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certain flux of Maxwell tPM :::: Is*F ("electric tubes"). (a) Show that the fluxes tPF and tPM depend only on the choice of loop, and not on the choice of the surface S bounded by the loop, if and only if dF :::: d*F :::: 0 (no magnetic charge; no electric charge). Hint: use generalized Stokes theorem, Boxes 4.1 and 4... |
ose the door on several later portions of track two, which lean heavily on material treated in this chapter. §5.2. THREE-DIMENSIONAL VOLUMES AND DEFINITION OF THE STRESS-ENERGY TENSOR The rest of this chapter is Track 2. It depends on no preceding Track-2 material. It is needed as preparation for Chapter 20 (conservati... |
ely spatial: AD = BO = Co = O. Hence, the volume I-form has components ~ i = 0 and Box 5.2 (continued) Al A2 A31 ~ 0 = (OiikAiBiCk = det BI B2 B3 CI C2 C3\ =A • (B X C), in the standard notation of 3-dimensional vector analysis; = + V (V = volume of box) if (A, B, C) are righthand ordered (positive sense of E toward fu... |
(£2.-t- B2)/87T, Tik =J... [_(£i£k + BiBk) + .!. (£2 + B2) 8ik]. TOi = TiO= (£ X B)i/47T, 47T 2 (5.23) Show that the stress tensor does describe a tension (£2 + B2)/87T along the field lines and a pressure (£2 + B2)/87T perpendicular to the field lines, as stated in the text. §5.7. SYMMETRY OF THE STRESS-ENERGY TENSOR ... |
l of Tf.W- into a volume integral of p.a,a: 0= f p a d 3I a = f Tf.W-,adtdxdydz. (J'jf" '"- (5.35) (See Box 5.3 for elementary discussion; Box 5.4 for sophisticated discussion.) If the integral of Tf.W-,a is to vanish, as demanded, for any and every 4-volume 'Y, then Tf.W-,a must itself vanish everywhere in spacetime: ... |
from time-rate of change following the fluid to time-rate of change as measured at a fixed location, finding rate of Change) . h . tIme r 11 fl'd . 10 owmg UI WIt ( or or (rate of Change) . h . = Wit fi xe tIme at + ocatlOn d l ' l ' ve OCIty (Of flUId) . ' . . . f h rate 0 c ange (WIth pOSItIOn) (2) (Latin indices ru... |
NTS AND JACOBIANS (a) Write out explicitly the sum defining d 2S01 in (5.54) (5.55) Thereby establish the formula 2 _ d Sp.v - €p.vlaf31 o(x a , x (3 o(a, b) ) _ ~ da db - 2! €p.vaf3 o(x a , x (3 o(a, b) ) d a db. (Expressions such as these should occur only under integral signs. In this exercise one may either supply ... |
hat he accelerates with himself by means of forces applied to their centers of mass (no torque!). Such a nonrotating latticework has "Fermi-Walker trans ported" basis vectors (§6.5), where u = 4-velocity, and a = duldT = 4-acceleration. (3) There are inertial forces of precisely the same type as are encountered in Newt... |
rthonormal tetrad." The essential idea lends itself to simple illustration for hyperbolic motion, as follows. §6.4. THE TETRAD CARRIED BY A UNIFORMLY ACCELERATED OBSERVER An infinitesimal version of a coordinate system is supplied by a "tetrad," or "moving frame" (Cartan's "repere mobile"), or set of basis vectors eo" ... |
s for e' less than _g-I. y 174 6. ACCELERATED OBSERVERS The coefficients of d~Il' de' in this expansion are not the standard Lorentz metric components. The reason is clear. The ~Il' do not form an inertial coordinate system. However, at the coefficients reduce to the standard form: Therefore these "local coordinates" a... |
81 = 0, where f/J f 1 = -m e tfJ dz'" dz/3 )112 dt.. -11"'/3 dt.. ( dt.., (7.3) Here m = (rest mass) and z"'(t..) = (parametrized world line) for a test particle in the scalar gravitational field f/J. By varying the particle's world line, derive differential equations governing the particle's motion. Write them using ... |
ysically equivalent. Bya specialization of the gauge analogous to the "Lorentz" specializationAOI.a = 0 of electromagnetism (equation 3.58a; exercise 3.17), one imposes the condition h/.La = o. ,a This reduces the field equations (4) to the simple d'Alembertian form (6) (7) (see exercise 18.2). Here and henceforth we s... |
