Waner S. Introduction to Differential Geometry and General Relativity(T)(128s).djvu: eKnigu

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Waner S. Introduction to Differential Geometry and General Relativity(T)(128s).djvu

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Date Dec 3, 2004

In the absence of a metric, the best
you can do is define a "relative tensor," which is not quite the same, and what Rund calls the "Levi-Civita
symbols" in his book...
This (moving) coordinate system is called the momentary comoving frame of
reference and corresponds to the "real life" coordinate systems...
More generally, define
total 4-momentum P exerted on the positive side of the (scaled) volume
P(e) = element e3 AV on the positive side of this volume element by the negative
side...
Conservation Laws
Let us now go back to the general formulation of T (not necessarily in a perfect fluid),
work in an MCRF, and calculate some covariant derivatives of T...
Using the local coordinate x4 as a parameter, we obtain a path in M given by
\a + vx if i = I, 2, 3
if j = 4
so that the tangent vector (velocity) dxldx has coordinates (v\ v2, v3, I) and hence square
magnitude
ll(v\ v2, v3, l)ll2 = Ivl2 - c\
It is timelike at sub-light speeds, lightlike at light speed, and spacelike at faster-than-light
speeds...
By definition,
j-, i ]_ ij, s
X44 - 2 8 U>4/,4 + ?/4,4 ~ 844j)•
We now evaluate this at a specific coordinate i = 1, 2 or 3, where we use the definition of
the metric g, recalling that g** = (g**), and obtain
\0 + 0 - bpj) « \(\
* Why don't we work in an inertial frame (the frame of the particle)? Well, in an inertial frame, we adjust
the coordinates to make g = diag[l, 1, 1, -1] at the origin of our coordinate system...
(See the
exercise set.) In other words, we have found that we can duplicate, to a good
approximation, the physical effects of Newton-like gravitational force from a simple
distortion of the metric...
Further, we take
as the definition of spherical symmetry, that the geometry of the surfaces r = t = const, are
spherical, thus foring us to have the central 2x2 block...
Heisenberg Uncertainty Principle: A quantum particle has no precise position, x, or
momentum, p...
We would like to express E in terms of N, like Ee—this will
make the resulting expressions easier to adapt to neutron stars later on...
The critical mass is
Mc = uNrmn =
If M > Mc, the star will continue to collapse and its electrons will be pushed closer and
closer to the nuclei...