## Gravitational waves The TT gauge

In this and the following two sections we develop some of the basic facts about gravitational radiation. We have, of course, already met an exact gravitational wave in Chapter 13, but that came somewhat out of the blue, and allowed few general conclusions. The systematic approach we shall follow here is the basis of most of the experimental work on wave detection currently in progress. Let us consider an arbitrary plane wave propagating in the x-direction, c,v cZv(u), u t - x. (15.35) This...

## Time dilation The twin paradox

Let us again consider two inertial frames S and S' in standard configuration. Let a standard clock be fixed in S' and consider two events at that clock when it indicates times differing by At'. We enquire what time interval At is ascribed to these events in S. From the A-form of (2.9)(iv) we see at once, since Ax' 0, that At y At', or, replacing At and At' by the more specific symbols T and T0, respectively, T YT0 (1 T c2)1 2 . (3.2) We can deduce from this quite generally that a clock moving...

## De Sitter precession via rotating coordinates

De Sitter precession (also known as geodetic precession) is a general-relativistic phenomenon without classical equivalent. Its mathematical existence was discovered as early as 1916 by de Sitter, who found that a gyroscope in free (and therefore 'geodetic') orbit around a massive body will precess. We shall here re-derive this result (for the special case of circular orbits) by the method of rotating coordinates that we already used in Section 9.7 to derive the Thomas precession. (This method...

## Plane waves

In this section we shall make the acquaintance of a third fundamental kinematical 4-vector, the wave 4-vector L. While U and A refer to the motion of a particle, L refers to the motion of a wave. Consider a series of plane 'disturbances' or 'wavecrests', a wavelength X apart, and progressing in a unit direction n (l,m,n) at speed w relative to an inertial frame S. A single stationary plane in S with normal n and at distance p from a point P0 (x0, y0, zo) satisfies the equation n r p, or, in...

## Preview of general relativity

Recall Einstein's philosophical objection to Newton's extended inertial frames they are absolute structures that act but cannot be acted upon. In his equivalence principle Einstein saw a way to rid physics of these objectionable pre-existing structures. For special relativity he had still needed them in order to specify the arena of validity of the theory. But the EP changed all that. No longer is there a need to assume any absolute structure. What undoubtedly exists in the physical world is...

## The coordinate lattice

Loosely speaking, a gravitational field is stationary if it does not change in time, and it is static if, additionally, 'there is no rotation'. Examples are the following the fields in a uniformly accelerating rocket or on a uniformly rotating disk in Minkowski space (these are a little unusual in having no sources) the fields of an arbitrary massive body at rest, or of an axially symmetric body rotating uniformly the field of a gravitating fluid flowing uniformly around an arbitrarily shaped...

## The energy tensor of the electromagnetic field

In this final section we give a brief account of Minkowski's energy tensor, and show how the electromagnetic field itself possesses energy, momentum, and stress just like a material continuum. In this way the well-validated conservation laws of mechanics can be extended to interactions between charged matter and electromagnetic fields. For example, if we simultaneously release two oppositely charged particles from rest and they accelerate towards each other, where does the kinetic energy come...

## A plan for general relativity

We have already given a preview of general relativity in Chapter 1 (cf. Sections 1.13 1.16), although at that stage it was necessarily rather vague. The reader may nevertheless wish to look at it again from our present vantage point. We are now in a much stronger position to sketch out the theory that must result (almost inevitably) from Einstein's equivalence principle. It is not yet full GR. That only results when Einstein's field equations are added. The field equations, however, involve the...

## The anthropic principle

It would be reassuring if one could believe that our universe (which seems destined to expand indefinitely and suffer a freezing death) is not the sum-total of all physical existence. A philosophically persuasive line of argumentation has, in fact, led some modern cosmologists to posit the existence of infinitely many alternative universes, all with possibly different physical constants and different initial conditions. For them that seems to be the only scientific way out of a profound puzzle...

## The axioms of the new mechanics

The mechanics for which we now seek a Lorentz-invariant substitute is the non-gravitational part of Newton's mechanics that is to say, that which is covered by Newton's basic three laws (cf. Section 1.2) and which primarily concerns itself with particle collisions, particle systems and particles in external fields. We specifically assume the absence of gravity since, according to GR, gravity distorts the Minkowskian spacetime of SR which here plays a key role. Though there are many approaches...