s negligible). On the contrary, it demands that free particles be accelerated relative to the Lorentz frame by the Earth's gravitational field. It is indifferent to the mathematical nature of that field (scalar, vector, tensor, ...), but it does insist that the gravitational accelerations agree with experiment. And, of... |
o a broadscale study of geometry. From this more advantageous viewpoint, one can then face the problem of discussing the relationship between the local inertial coordinate systems appropriate to two nearby regions that have slightly different gravitational fields. There are actually two distinguishable ways in which ge... |
rasping fully the Newtonian-based motivation for the Einstein field equations (Chapter 17); (3) no possibility of understanding fulry the mathematical interrelationships of "geodesic," "parallel transport," "covariant derivative," "cur vature," and "metric" (Chapters 9,10,11,13); (4) no possibility of introducing the m... |
ric symbol + 1 if 0:,8y8 Is' an even permutation of 0123, [0:,8y8]_ -1 if 0:,8y8 is an odd permutation of 0123, 1 o if 0:,8y8 are not all different, and where g is the determinant of the matrix IgafJ " (8.10a) (8.10b) (8.11) Read Box 8.4 for full discussion and proofs; work exercise 8.1 below for fuller understanding a... |
ions reduce to (4) Show, from the transformation law for the metric components, that (detIUit")2 detllga {311 = -I. (5) Combine these results to obtain expressions (8.10).] (b) Show that the components of the permutation tensors [defined by equations (3.50h) (3.50j)] have the same values [equations (3.50k)-(3.50m)] in ... |
face (great circles) are a foil against which one can visualize connection coefficients; see Figure 8.3. The material of this section is presented more deeply and from a different view point in Chapters 10 and 13. The Track-2 reader who plans to study those chapters is advised to ignore the following exercises. The Tra... |
33) holds in this frame, and-as a tensor equation-in all frames. (2) The product rule is also a tensor equation, true immediately via components in a local Lorentz frame. (3) Prove the product rule also the hard way, to see where equation (8.33) enters. Use the chain rule of exercise 8.9 to write Vu(g 0 A ® B) = (Vug) ... |
was still, for them, a rigid, homogeneous something, susceptible of no change or conditions. Only the genius of Riemann, solitary and uncomprehended, had already won its way by the middle of the last century to a new conception of space, in which space was deprived of its rigidity, and in which its power to take part ... |
sely, tangent space at '3'0 depend on how the embedding is done, or depend for their existence on the em bedding process. They do not. And to make clear that they do not is one motivation for defining the directional derivative operator "dldA." to be the tangent vector, rather than using Cartan's more pictorial concept... |
a' = ax f3 laxa'; also, it provides a good way to remember the signs in the A matrices.) §9.4. 1-FORMS When the Lorentz metric is removed from spacetime, one must sharpen up the concept of a I-form u by insisting that it, like any tangent vector u, be attached to a specific event '3'0 in spacetime. The family of surfac... |
237 '3'4 - '3'3 = [u('3'O) + v('3'l)] = [v('3'1) - v('3'O)] = (V f3 uae,,)<? = [U, v]<? + errors. ,a fJ 0 [U('3'2) + v('3'O)] [U('3'2) - u('3'O)] (UP aVae ,,)<? + errors fJ 0 , o 4terms such as vf3 ,l' p ullu Pe ] f3 5. Notice that if u and v are halved everywhere, then [u, v] is cut down by a factor of 4, while the er... |
it is the set of points € of R3 satisfying (~1)2 + (~2)2 + (~3)2 = 1. Clearly a different point '3' of S2 (one ray in R3) intersects each point of this standard 2-sphere surface, and the correspondence is continuous and differentiable in either direction (ray to point; point to ray). The same is true for <:tny ellipsoi... |
others, are summarized in Figure 10.1 and Box 10.1. §10.2. PARALLEL TRANSPORT AND COVARIANT DERIVATIVE: PICTORIAL APPROACH Two test bodies, initially falling through spacetime on parallel, neighboring geodesics, get pushed toward each other or apart by tidal gravitational forces (spacetime curvature). To quantify this ... |
