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Proposition 1.1.3. Let \( W \subset V \) be a subspace. (a) \( W \) timelike \( \Leftrightarrow {W}^{ \bot } \) spacelike, and \( W \) spacelike \( \Leftrightarrow {W}^{ \bot } \) timelike. (b) \( W \) lightlike \( \Leftrightarrow W \cap {W}^{ \bot } \neq \{ \) the zero vector \( \} \Leftrightarrow {W}^{ \bot } \) ligh... | Proof. The notation \( {W}^{ \bot } \) is defined in Exercise 0.0.11, and we will make use of this exercise without further comments in the following.\n\nNow, \( W \) timelike \( \Rightarrow W \) contains a timelike vector \( \Rightarrow W \) contains a unit timelike vector \( w \Rightarrow \) there exists an orthonorm... | Yes |
Corollary 1.1.5. Two lightlike vectors are orthogonal iff they are proportional. | Proof. Let \( v, w \in V \) be lightlike, \( e \in V \) be timelike, and suppose \( g\left( {v, w}\right) = 0 \) . Then \( g\left( {e, v}\right) \neq 0 \) by Corollary 1.1.4, so that for some \( a \in \mathbb{R}g\left( {e, w + {av}}\right) = 0 \) . Then \( w + {av} \) is spacelike, again by Corollary 1.1.4. But \( g\le... | Yes |
Proposition 1.1.7. The lightcone \( {\mathcal{L}}_{0} \) is a lightlike submanifold. | Proof. Suppose \( v \in {\mathcal{L}}_{0} \) ; then \( g\left( {v, v}\right) = 0 \) and \( v \neq 0 \) . Let \( \mathcal{U} \) be a neighborhood of \( v \) that does not contain the origin and define \( \widetilde{g} : \mathcal{U} \rightarrow \mathbb{R} \) by \( \widetilde{g}w = g\left( {w, w}\right) \forall w \in \mat... | Yes |
Proposition 1.2.1. The set \( \mathcal{T} \subset {TM} \) of timelike points is an open submanifold. \( \mathcal{T} \) has either one (connected) component or two. | Proof. Define \( K : {TM} \rightarrow \mathbb{R} \) by \( K\left( {x, X}\right) = g\left( {X, X}\right) \) . Then \( K \) is \( {C}^{\infty } \) . As the complete inverse image of \( \left( {-\infty ,0}\right) \) under \( K,\mathcal{T} \) is open. Let \( \mathcal{A} \) be a component of \( \mathcal{T} \) . If \( \psi :... | Yes |
Proposition 1.3.2. Suppose \( \left( {N, h}\right) \) contains \( \left( {M, g}\right) \) but, \( \forall \) lightlike geodesic \( \lambda : \mathcal{E} \rightarrow N \) such that \( \left( {\lambda \mathcal{E}}\right) \cap M \neq \phi ,\lambda \mathcal{E} \subset M \) . Then \( M = N \) . | Roughly, the proposition says that a spacetime is maximal iff one cannot see into it or out of it. Like many other results, it indicates the key role played by lightlike geodesics. The proof uses techniques more advanced than have been discussed here. The idea is to assume a point \( p \) on the boundary of \( M \) and... | No |
Let \( \left( {{\mathcal{S}}^{2}, h,\zeta }\right) \) be the unit 2-sphere (Section 0.0.9). Let \( \mu \in \left( {0,\infty }\right) \) be given. Define \( \mathcal{A} \subset {\mathbb{R}}^{2} \) by \( \mathcal{A} = {\left( {u}^{1}\right) }^{-1}\left\lbrack {\left( {0,{2\mu }}\right) \cup \left( {{2\mu },\infty }\right... | Now define \( N \subset M \) by \( N = {r}^{-1}\left( {{2\mu },\infty }\right) .N \) is connected, and \( {\left. g\right| }_{N} \) is a Lorentzian metric. Relative to \( {\left. g\right| }_{N},{\left. \left( \partial /\partial t\right) \right| }_{N} \) is timelike. When \( \left( {N,{\left. g\right| }_{N}}\right) \) i... | Yes |
