THE EINSTEIN-GROSSMANN COLLABORATION 215we haveThe pu are in general time-dependent fields which satisfy pu(t;fi) = (^5) W. In the
Newtonian case, Bern's infinitesimal rigidity condition is given by dpu/dt = 0.
This is an invariance condition: ds^2 remains the same at all times and has forever
the magnitude given by the Euclidean expression (Eq. 12.2).
Born attempted next to generalize from the Newtonian to the special relativistic
case by means of a 'relativistic Lagrangian method.' Instead of the x'(g,t), he intro-
duced "(£")) x = ict(£ = icr is the proper time) and we have
The Minkowskian line element
becomes, when expressed in Lagrangian coordinates,
Consider those world points which are simultaneous as seen by an observer mov-
ing with four-velocity w*^1 with the volume element df?: u^d^dx^/d^^1 = 0. Use
this relation to eliminate d£ from Eq. 12.8. Then Eq.12.8 can be written in the
form of Eq. 12.4 with the rigidity condition dpu/dr = 0. Ehrenfest [E26] and
Herglotz [HI] noted that Bern's relativistic local rigidity criterion for a volume
element of a body in general motion can be phrased as follows. Relative to an
observer at rest, that volume element suffers a Lorentz contraction corresponding
to the instantaneous velocity of the center of that volume element. For our pur-
poses, it is of no relevance to discuss the paradoxes to which this approach gives
rise for the case of a finite body—the case to which Einstein referred in his June
7 letter to Ehrenfest. The interested reader can find more on this subject in Pauli's
encyclopedia article [PI].
Bern's reasoning can be transcribed as follows. In the Newtonian case, intro-
duce a three-dimensional manifold on which Eq. 12.4 defines a Riemannian met-
ric. The transformations (Eq. 12.3) are point transformations linear in the differ-
entials which leave ds^2 invariant. The pkl are determined by the dynamics that
governs the motions of the medium. Generalize to four dimensions.
I now return to Einstein. In his papers, he remained silent on the specific prob-