## At last the full field equations

In Section 10.6 we saw reasons why the GR analog of Laplace's equation Y 0 is Rzv 0. What, then, is the analog of Poisson's equation J2 ,ii 4nGp In Newtonian gravity the only source is the mass density p the instantaneous field is the same whether the mass moves or not, or whether it is squeezed or not. Already in Maxwell's theory the motion of a charge makes a difference to the field. And now it looks as though in GR even the state of stress of the matter will affect the field, since only Tzv...

## Rigid motion and the uniformly accelerated rod

Look again at Fig. 3.3 and consider, in fact, the equation for a continuous range of positive values of the parameter X. For each fixed X it represents a particle moving with constant proper acceleration c2 X in the x -direction. Altogether it represents, as we shall presently show, a 'rigidly moving' rod. By the rigid motion of a body one understands a motion during which every small volume element of the body shrinks always in the direction of its motion in inverse proportion to its...

## Schwarzschild black holes

We have already on several occasions noted the special role that the locus r 2m (the Schwarzschild horizon) plays in the Schwarzschild metric. It is now time to study it systematically. In full units cf. (11.16) the Schwarzschild radius r for a spherical mass M of uniform density p and radius r is given by 2GM 8n G 3 r --pr , (12.1) c2 3 c2 if we ignore curvature corrections. So we have which shows that, for any constant density p, however small, we can have - > r if only r is large enough...

## Antide Sitter space

The cosmological significance of the spacetime (14.26) with A < 0 (known as anti-de Sitter space) is distinctly inferior but still not negligible. This spacetime has no coordinate singularity and no horizon it is globally static and it is maximal (that is, inextensible). And in its simplest topological form it possesses, as we shall see, an interesting feature shared with a few other GR universes closed timelike lines, which allow one to travel into one's past. To study it, we again...

## Caveats on the equivalence principle

Consider the following notorious paradox an electric charge is at rest on the surface of the earth. By conservation of energy (or just by common sense ), it will not radiate. And yet, relative to an imagined freely falling cabin around it, that charge is accelerating. But charges that accelerate relative to an IF radiate. Why doesn't ours Again, consider a charge that is fixed inside an earth-orbiting space capsule. Now, circularly moving charges do radiate, and one cannot imagine how the...

## The uniformly accelerated lattice

In preparation for the discussion of Kruskal space, we return to special relativity and the uniformly accelerated rocket of Section 3.8. Looking at Minkowski space, M4, from the lattice of such a rocket will shed much light on the relation between Schwarzschild and Kruskal space. Recall that the motion of the entire rocket was characterized by eqn (3.19), (now written with c 1), each point of the rocket corresponding to some fixed value of the parameter X > 0, and having constant proper...

## The Friedman RobertsonWalker metric

To Friedman belongs the great distinction of having been the first (in 1922) to contemplate and analyze a dynamic universe that moves under its own gravity. This was one of the few momentous leaps forward against received opinion that was in the air and yet was missed by Einstein. It was also completely ignored for several years. Although Friedman's derivation was to some extent not quite rigorous, the metric he found lives on and well deserves to be known by his name. It is, however, often...

## The Riemann curvature tensor

Let us introduce the following notation for the repeated covariant derivative of a tensor and similarly for higher derivatives. (This parallels the notation we have already introduced in Section 7.2E for repeated partial derivatives T' ' ' i .) From elementary calculus we are familiar with the commutativity of partial derivatives T''' T''' But the corresponding statement for covariant derivatives is generally false. Only in flat space is it true that For then we can always choose...

## Graphical representation of the Lorentz transformation

In this section we concern ourselves solely with the transformation behavior of x and t under standard LTs, ignoring y and z which in any case are unchanged. What chiefly distinguishes the LT from the classical GT is the fact that space and time coordinates both transform, and, moreover, transform partly into each other they get 'mixed,' rather as do x and y under a rotation of axes in the Cartesian x, y plane. We have already remarked on the formal similarity of the invariance of (2.15), which...

## Particle orbits in Schwarzschild space

The variety of qualitatively different orbits in Schwarzschild space is much larger than in the case of a Newtonian central field, where all the orbits are conics. In Schwarzschild space, for example, there are spiraling orbits. While a detailed discussion of all possible orbits is beyond our present scope, we shall nevertheless provide an overview. Recall that the 'plane' 0 n 2 (and every other such central plane obtainable from this one by a rotation) is a symmetry surface. (The metric is...