f u = dld'A; ,also sometimes denoted V,J, (for proof, see exercise 10.2.) E. In the real physical world, be it Newtonian or relativistic, parallel transport of a triangle can not break its legs apart: (l) A, S, C initially such that A + S = C; (2) A, S, Ceach paral lel transported with himself by freely falling then A ... |
hood: r"f3Y components of V = V(w", ef3('3'), e y) (w", V eyef3) ~ ("a-component of change in basis vector ef3' when) in evaluating ef3 one moves from tail to tip of e y . - These components of V are called the "connection coefficients" of the basis {e,,}. They are the "coordinate representation" of the covariant deriv... |
sing both sides of the equation in terms of directional derivatives (, e) plus connection-coefficient corrections. Hint: the left side becomes ( S"/3 M Y) Y /3 ;a - (sa/3 M Y) + Fa (SI'/3 M Y) , Y /3 Y /3 I'a .Ct, . t ""sa/3 y,a M flY + S"/3y M/3Y,a 1 L{~y chain rule for directional derivativ':.J y 262 10. GEODESICS. P... |
a/3Y _ ar a/38 - - - - ax8 axY + r a/Ly r /L/38 - r a/L8 r /L/3Y' (3) This curvature tensor not only quantifies the concept of "tidal gravitational force," but also enters into Einstein's law, by which "matter tells spacetime how to curve." That law, to be studied in later chapters, takes the following operational com... |
ector parameter." i.e., 2. Equation of motion for each trajectory: 2. Geodesic equation for each trajectory: aZx i atZ n ) + al/J. = 0 ax' ' ( where l/J is Newtonian potential. 3. Take difference between equations of motion for neighboring trajectories, nand n + LIn, and take limit as LIn -+ O-i.e., take derivative (~)... |
ossible experiment. For planning an abstract and coordinate-free calculation (the present line of action), introduce a "fiducial field," only to take it away at the end of the calculation. Plan: Conceive of A, not as a localized vector defined solely at the start of the trip, but as a vector field (defined throughout t... |
(for routes obtainable one from the other by any contiimous sequence of deformations). As for !!!., so for all points of the manifold; and as for the one base vector e JL , so for a complete set of base vectors (p. == 0, 1,2,3): Parallel transport of a basis {ea(g'o)} yields everywhere a field of frames ("frame field")... |
CHI Show that the Riemann curvature tensor satisfies the following "Bianchi identities" IDENTITIES Ra p[y8;<l = O. (l1.38) The geometric meaning of these identities will be discussed in Chapter 15, [Hint: Perform the calculation at the origin of a Riemann normal coordinate system,) y 288 11. GEODESIC DEVIATION AND SPAC... |
- dx!3 nY dx 8 _ Ri d"A. .!!!...- nk .!!i. - Ri OkO d"A. d"A. 40 unless y is space index] for y a space index: 0 unless f3 = /) = 0) OkO l nk _ o2l/J - oxi oxk nk Resultant equation of geodesic deviation: agrees with result nO = 0 always, which followed from choice "A. = t for all particles agrees with Newton-type cal... |
's. yes: spacetime metric gu," no: r i - 21' ('-123)' 00 - - - . 1_ t • • 2x' all other r a#,. vanish r a u,.'s have no independent existence; all derived from ra = gaP ~ (. 2gB> ax# 2 #> + agB# ax" _ 2g#» 2x P ("metric theory of gravity") 298 12. NEWTONIAN GRAVITY IN LANGUAGE OF CURVED SPACETIME EXERCISES x' ________ ... |
al, a result that conflicts with the symmetries of the Riemann tensor [eq. (8.45)) in a manifold with compatible metric and covariant derivative. §12.5. THE GEOMETRIC VIEW OF PHYSICS: A CRITIQUE An important digression is in order. "Every physical quantity must be describable by a (coordinate-free) geometric object, an... |
(from point 1) and (2m) (from point 2) for each point m close to 0 (m = 3,4, ... , N + 2) to be able to work out 'If the distance (03) is given arbitrarily, the resulting four-vertex figure will burst out of the plane. Regarded as a tetrahedron in a three-dimensional Euclidean space. it has a volume given by the formu... |
