in the sense of "invariability of shape", to define and describe relations between "ends" that actually are "spatially separated". Different notions of "(spatial) shape" thereby lead to different, generally inequvalent definitions of "rigidity" or "rigid motion".
Historically important is Born rigidity which considers "rest shape" remaining unchanged in the course of the "motion" of the participating "ends". (Born rigidity therefore depends on the existence of suitable inertial "instantaneous rest frames" in the region in which the "motion" is being considered.)
Another, more broadly applicable notion of "chronogeometric rigidity" is described by Synge [2] as

  [...] sending photons from one timelike curve to another, and receiving back the scattered or reflected photons. The criterion for rigidity is that the elapsed time [duration of the signal source] from emission to return [...] should be constant. 

 [...] this test of rigidity by measuring trip-times is really the same as the testing of length by means of an interferometer  
(which thereby refers to invariability of "chronogeometric shape" and which can be determined even in regions where inertial frames can't be found).
References:
[1] J. L. Synge, Relativity: The General Theory, p. 108
[2] ibid., p. 115
A: If you take an aluminum rod and tap one end with a hammer, the disturbance travels along the rod at the speed of sound in aluminum, which is about 5000 m/s. This speed is what determines the frequency of the ringing that you hear. The speed is many orders of magnitude less than c. If it were higher than c for some other substance (one that was very stiff and had a very low density), then it would be possible to use the vibrations to transmit information faster than the speed of light, which is forbidden by relativity; it leads to paradoxes, since there would be frames of reference in which the signal was received earlier than it was transmitted. This tells us that relativity imposes limits on the properties of materials. It isn't surprising that such limits exist, since the properties of materials are determined by the electrical interactions between atoms, and those interactions propagate at no more than c.
A similar example from general relativity is that we can't use a rope to retrieve an object from inside the event horizon of a black hole. If the rope were to be strong enough to support even its own weight, then the speed of sound in the rope would be greater than c, which is impossible.
Although relativity doesn't allow the existence of perfect rigidity as a passive property of a substance, it still allows us to define a notion of rigidity called Born rigidity (Born 1909). In a Born-rigid object, an observer at rest relative to a certain part of the object sees that part of the object as always maintaining a constant distance between neighboring parts. Born rigidity can't be a passive property of a material; to achieve Born rigidity, one has to carry out a pre-planned program of applying forces to different parts of the object as a function of time.
The Herglotz-Noether theorem says that Born rigidity is incompatible with the kinds of free rotations and translations that we expect nonrelativistically to be able to apply to a rigid body. It is not possible for a Born-rigid body to change its angular velocity, and if such a body is rotating, its center of mass can't be accelerated.
Historically, this kind of thing was studied intensively ca. 1910 both because of the desire to resolve paradoxes such as the Ehrenfest paradox and because people were trying to make a theory of electrons as extended objects, in order to avoid the infinite energy inherent in the field of a point charge.
Max Born. Die Theorie des starren Elektrons in der Kinematik des Relativitätsprinzips. (The theory of the rigid electron in relativistic kinematics) Annalen der Physik (Leipzig), Annalen der Physik 30, 1; also referred to as 335 (11), 1-56 (vierte folge, band 30), 1909.
A: In reality there's no such thing as a perfect rigid body. There will always be a delay in the motion propagating along the body.
Under "normal" conditions you don't notice this delay as it's infinitesimal when compared to the size of everyday objects you interact with.
However if you had a rod several light years long (assuming that's at all possible) the delay in transmitting the motion (rotation or translation) along the body would mean that the far end wouldn't move instantaneously, but only at some time later consistent with the information not travelling faster than light.