Corollary 1.4.5. Let \( Z \) be a future-pointing unit timelike vector field on Einstein-de Sitter spacetime which is an eigenvector of \( \mathbf{G} \) in the following sense: \( G\left( {Z, \cdot }\right) = {fg}\left( {Z, \cdot }\right) \) for some function \( f \) . Then \( Z = {\partial }_{4} \) . | Proof. Suppose \( G\left( {Z, \cdot }\right) = {fg}\left( {Z, \cdot }\right) \) . At each point, \( d{u}^{4}\left( Z\right) d{u}^{4} = {ag}\left( {Z, \cdot }\right) \) for some \( a \in \mathbb{R} \) . \( d{u}^{4}\left( Z\right) \neq 0 \) since both \( d{u}^{4} \) and \( Z \) are timelike (Exercise 1.1.10b). Thus \( d{... | No |
Corollary 1.4.6. Let \( \left( {\widetilde{M},\widetilde{g}}\right) \) be a spacetime and \( \phi : M \rightarrow \widetilde{M} \) an isometry of \( \left( {M, g}\right) \) onto \( \left( {{\phi M},{\left. \widetilde{g}\right| }_{\phi M}}\right) \) . Then \( \left( {\widetilde{M},\widetilde{g}}\right) \) is incomplete.... | Proof. By Section 1.0.3 and the proof of Proposition 1.4.4,\n\n\[ \n{D}_{Z}Z = \mathop{\sum }\limits_{{i = 1}}^{4}Z\left( {{\omega }^{i}Z}\right) {X}_{i} + \mathop{\sum }\limits_{{i, j = 1}}^{4}{\omega }_{j}^{i}\left( Z\right) {\omega }^{j}\left( Z\right) {X}_{i} = \left( {Z1}\right) {X}_{4} + 0 = 0. \n\] \n\nThus each... | No |
Proposition 2.2.1. There is precisely one connection \( F \) over \( \gamma \) such that \( {F}_{\mathbf{Y}}\mathbf{X} = \) \( \left\lbrack {p{\left( {\gamma }^{ * }D\right) }_{\mathbf{Y}}p + q{\left( {\gamma }^{ * }D\right) }_{\mathbf{Y}}q}\right\rbrack \mathbf{X}\forall \) vector field \( \mathbf{Y} \) on \( \mathcal... | Proof. \( {F}_{Y}X \) is \( \mathbb{R} \) -linear in \( X \) and linear (with respect to \( {C}^{\infty } \) functions over \( \gamma \) ) in \( Y \) . Moreover, if \( f \) is a \( {C}^{\infty } \) function over \( \gamma \), then \( {F}_{\mathbf{Y}}\left( {f\mathbf{X}}\right) = f{F}_{\mathbf{Y}}\mathbf{X} + \left( {\m... | Yes |
Proposition 2.2.2. Let \( X, Y \) be vector fields over \( \gamma \) . The Fermi-Walker connection \( F \) over \( \gamma \) satisfies the following:\n\n(a)\n\n\[ \n{F}_{{\gamma }_{ * }}X = {D}_{{\gamma }_{ * }}X + g\left( {{\gamma }_{ * }, X}\right) {A}_{\gamma } - g\left( {{A}_{\gamma }, X}\right) {\gamma }_{ * }, \n... | Since \( \gamma \) is a curve, a standard property of connections tells us that if \( V \in {M}_{vu} \) for some \( u \in \mathcal{E} \), then there is a unique \( F \) -parallel vector field \( V \) over \( \gamma \) such that \( {Vu} = V \) (Bishop-Goldberg 5.7). Thus, if \( {u}_{0} \in \mathcal{E} \) and \( \left\{ ... | No |