(v' v > 0), then spacetime is locally Lorentz. Otherwise it is not. Metric must be locally Lorentz y EXERCISES 312 13. RIEMANNIAN GEOMETRY: METRIC AS FOUNDATION OF ALL Exercise 13.1. TEST WHETHER SPACETIME IS LOCAL LORENTZ Prove that the above two-step procedure for testing whether spacetime is locally Lorentz is valid... |
cetime length in curved 7' = f 'N 'N d7' = f' (-'I)p.v dxp. dx v)1/2 {/ {/ a maxim urn for straight line = as compared to any variant of ) . (13.24 ) ( the straight line y 316 13. RIEMANNIAN GEOMETRY: METRIC AS FOUNDATION OF ALL Such a test for straightness can be carried out separately in each local Lorentz region alo... |
rough all those points (in path space) for which the corresponding world lines (in spacetime) r.ack up the indicated lapse of proper time 'T. Foregoing description is classicaL according to quantum mechanics, all the timelike world lines connecting t! and !l3 occur with the same probability amplitude ("principle of dem... |
iemann* (analog of Maxwell The curvature tensor 6 *Faraday), which has components L'af3 = 1- af3p. PR pCl 12 f ya - p.p U 2 fpClya - - _ 1-saf3p.p R pCl 4 pClya p.y (13.46) (exercise 13.11). (2) The Einstein curvature tensor, which is symmetric (exercise 13.11) Einstein tensor (13.47) (3) The Ricci curvature tensor, wh... |
line. The geodesic through it originated on the observer's world line at a specific time T, had original direction n = njej, and needed to extend a distance s before reaching ':P. Hence, the four numbers (13.65) are a natural way of identifying the event 9. These are the coordinates of ,:P in the observer's proper ref... |
mputer (see Box 14.3). Exercise 14.1. CURVATURE OF A TWO-DIMENSIONAL HYPERBOLOID Compute the curvature of the hyperboloid 12 - x 2 - y2 == jZ == const in 2 + 1 Minkowski 2 == _d12 + dx2 + dy 2. First show that intervals within this two-dimen spacetime with dS3 sional surface can be expressed in.the form ds 2 == P(da2 +... |
and v. The factors that multiply R a{3/l' are (1) the component of the vector A in the 11th direction and (2,3) the Il" component of the extension of the parallelogram, (u/lVV - u'vJI-). Thus Box 14.2 STRAIGHTFORWARD CURVATURE COMPUTATION (IIIuatflited for • Globel The elementary and universally applicable method for ... |
e world of mathematics and physics. In contrast, the method of curvature 2-forms is efficient, but demands a heavier investment in the mathematics of I-forms and 2-forms than anyone would normally find needful for any introductory survey ofrelativity. Anyone facing several days' work at computing curvatures, however, w... |
s not new; it is the covariant derivative of the vector v taken in the direction of the vector u. When one abstracts away from any special choice of the direction of differentiation u, one finds an expression that one has encountered before, though not under its new name of "vector-valued I-form." This expression measu... |
nds the symmetry of the covariant derivative. The other principle basic to the forthcoming computa tions is "compatibility of covariant derivative with metric," as expressed in the form d(u' v) = (du) . v + u· (dv). (14.30) It is essential here to ascribe to the metric (the "dot") a vanishing covariant deriva tive; thu... |
n de' = e'il' to deduce the transformation law Rewrite this in decompressed notation for coordinate frames with A'JJ.' = ox' j(: x JJ.' as a formula of the form rJJ.'a'/3' = (?). (14.40) 360 14. CALCULATION OF CURVATURE Exercise 14.11. SPACE IS FLAT IF THE CURVATURE VANISHES (see §11.5) If coordinates exist in which al... |
ram, where they can be inspected in detail. The boundary of the 4-cube is composed of eight oriented hyperfaces, each of them three-dimen sional (top hyperface with extension L1x L1y Liz, for example; a "front" hyperface with extension Lit L1y Liz; etc.) ~Y x +--- ++++ ~z x - - - - - --.---- r--~-+++ 1:= -1/2411 § 15.1... |
. To the lowest relevant order of small quantities one can write (change in A) = - 41)' 41z !il(ev' e z ) A in operator notation; or in coordinate language, -SA" = R"/lv.<at x + 41x)A/l 41)' 41z. §15.2. BIANCHI IDENTITY dtJl = 0 AS A MANIFESTATION OF "BOUNDARY OF BOUNDARY = 0" Bianchi identity, d(>i =0, interpreted in ... |