## The cosmological frequency shift

The redshift is the primary piece of information we can gather from the galaxies in the universe. We shall now derive the relevant formula as a first example on the use of the 'rubber' models. The cosmological frequency shift is traditionally denoted by 1 + z, and it is given by 1 + z 1 + (17.3) where Xe and X0 Xe + AX are the wavelengths at emission and reception, respectively, of light emitted and received at cosmic times te and t0, respectively. Now for the proof If two beetles crawl in...

## Fourforce and threeforce

The only influences on the motion of particles that we have so far considered were collisions, and we have come quite a long way without recourse to the concept of force. But that, too, plays an important role in relativistic mechanics, as when fast charged particles move through an electromagnetic field. Even without such practical need it would be desirable to have a relativistic version of the force concept, so that relativistic mechanics might 'contain' all of Newtonian mechanics in a...

## Ggay gay gya

GR is thus a field theory with a tensor potential. That is why its presumed quantum, the graviton, would have spin 2. One enormous difference from the other two theories is that GR is non-linear though its field equations are linear in the second derivatives of the gz , as is seen from eqn (10.61), they are non-linear in the first derivatives. This not only greatly complicates their mathematical solution, but it also robs us of the possibility of 'adding' solutions. One reason for the...

## De Sitter space

As a second application of the metric (14.22) we specialize it to the case m 0, when it is called the de Sitter metric ds2 (1 i Ar2) dt2 (1 1 Ar2) 1dr2 r2(d02 + sin2 0 dp2). (14.26) This represents the unique spherically symmetric spacetime that satisfies the modified vacuum field equations and has a center (r 0) where it is regular. Since we expect the undisturbed vacuum to have the maximal symmetry of a spacetime of constant curvature 1A cf. after (14.19) , we expect the metric (14.26) to...

## Maxwells equations in tensor form

Having prepared the mathematical groundwork, we are now ready to discuss electro-magnetism. Any such discussion must inevitably begin with a choice of units. The currently favored SI system (Systeme International) is extremely inconvenient in relativity, where it masks the inherent symmetry between the electric and the magnetic fields. Accordingly we choose the older Gaussian or cgs (centimeter-gram-second) system of units in which the charge q is defined so that the Coulomb force is q1q2 r2...

## Gravitational frequency shift and light bending

Einstein's EP leads directly that is, without the field equations of GR and also without the photon concept to two interesting predictions about the behavior of light in the presence of gravity. The first is that, as light climbs up a gravitational gradient, its frequency decreases and the other, that light is deflected 'ballistically' in a parallel gravitational field. Of course, both these effects are intuitive once we know that light consists of photons We 'only' need to know the...

## Objections to absolute space Machs principle

Newton's concept of an absolute space has never lacked critics. From Huyghens and Leibniz and Bishop Berkeley, all near-contemporaries of Newton, to Mach in the nineteenth century and Einstein in the twentieth, cogent arguments have been brought against AS. There are two main objections i Absolute space cannot be distinguished by any intrinsic properties from all the other inertial frames. Differences that do not manifest themselves observationally should not be posited theoretically. ii 'It...

## Black Hole Thermodynamics and Related Topics

In this section we report, in a purely anecdotal way, some of the later developments in black hole theory, of which the reader should at least be aware.1 A kind of revolution began in theoretical general relativity in 1965 with the introduction of global topological techniques by Penrose, and it continued later with the introduction of quantum-mechanical methods by Hawking in 1974. Already in 1963, general relativity had been enriched by Kerr's discovery of a new exact stationary vacuum...

## From the mechanics of the field to the mechanics of material continua

It was Maxwell, Poynting, Heaviside, and J. J. Thomson who, elaborating on earlier ideas of Faraday and W. Thomson, found the mathematical expressions for the 'mechanical' characteristics of the electromagnetic field, namely its conserved energy and momentum, and its stress. Then, after the advent of special relativity, Minkowski in 1908 discovered that these mechanical field quantities combine to form a single 4-tensor and one, moreover, whose divergence links the field to any charged fluid...

## R1

For a moving point-charge q, where the brackets indicate that the enclosed expressions are to be evaluated at the retarded event at the charge r is the distance of that event from the observer and ur the radial velocity of the charge away from the observer. Hint assume that the charge is moving uniformly and 'spot' that the Coulomb potential in its rest-frame satisfies the tensor equation where U is the 4-velocity of the charge and Rv the null connecting vector from the retarded event at q to...