Proposition 2.3.3. Let \( Q \) be a geodesic reference frame and let \( W \) be a neighbor of an observer \( \gamma : \mathcal{E} \rightarrow M \) in \( Q \) . Then the 3-acceleration of \( W \) relative to \( \gamma \) satisfies: \( {F}_{{\gamma }_{ \bullet }}{}^{2}W = {\psi W} \), where \( \left( {\psi W}\right) u = ... | Proof. By Exercise \( {2.0.4}{L}_{Q}W = 0 \) . Fix a \( u \in \mathcal{E} \), and let \( \widetilde{W} \) be a vector field defined in some neighborhood of \( {\gamma u} \) such that \( \left\lbrack {\widetilde{W}, Q}\right\rbrack = 0 \) and \( \widetilde{W} \circ \gamma = W \) (Section 2.0.3). Now \( {D}_{Q}{}^{2}\wid... | No |
Proposition 2.3.4. Let \( Q \) be a reference frame and let \( \gamma : \mathcal{E} \rightarrow M \) be an observer in \( \mathbf{Q} \) . The assignment \( X \rightarrow - {D}_{X}\mathbf{Q} \) then defines a linear transformation \( {A}_{Q} : {R}_{u} \rightarrow {R}_{u} \) which assigns to each neighbor of \( \gamma \)... | Proof of 2.3.4. We first show that \( {A}_{Q}{R}_{u} \subset {R}_{u} \) . This is because \( g\left( {{A}_{Q}X, Q}\right) \) \( = - g\left( {{D}_{X}Q, Q}\right) = - \frac{1}{2}{Xg}\left( {Q, Q}\right) = \frac{1}{2}{X1} = 0,\forall X \in {R}_{u} \) . Next let \( \breve{W} \) be a neighbor of \( \gamma \) in \( \mathbf{Q... | Yes |
Proposition 2.3.5. A reference frame is irrotational iff it is locally synchronizable. | Proof. Let \( \mathbf{Q} \) be the reference frame as usual and let \( \omega \) be the 1 -form physically equivalent to \( \mathbf{Q} \) . In a sufficiently small neighborhood, let \( \mathbf{X},\mathbf{Y} \) be vector fields such that \( g\left( {\mathbf{Q},\mathbf{X}}\right) = g\left( {\mathbf{Q},\mathbf{Y}}\right) ... | Yes |
Proposition 2.3.7. Let \( Z \) be the comoving reference frame, \( \gamma \) an observer in \( Z \) , and \( W \) a neighbor of \( \gamma \) in \( \mathbf{Z} \) . Then: (a) \( \mathbf{Z} \) is proper time synchronizable. (b) \( Z \) is geodesic. (c) \( W \) ’s 3-velocity relative to \( \gamma \) is \( \left( {2/\left( ... | Proof. (a) The 1-form physically equivalent to \( Z \) is \( - d{u}^{4} \), so \( {u}^{4} \) is a proper time function for \( \mathbf{Z} \) . (b) was proved in Corollary 1.4.6. (c) Let \( W = \left( {\mathop{\sum }\limits_{{\mu = 1}}^{3}{a}_{\mu }{\partial }_{\mu }}\right) \circ \gamma \) (Exercise 2.3.10a). Then (b) a... | No |
Proposition 3.2.3. There is at most one particle flow \( \left( {\mathbf{P},\eta }\right) \) such that \( \operatorname{div}\left( {\eta \mathbf{P}}\right) \) \( = 0 \) and \( \eta \circ \phi = {\eta }_{0} \) . | Proof. Suppose \( x \in M \) . There is exactly one \( \gamma \) as above such that: (a) \( ⓾ \in {\phi N} \) , and (b) \( \gamma {u}^{1} = x \) for at least one \( {u}^{1} \in \mathcal{E} \) . In fact, \( {u}^{1} \) is then unique: since \( \gamma \) is an integral curve of a vector field, \( \gamma {u}^{1} = \gamma {... | No |