the starting volume with the final moment, the "contracted double-dual" of Riemann, is so important that it deserves and receives a name of its own, G Einstein; thus This tensor received attention in §§13.5 and 14.2, and also in the examples at the (15.18) § 15.5 CONSERVATION OF MOMENT OF ROTATION FROM "00 = 0" 377 en... |
) This source is conserved (no creation in an elementary spacetime 4-cube). These principles form the background for the probe in this chapter of the Bianchi identities. That is why two otherwise most interesting identities [Allendoerfer and Weil (1943); Chern (1955,1962)] are dropped from attention. One deals with the... |
t the equations p.P;p = 0 for this system reduce to the familiar Newtonian law of mass conservation, and the Newtonian equation of motion for a fluid in a gravitational field: (l6.2c) avi dp Cit = -p axi ' where dldt is the time derivative comoving with the matter . a a d -=-+v'dt - at Oxi ' (l6.3a) (l6.3b) 388 16. EQU... |
n. The result should be equation (16.8).] (b) Rewrite equation (16.8) in the Earth's local Lorentz frame, using the equation RiOko = a2tPlax; ax k for the components of Riemann in terms of the Newtonian gravitational potential. (Newto nian apprOXimation to Einstein theory. Track-2 readers have met this equation in Chap... |
lar system that the clock manufacturer can ignore them. However, a 1973 atomic clock, subjected to the tidal acceler ations near a spacetime singularity, should break the "lock" to its atomic process long before the tidal forces become strong enough to influence the atomic process itself. B. Influence ofthe acceleratio... |
gravity gradiometer was designed and built by Robert M. Forward and his colleagues at Hughes Research Laboratories, Malibu, California. It measures the Riemann curvature of spacetime produced by nearby masses. By flying a more advanced version of such a gradiometer in an airplane above the Earth's surface, one should ... |
he equation Roo = 4'ITp (derived by comparing relative accelerations in the Newton and Einstein theories) and the equation Roo = !Kp (derived directly from the Einstein field equation) can agree only if the proportionality constant K is 8'IT. Thus, the Einstein field equation, describing the generation of curvature by ... |
metry and would thus violate the spirit of the theory. After much anguish, one concludes that the assumption which one might drop with least damage to the beauty and spirit of the theory is assumption (1), that "G" _vanish when spacetime is flat. But even dropping this assumption is painful: (1) although "G" might stil... |
GY GENERATES CURVATURE velocities in the system, relative to its center of mass and also relative to the Newto nian coordinates, must be small compared to the speed of light v ~ 1. (17.15a) As a particle falls from the outer region of the system to the inner region, grav~ty Im<Pl max. [Here <P < 0 is Newton's gravita a... |
; "principle of extremal action"). e. Thus arrive at the Einstein field equation for empty space, GIJ.P = O. (5) f. The continuation of the reasoning leads to the identity Chapter 21, on the variational principle, gives more detail and takes up the additional term that appears on the righthand side of(5) when matter or... |
" translates to the formula (3)R + (Tr K)Z - Tr K2 = 16r.p, (II) §17.5. AXIOMATIZE EINSTEIN'S THEORY] 423 or, in shorthand form, moment of) rotation ( ( intrinsic) ( extrinsiC) (density of ) = curvature + curvature = mass-energy , (12) valid for every spacelike slice through spacetime at any arbitrary point 9. k. All o... |
, for example, Hepp (1969)]. f. Similar divergences appear when one counts up formally the energy associated with other fields and with vacuum fluctuations in number of pairs of electrons, /L-mesons, and other particles in the limit of quantum energies large in com parison with the rest mass of any of these particles. ... |
esics of the "physical metric" g, and get deflected by the gravitational fields ofihe stars. But gravita tional waves propagate along geodesics of a flat background metric '1, and are thus unaffected by the stars. Consequently, Hertz's captains must ma neuver to keep on station; and they hear a chang ing beat pattern b... |
