## Completely Antisymmetric Part Vanishes

From the complete set of symmetries in the presence of a metric, a(3y5 R al3 78 and Raipytt 0, derive (a) symmetry under pair exchange, Ra y& Rysap, and (b) vanishing of completely antisymmetric part, ta 3y5l 0. Then (c) show that the following form a complete set of symmetries Ra 3yS MUyS RySct0> R a yS 0- (13.55) Exercise 13.11. DOUBLE DUAL OF RIEMANN EINSTEIN (a) Show that G *Riemann* contains precisely the same amount of information as Riemann, and satisfies precisely the same set of...

## L tiGcjxn 16 X 1033 cm T fiGc512 54 x 1044 sec M ticGx2 22 x 105 g

The only system of units, Planck (1899) pointed out, free, like black-body radiation itself, of all complications of solid-state physics, molecular binding, atomic constitution, and elementary particle structure, and drawing for its background only on the simplest and most universal principles of physics, the laws of gravitation and black-body radiation. Relative to the Planck units, every constant in every other part of physics is expressed as a pure number. Three hierarchies of fossils...

## Gravitational Redshift Experiments

The uniqueness of free fall, as tested by the Dicke-Eotvos experiments, implies that spacetime is filled with a family of preferred curves, the test-body trajectories. There *The experiment of Farley et al. is a 2 percent check of acceleration-independence of the muon decay rate for energies E m (1 ir) l 2 12 and for accelerations, as measured in the muon rest frame, of a 5 X 10'-' cm sec2 0.6 cm'1. Physical meaning of a comparison between test-body trajectories and geodesies of metric...

## Lorentz Frame

By definition, a local Lorentz frame at a given event is the closest thing there to a global Lorentz frame. Thus, it should be a coordinate system with gM (90) and with as many derivatives of g as possible vanishing at Prove that there exist coordinates in which o) V and g pC 0) - 0, but that cannot vanish in general. Hence, such coordinates are the mathematical representation of a local Lorentz frame. Hint Let x' ) be an arbitrary but specific coordinate system, and be a local Lorentz frame,...

## And Ppn Frame

Carry out the details of the derivation of the transformation matrix (39.41) and in the process calculate the correction of 0(c4) to The post-Newtonian corrections to the Newtonian equations of motion (39.15) and (39.16) are derived from the law of conservation of baryon number (p0ua).a 0, and from the law of conservation of local energy-momentum, T 0. The simplest of the equations of motion is the conservation of baryon number. Its exact expression is (p0ua).a - (1 V g)(V gp0a)r0l 0. Define a...

## In The Equivalence Principle

Factor-ordering problems and On occasion in applying the equivalence principle to get from physics in flat space-coupling to curvature tjme to physics in curved spacetime one encounters factor-ordering problems analogous to those that beset the transition from classical mechanics to quantum mechanics.* Example How is the equation (3.56) for the vector potential of electrodynamics to be translated into curved spacetime If the flat-spacetime equation is written then its transition (comma goes to...

## Weak Gravitational Fields

The way that can be walked on is not the perfect way. The word that can be said is not the perfect word. 18.1. THE LINEARIZED THEORY OF GRAVITY Because of the geometric language and abbreviations used in writing them, Einstein's field equations, Gii 8777 , hardly seem to be differential equations at all, much less ones with many familiar properties. The best way to see that they are is to apply them to weak-field situations Linearized theory of gravity (1) as weak-field limit of general...

## Why Black Hole

What is all this talk about black holes When an external observer watches a star collapse, he sees it implode with ever-increasing speed, until the relativistic stage is reached. Then it appears to slow down and become frozen, just outside its horizon (gravitational radius). However long the observer waits, he never sees the star proceed further. How can one reasonably give the name black hole to such a frozen object, which never disappears from sight Salvatius. Let us take the name...

## P2 m2u2 m2

Second Solution In the frame of the observer, where E and p are measured, the 4-momentum splits into time and space parts as P - E, p1e1 + p2e2 + p3e3 p, hence, its squared length is But in the particle's rest frame, p splits as hence, its squared length is p2 m2. But the squared length is a geometric object defined independently of any coordinate system so it must be the same by whatever means one calculates it The vector separation v 9 between two neighboring events i 0 and a 1-form cr and...

## An Anisotropic Universe

Adiabatic cooling and viscous dissipation might not be the chief destroyers of anisotropy in an expanding universe. More powerful still might be another highly dissipative process, which might occur at still earlier times, very near the initial singularity. This is a process of particle creation which was first treated by DeWitt Creation of particles by (1953), then explored by Parker (1966 and 1969) for isotropic cosmologies and finally anisotropy of expansion by Zel'dovich (1970) in the...

## Info

For the PPN parameter (Recall 3 measures the amount of nonlinearity in the superposition law for g00.) The periodic effects in the planetary orbits have not yet (1973) been studied experimentally. The above discussion and Box 40.3 have ignored the motion of the solar system relative to the preferred frame (if one exists) i.e., they have ignored the terms (40.3') Bo* 40.3 PERIHELION SHIFTS EXPERIMENTAL RESULTS Relativistic corrections to Newtonian theory are not the only cause of shift in the...