Proposition 3.3.4. Let \( E,{E}^{\prime } \) be symmetric \( \left( {0,2}\right) \) tensor fields on \( M \) and suppose \( \mathbf{E}\left( {Z, Z}\right) = {\mathbf{E}}^{\prime }\left( {Z, Z}\right) \forall \) instantaneous observer \( \left( {z, Z}\right) \) . Then \( \widehat{\mathbf{E}} = {\widehat{\mathbf{E}}}^{\p... | Proof. Fix an instantaneous observer \( \left( {z, Z}\right) \) . \( \forall \) future pointing timelike \( X \in {M}_{z}, E\left( {X, X}\right) = {E}^{\prime }\left( {X, X}\right) \) since \( \left( {z, X/\left| X\right| }\right) \) is an instantaneous observer at \( z \) . Now the set \( {\mathcal{T}}_{x}{}^{ + } \su... | Yes |
Proposition 3.3.6. Let \( \widehat{I} \) be the stress-energy tensor of a particle flow and \( X \) be a causal vector field. Then \( T\left( {X, X}\right) - \frac{1}{2}\left( {\operatorname{trace}T}\right) g\left( {X, X}\right) \geq 0 \) . | Proof. \( T\left( {X, X}\right) - \frac{1}{2}\left( {\operatorname{trace}T}\right) g\left( {X, X}\right) = \eta \left\{ {{\left\lbrack g\left( X, P\right) \right\rbrack }^{2} - \frac{1}{2}g\left( {P, P}\right) g\left( {X, X}\right) }\right\} \) . \( \eta \) is nonnegative by definition and the factor in curly brackets ... | Yes |
Proposition 3.5.2. Let \( \left( {M,\mathcal{M}, F}\right) \) be a relativistic model with \( \mathcal{M} \) a finite collection of particle flows, \( \widehat{T} \) be the stress-energy tensor of \( \mathcal{M} \) . Then \( T\left( {Z, Z}\right) = 0 \) for one instantaneous observer \( \left( {z, Z}\right) \) iff \( \... | Proof. \( \mathbf{T}\left( {Z, Z}\right) = \sum \left( {{\eta }_{A}z}\right) {\left\lbrack \mathbf{g}\left( {\mathbf{P}}_{A}, Z\right) \right\rbrack }^{2} \) . \( \forall A,{\eta }_{A}z \) is nonnegative by definition, and \( {\mathrm{e}}_{A} = - g\left( {{P}_{A}, Z}\right) > 0 \) as usual.\n\nSuppose \( N \neq 0 \) . ... | Yes |
Proposition 3.7.5. \( \left( {M, F, J}\right) \) obeys Maxwell’s equations iff: (a) \( {\int }_{\partial \mathcal{D}}i\left( \widehat{F}\right) \Omega = \) \( {4\pi }{\int }_{\mathcal{D}}i\left( J\right) \Omega \) for every space-section \( \mathcal{D} \) ; and (b) \( {\int }_{\partial \mathcal{D}}F = 0 \) for every sp... | Proof. div \( \widehat{F} = {4\pi J} \) iff \( d\left\lbrack {i\left( \widehat{F}\right) \Omega }\right\rbrack = {4\pi i}\left( J\right) \Omega \) iff (a) holds; \( {dF} = 0 \) iff (b) holds (Sections 3.0.1 and 3.6.2). Moreover, if \( \operatorname{div}\widehat{\mathbf{F}} = {4\pi }\mathbf{J} \) then\n\n\[ \operatornam... | No |
Theorem 3.8.3. (a) There is precisely one inextendible curve \( \gamma : \mathcal{E} \rightarrow M \) such that (1) \( ⓾ = x,\left( 2\right) {\gamma }_{ * }0 = W \), and (3) \( {D}_{y},{\gamma }_{ * } = e\widetilde{F}{\gamma }_{ * } \) . (b) \( \left( {\gamma ,\left| W\right|, e}\right) \) is a particle on \( M \) . (c... | Proof. We first prove (a). Let \( \left\{ {x}^{i}\right\} \) be coordinate functions around \( x \) such that \( {x}^{i}\left( x\right) = 0\forall i \), and let \( W = {W}^{i}\left( {{\partial }_{i}x}\right), F = {F}_{ij}d{x}^{i} \otimes d{x}^{j}\left( {{F}_{ij} = - {F}_{ji}}\right) \) . If \( \gamma \) is the sought-f... | Yes |