le with special relativity can be aligned into the system of general relativity by means of the absolute differential calculus, without [general relativity] supplying any criterion for the acceptability of that theory." Hilbert (1915): "Axiom I [notation changed to conform to usage in this book]. The I) I 434 17. HOW M... |
only if its generating functions satisfy the sourceless wave equation ~Q./3/3 = O. Exercise 18.3. EXTERNAL FIELD OF A STATIC. SPHERICAL BODY Consider the external gravitational field of a static spherical body, as described in the body's (nearly) Lorentz frame-i.e., in a nearly rectangular coordinate system Ih/lvl ~ I... |
ne would calculate using special relativity (with gp.. = Tlp..)' As a result, the energy-momentum conservation formulated here contains no contributions or effects of gra vity! From this one sees that linearized theory assumes that gravitational forces do no significant work. For example, energy losses due to gravitati... |
e region far outside the system, and expand hp. p in powers of x'/, = x'/Ixl, using the relations _ T (t p.p " Ix - x'l, x') = "" L.. n! n=O 1 [on _ atn T (t p.p ] " x') (, - Ix - x'l)n, (19.3a) , - Ix - x'i = Xi ( 1 xix k (xi'x k x,.) , 2 , + - - - ' - ,2 ,'2 Q'k) , + .. " 1 ---=-+--+--Ix - X'I 1 xix k (3 Xi'x k' ,2 2... |
this, but it is also obvious that this is not the way the mass of the sun is, in fact, determined by astronomers! Theories of stellar structure are adjusted to give the observed mass; they are not constructed to let one deduce the mass from nongravitational observa tions. The mass of the sun is measured in practice by... |
ne the constants AO and Fl to agree with equation (19.13). (9) All linear terms in the metric are now accounted for. The dominant nonlinear terms must be proportional to the square, (M/r)2, of the dominant linear term. The easiest way to get the proportionality constant is to take the Schwarzschild geometry for a fully... |
sense. 462 20. CONSERVATION LAWS FOR 4-MOMENTUM AND ANGULAR MOMENTUM Linearized field equations in terms of HI",ull In terms of H Wfl'{3, the linearized field equations (18.7) take on the much simplified form Gaussian flux integrals in linearized theory: (1) for 4-momentum (20.5) and from these, by antisymmetry of HIl... |
-ifttegrats fofPTL anClJ1l" will transform like special relativistic tensors, and that under infinitesimal coordinate transformations (gauge changes) they will be invariant. Because t/l Vare not tensor components, they can vanish at a point in one coordinate system but not in another. The resultant ambiguity in defini... |
die out as 1/r2 or 1/r3 at large r can contribute to the flux integrals (20.25), (20.26). For const. + O(1/r)]. teL dies out as l/r-l. Hence, the only contri static solutions [!Ip. p butions come from dynamic parts of the metric. which, at these large distances, are entirely in the form of gravitational waves. The stu... |
s velocity negligibly compared to the speed of light during the light-travel time across itself see, e.g., Burke (1970)] (20.42) Every acceptable line of reasoning has always led to expression (20.42). It also represents the field required to reproduce the long-known and thoroughly tested law of radiation damping. (2) ... |
correction terms which may be symbolized as Coupling of spin to curvature 478 and 20. CONSERVATION LAWS FOR 4-MOMENTUM AND ANGULAR MOMENTUM 8(metric) - (Slr 2 ) (spherical harmonic of order one) (20.46) 8(metric)- (f/r 3 ) (spherical harmonic-of order two) (20.47) and higher-order terms. Here S(cm2) is a typical compo... |
~, is invariant with respect to general coordinate transformations. (Hint Use the theorem that the determinant of the product of three matrices is equal to the product of the determinants of those three matrices.) (i) Show that this determinant has the value - (E· B)2 by evaluating it in the special local inertial fram... |