## Coordinates And Schwarzschild Coordinates

Kruskal-Szekeres coordinates reveal that Schwarzschild spacetime has two r 0 singularities and two r - oo exterior regions This is not the only surprise that lies hidden in the Kruskal-Szekeres line element (31.14). Notice also that r > 2A (the region of spacetime far outside the gravitational radius) is given by u2 > v2. Thus there are actually two exterior regions* both In the Kruskal-Szekeres coordinate system, the singularity r 0 is located at v2 u2 1. Thus there are actually two...

## Realistic Gravitational Collapse An Overview

Review of spherical collapse Instability, implosion, horizon, and singularity these are the key stages in the spherical collapse of any star. Instability The star, having exhausted its nuclear fuel, and having contracted slowly inward, begins to squeeze its pressure-sustaining electrons or photons onto its atomic nuclei this softens the equation of state, which induces an instability see, e.g., 10.15 and 11.4 of Zel'dovich and Novikov (1971) Stage 3 body gets crushed and distended for details ....

## Objection to the Answer a Reply and a Conjecture

One can object that the collapses of both pancake and cylinder can be halted short of their endpoints, especially that of the pancake. As the thickness of the pancake approaches zero, the vertical pull of gravity remains finite, but the pressure gradient caused by any finite pressure goes to infinity. Hence, pressure halts the collapse. Subsequently the rim of the pancake contracts toward the relativ-istic regime 6 2m < 2M. In the collapse of a cylinder according to Newtonian theory, with a...

## Tensors For A Perturbed Metric

In a specific coordinate frame of an arbitrary spacetime, write the metric coefficients in covariant representation in the form (At the end of the calculation, one can split h into two parts, h *- h , + and out of this split obtain the formulas used in the text.) Assume that the typical components of h are much less than those of so one can expand Christoffel symbols and curvature tensors in hur. Raise and lower indices of h with g > and denote by a covariant derivatives relative to and by a...

## Regge Calculus

Gravitation theory is entering an era when situations of greater and greater complexity must be analyzed. Before about 1965 the problems of central interest could mostly be handled by idealizations of special symmetry or special simplicity or both. The Schwarzschild geometry and its generalizations, the Friedmann cosmology and its generalizations, the joining together of the Schwarzschild geometry and the Friedmann geometry to describe the collapse of a bounded collection of matter, the...

## XiPa j[2 p fX 7a

The variational principle gives Hamilton's equations for the rates of change From these equations, one discovers that X itself must be a constant, independent of the time-like parameter X. The value of this constant has to be imposed as an initial condition, X 0 (specification of particle mass), thereafter maintained by the Hamiltonian equations themselves. This vanishing of X in no way kills the partial derivatives, that enter Hamilton's equations for the rates of change, Whether derived in...

## Incompatibility Of Gravity

ATTEMPTS TO INCORPORATE GRAVITY INTO SPECIAL RELATIVITY The discussion of special relativity so far has consistently assumed an absence of gravitational fields. Why must gravity be ignored in special relativity This chapter describes the difficulties that gravitational fields cause in the foundations of special relativity. After meeting these difficulties, one can appreciate fully the curved-space-time methods that Einstein introduced to overcome them. Start, then, with what one already...

## R

Box 17.2 SIX ROUTES TO EINSTEIN'S GEOMETRODYNAMIC LAW OF THE EQUALITY OF CURVATURE AND ENERGY DENSITY (EINSTEIN'S FIELD EQUATION) Recommended to the attention of Track-1 readers are only route 1 (automatic conservation of the source, plus correspondence with Newtonian theory) and route 2 (Hilbert's variational principle) and even Track-2 readers are advised to finish the rest of this chapter before they study route 3 (physics on a spacelike slice), route 4 (going from superspace to Einstein's...

## F AVfi

And the conditions under which each applies for the meaning and answer to this exercise, see Lascoux (1968) . Exercise 4.6. THE FIELD OF THE OSCILLATING DIPOLE Verify that the expressions given for the electromagnetic field of an oscillating dipole in equations (4.23) and (4.24) satisfy dF 0 everywhere and d*F 0 everywhere except at the origin. Exercise 4.7. THE 2-FORM MACHINERY TRANSLATED INTO TENSOR MACHINERY This exercise is stated at the end of the legend caption of Figure 4.1. Exercise...

## Y aV34 jPV374

For this super-Hamiltonian both_ pa and p_ are constants of motion, whereas the -Hamiltonian, p 2 represents a simple bounce against a one-dimensional potential wall with the initial and final values of p different only in sign. The behavior of the anisotropy parameters near the singularity thus consists of a simple Kasner step where d i da const., with the dfi da. gt or conditions equivalent by symmetry, satisfied relative to one of the three walls , followed by a...

## Storage And Removal Of Energy From Black Holes [Penrose 1969

When a small object falls down a large hole negligible compared to object's rest mass 2 hole's mass, charge, and angular momentum change by AM E, AQ e, AS L, When an object falls into a black hole, it changes the hole's mass, charge, and intrinsic angular momentum first law of black-hole dynamics Box 33.4 . If the infalling object is large, its fall produces much gravitational and electromagnetic radiation. To calculate the radiation emitted, and the energy and angular momentum it carries away...