## Geometric Form of Einsteins Equations

Let V3 be a space-like hypersurface, let e0 N be a unit normal field, and let elt e2, e3 be unit orthogonal tangent vectors to V3. Then from Equation 4-8, 8tt7< T(< 0, e0) K( i a e2) + a e3) + K(< 2 a e3). Choose e , e2, e3 to be principal directions on V3 corresponding to principal curvatures kx, k2, k3. Then irKT(e0,e0) kf(f a e2) + Kf(eja e3) + Kf (e2Ae3) - l rys + k2 + k1k3 + k2k3, (4-12) where 1 3 is the scalar curvature of V3 in its own (induced) Riemannian metric. H tr(b) ki + k2 +...

## Electromagnetism in Three Space and Minkowski Space

Twisted Forms and the Vector Product A twisted (exterior) differential form ak on a manifold Mn assigns in a continuous fashion an ordinary exterior form to each oriented tangent space Mp in such a way that if the orientation of Mp is reversed, the form is replaced by its negative. (These forms are called, by de Rham, forms of odd kind', ordinary forms are then forms of even kind.) For example, the volume form & > rt on a (pseudo) Riemannian Mn assigns to the coordinate patch (U x1, , xn,...

## The Schwarzschild Solution

The Einstein equations give a relation between matter (Ta) and the Ricci tensor (i y) of space-time, not the full Riemann tensor. For example, a region of space-time is said to be empty if 7y 0 there. The Einstein equations, R Sttk(Tfj VzguT), then say that the Ricci tensor vanishes in this region the region need not be flat but can be curved because of matter elsewhere. The simplest and most important case is concerned with a spherically symmetric static mass distribution, like an idealized...

## Cosmology

To consider the universe on a cosmic scale, Einstein made a first approximation by smoothing out all local irregularities (planets, stars) into one cosmic fluid. One can imagine a fluid in which each molecule is an entire galaxy or group of galaxies. The simplest case would assume zero pressure, i.e., an incoherent dust galaxies would attract each other but no pressure would result from collisions. From Equation 7-26, the world lines of the galaxies would be geodesies in space-time. Can one...

## The Relativistic Equations of Motion

Fermi Transport and the Relative Velocity Vector The unit tangent vectors along a family of world lines arise from using proper time as parameter. For many purposes, however, it is convenient to introduce a new parameterization. Consider two nearby world lines and each parameterized by its proper time r, and suppose that a geodesic segment y0 that leaves orthogonally at (0) strikes at (0). Follow each world line near for proper time r. Then the geodesic segment y0 will be sent into a new...