of spacetime. In effect, it slices spacetime into a great number of spacelike slices. It finds it most convenient (§21.4) to do separate bookkeeping on (I) the 3-geometry of the individual slices and (2) the relation between one such slice and the next, as expressed in a "lapse function" N and a 3-vector "shift func ti... |
cale of lengths at that point. This breakdown of the 3-geometry into two parts provides a particularly simple way to deal with two special initial-value problems known as the time-symmetric and time-antisymmetric initial-value problems (§21.1 0). The ADM formalism is today in course of development as summarized in §21.... |
of variation principle. The variation of (I), as defined and calculated in this new way, becomes OJ =p ox I :r",t" :r',t' + f :r',t' :r",t" [( i - OH) 'ap ( op + - p - - 'aH) J ox dt. ox (2) Demand that the coefficient of op vanish and have the sought-for new version, , 'aH(p, x, t) x = ---"--- op Box 21.1 (continued) ... |
... , upon which geometry is founded, belongs to mechanics. Geometry does not teach us to draw these lines, but requires them to be drawn." Newton's remark is also a question. Mechanics moves a particle along a straight line, but what is the machinery by which mechanics accomplishes this miracle? The quantum principle... |
he change in each element of that determinant by its cofactor and adding the resulting products (exercise 5.5) prove that c'J( - g)1/ 2 =t (- g)1/2g !'v c'Jg!'v and c'J( - g)1/2 = - t (- g)1!2g!,v c'Jg!'v. Also show that g = det 11!I!'v" and Action unaffected by mere change in coordinatization EXERCISE 504 21. VARIATIO... |
confusion. Comparing (21.41) and (21.40), one arrives at the following construction of the 4-metric out of the Details of the 4-geometry 3-metric and the lapse and shift functions [Arnowitt, Deser, and Misner (1962)]: (21.42) The welded connectors do the job! In (21.42), the quantities Nm are the components of the shi... |
he tips of the normals in Figure 21.3 are closer than their bases, as they are, for example, during the recontraction of a model universe, in agreement with the conventions employed by Eisenhart (1926), Schouten (1954), and Arnowitt, Deser and Misner (1962), but opposite to the convention of Israel (1966). Into the slo... |
akes work. It demands that one evaluate the remaining type of object, !Jl(ej, n)e j • One step towards that calculation will be found in exercise 21.7. Sachs does the calculation (1964, equation 10) but only after specializing to Gaussian normal coordinates. These coordinates presuppose a very special slicing of spacet... |
Lie derivative as defined in exercise 21.8. Second, the divergence (21.85), which has to be added to the Lagrangian of (21.86) to obtain the full Hilbert Lagrangian, is -2[( -(4)g)l/2(n a'Tr K + aa')l,a., where the coordinates are general (see exercise 21.10), and a a' - n a' - ;f3' nf3' (21.88) (21.89) is the 4-accele... |
equations, one obtains a term at limits L [,"i 8Ai d 3x - f [,"i 8Ai d 3x. (21.111 ) """initIal I final One demands that both these terms at limits must vanish in order to have a well defined variational problem. Go from the given vector potential to another vector potential, A inew ' by the gauge transformation (21.11... |
TIONAL PRINCIPLE AND INITIAL-VALUE DATA F t)r-------rE ~ c,---D=4) Figure 21.4. Some of the many ways to make distinct spacelike slices through one and the same (4)~, the complete Schwarzschild 4-geometry. functions N, ~, means (a) an altered laying down of coordinates in spacetime, and therefore (b) altered results fo... |
solution is guaranteed by the circumstance that (3)~ is a Riemannian geometry. However, one could have started with different coordinates and ended up with different metric coefficients for the description of the same 3-geometry. No matter. Pick one set of coordinates, take the resulting metric co efficients, and stick... |
IME-SYMMETRIC AND TIME-ANTISYMMETRIC INITIAL-VALUE PROBLEMS Turn from the general initial-value problem to two special initial-value problems that lend themselves to detailed treatment, one known as the time-symmetric ini tial-value problem, the other as the time-antisymmetric problem. A 4-geometry is said to be time-s... |
