Because of the principle of equivalence, one cannot ascribe a direct physical interpretation to the gravitational field insofar as it is characterized by Christoffel symbols
Now the relative motion of free particles is given by the equation of geodesic deviation
Here is the infinitesimal orthogonal displacement from the (geodesic) worldline of a free particle to that of a neighboring similar particle. is the 4-velocity of the first particle, and the proper time along . If now one introduces an orthonormal frame on , being the timelike vector of the frame, and assumes that the frame is parallelly propagated along (which insures that an observer using this frame will see things in as Newtonian a way as possible) then the equation of geodesic deviation (14.1) becomes
Here are the physical components of the infinitesimal displacement and some of the physical components of the Riemann tensor, referred to the orthonormal frame.
By measurements of the relative accelerations of several different pairs of particles, one may obtain full details about the Riemann tensor. One can thus very easily imagine an experiment for measuring the physical components of the Riemann tensor.
Now the Newtonian equation corresponding to (14.2) is
It is interesting that the empty-space field equations in the Newtonian and general relativity theories take the same form when one recognizes the correspondence between equations (14.2) and (14.3), for the respective empty-space equations may be written and . (Details of this work are in the course of publication in Acta Physica Polonica.)
PIRANI: I have not put in an absorption term, but I have put in a “spring.” You can invent a system with such a term quite easily.
PIRANI: It is the same as the stability problem in classical mechanics, but I haven't tried to see for which kind of Riemann tensor it would blow up.
Interaction of Neutrinos with the Gravitational Field D. Brill
The wave equation of a neutrino in a centrally symmetric gravitational field was derived, using the formalism of Schrödinger and Bargmann
WHEELER: This already comes up in the electromagnetic case; what should one use for the stress energy tensor? One counts as gravitation producing only the excitations above the vacuum state, in order to avoid infinities. Here the only neutrino states one considers are the positive energy states; the negative energy states are all filled.
Linear and Toroidal Geons F. J. Ernst
Unified, Non-Symmetrical Field Theory M. Tonnelat
In order to avoid a point singularity, it is necessary in the static case to define an electric field which is finite at the origin. The formalism of the unified and non-symmetrical field theory immediately introduces the characteristic expressions of Born-Infeld theory.
By writing the fundamental tensor as the sum of two parts, we can relate it to the invariant Lagrangian that appears in the Born-Infeld electrodynamics and the two invariants of the Maxwell theory.
On the other hand, the Einstein theory introduces the contravariant tensor, which may also be written as a sum of symmetric and skew symmetric parts, which shall be called inductions. The relations foreseen partly by the Born-Infeld
We apply the equations of the field to the calculations of a static and
spherically symmetric solution of the form chosen by Papapetrou. We can then deduce the radial component of the electrostatic field,
. Now introducing like Born-Infeld a field
, we set
. Then, when
, a finite value. The originality of this conclusion comes from the fact that the values of the field and of the metric are not given a priori, but result from the field equations in the particular case of the Schwarzschild solution. We are now led to define a current following the principles of the Born-Infeld theory.
We find the charge density, after substitution of the values obtained in the