.11. YORK'S '"HANDLES" TO SPECIFY A 4-GEOMETRY 539 §21.11. YORK'S "HANDLES" TO SPECIFY A 4-GEOMETRY On a simultaneity-or on the simultaneity-of extremal proper volume, give the conformal part of the 3-geometry and give the two inequivalent components of the dynamically conjugate momentum in order (l) to have freely spe... |
for an entire conformal equivalence class of metrics; that is, for a given -" (21.151) Unique solution for conformal factor no matter how different the gab and If; themselves may be. The conformal 3-geometry and the "momentum density of weight 5/3" once the remaining initial-value equation (21.116) then becomes the "s... |
ative to the far-away stars. If "mass there governs inertia here," as envisaged by Mach, how can this be? Enlarge the question. By the democratic principle that equal masses are created equal, the mass of the earth must come into the bookkeeping of the Foucault pendulum. Its plane of rotation must be dragged around wit... |
st vanish identically for arbitrary choice of the ~1(X, y, z), which measure the equivalent of the sliding of a ruled transparent rubber sheet over an automobile fender. §21.13. JUNCTION CONDITIONS 551 Exercise 21.24. THE EXTREMAL ACTION ASSOCIATED WITH THE HILBERT ACTION PRINCIPLE DEPENDS ON CONFORMAL 3-GEOMETRY AND E... |
t and children of darkness is the vision of physics that emerges from this chapter, as from other branches of physics. The children of light are the differential equations that predict the future from the present. The children of darkness are the factors that fix these initial conditions. Exercise 21.25. EQUATION OF MO... |
y conservation: Second law of thermodynamics Shock waves and heat flow First law of thermodynamics d( energy in a volume element containing) a fixed number. A, of baryons ) ) = - p vO ume + U' en ropy ; T J( d( I t . 560 i.e., Le., 22. THERMODYNAMICS. HYDRODYNAMICS, . . ., AND KINETIC THEORY d(pA/n) = -pd(A/n) + Td(As)... |
er Euler equations (p + p)Vuu = -(g + u ® u)' Vp, (7), (8), (9) which determine the flow lines to which u is tangent. Normalization of 4-velocity which can be integrated to give p(n, s). u·u=-1. (10) (See exercise 22.4.) Contracting P with V . T = 0 [equation (22.10)] gives (3) Euler equation (p + p) Vuu = _po (Vp) -[V... |
relativity) that the Maxwell equations (22.17b) are satisfied. Indeed, it does, as one sees in exercise 22.8. To derive the wave equation that governs the vector potential, insert expression (22.19a) into the remaining Maxwell equations (22.17a), obtaining (22.19b) then commute covariant derivatives in the first term u... |
larization vector, if not familiar from electrodynamics, can be developed later in Exercise 22.12. So much for the foundations. Now for the calculations. First insert the geometric optics vector potential (22.25) into the Lorentz gauge condition: o = AIL;IL = 1\ {[~ k/a lL + eblL + ... ) + (aIL + eblL + ... );ILJ eI8/e... |
vectors ("wave vectors")k: The rays both lie in and are perpendicular to surfaces of constant phase, 0 ::: const.; and their tangent vectors are the gradient of 0: k ::: VO. In a local Lorentz frame, kO is the "angular frequency" and kO/2'77 is the ordinary frequency of the waves, and is a unit 3-vector pointing along ... |
ude that 1/2 (fOr any bundle of rays, all in the same) d 2 ~2 :::;; 0 surfac~ of constant phase, anywhere in (22.40) spacetIme (focusing theorem). This theorem plays a crucial role in black-hole physics (§34.5) and in the theory of singularities (§34.6). §22.6. KINETIC THEORY IN CURVED SPACETIME* The stars in a galaxy ... |
cles). (5) Hence, at any A along the initial particle's world line, the particle is in a phase-space region of unchanged volume 'V, unchanged number of particles N, and unchanged ratio 9l = NI'V: d91[9(A), p(A)] _ 0 - . dA (22.47) Collisionless Boltzmann equation (kinetic equation) This equation for the conservation of... |
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