Proposition 3.9.2. Suppose \( \left( {M,\mathcal{M}, F}\right) \) obeys the simple matter equations and Maxwell's equation. Then \[ \operatorname{div}\left( {\widehat{\mathbf{T}} + \widehat{\mathbf{E}}}\right) = 0 \] | Proof. \( \operatorname{div}\widehat{E} = - \widetilde{F}J = - \widetilde{F}\sum {e}_{A}{\eta }_{A}{P}_{A} \) (Sections 3.7 and 3.5). \( \operatorname{div}\widehat{T} = \) (by Section 3.5) \( \operatorname{div}\sum {\eta }_{A}{\mathbf{P}}_{A} \otimes {\mathbf{P}}_{A} = \) (by Exercise 3.6.4d) \( \sum \left\{ {\left\lbr... | No |
Proposition 3.10.1. Let \( X \) be a Killing vector field and suppose \( \operatorname{div}\widehat{\mathbf{D}} = 0 \) . Then \( \operatorname{div}\left( {\widetilde{D}X}\right) = 0 \) . | Proof. Since \( \widehat{\mathbf{D}} \) is symmetric, \( {D}^{ij} = {D}^{ji} \) . We have\n\n\[ \operatorname{div}\left( {\widetilde{D}X}\right) = {\left( {D}^{i}{}_{j}{X}^{j}\right) }_{\mid i} = {\left( {D}^{ij}{X}_{j}\right) }_{\mid i} = {D}^{ij}{}_{\mid i}{X}_{j} + {D}^{ij}{X}_{j \mid i} \]\n\n\[ = {D}^{ji}{}_{|i}{X... | No |
Theorem 3.11.1. Given (a)-(e). For \( \mathcal{U} \) sufficiently small, there exists a unique 2-form \( F \) on \( \mathcal{U} \) such that \( F \circ \phi = {F}_{0} \) and \( \left( {\mathcal{U}, F, J}\right) \) obeys Maxwell’s equations. Conversely, if \( \left( {\mathcal{U},\mathbf{F},\mathbf{J}}\right) \) obeys Ma... | Proof of Theorem 3.11.1. Let us first dispose of the second statement. Suppose \( \left( {\mathcal{U},\mathbf{F},\mathbf{J}}\right) \) obeys Maxwell’s equations. Since \( {\mathbf{F}}_{0} = \mathbf{F} \circ \phi \) , \( {\phi }^{ * }{F}_{0} = {\phi }^{ * }F \), where the right side is just the usual pull-back of forms.... | Yes |
Proposition 3.14.1. If a stress-energy \( \widehat{T} \) is normal at \( x \in M \), then \( \widehat{T} \) has a time-like eigenvector which is unique up to nonzero multiples. | Proof. Let \( \mathcal{S} \) be the unit sphere on \( {M}_{x} \) relative to an arbitrary positive definite inner product on \( {M}_{x} \) . Let \( \mathcal{A} \) be the set of nonspacelike one-dimensional subspaces of \( {M}_{x} \) . Then \( \mathcal{A} \cap \mathcal{S} \) has two components, each diffeomorphic to the... | Yes |
Proposition 3.15.3. \( \operatorname{div}\widehat{T} = 0 \) iff: (a) \( \operatorname{div}\left( {\rho Z}\right) = - p\operatorname{div}Z \) ; and (b) \( {D}_{Z}Z = \n- {\left( \rho + p\right) }^{-1}\{ \left( {Zp}\right) Z + \operatorname{grad}p\} \text{.} | Proof. div \( \widehat{T} = 0 \n\n\[ \Leftrightarrow {\left( \rho {Z}^{i}{Z}^{j} + p{g}^{ij} + p{Z}^{i}{Z}^{j}\right) }_{\mid j} = 0, \]\n\n\[ \Leftrightarrow {Z}^{i}{}_{|j}\left( {\rho {Z}^{j}}\right) + {Z}^{i}{\left( \rho {Z}^{j}\right) }_{|j} + {p}_{|j}{g}^{ij} + {p}_{|j}{Z}^{i}{Z}^{j} + p{Z}^{i}{}_{|j}{Z}^{j} + p{Z... | Yes |
Proposition 4.2.1. Suppose \( \left( {M,\mathcal{M}, F}\right) \) obeys Einstein’s equation. \( M \) is Ricci flat iff \( \mathbf{T} = 0 = \mathbf{F} \) . | Proof. Suppose \( \mathbf{T} = 0 = \mathbf{F} \) . Then \( \mathbf{E} = 0 \), so \( \mathbf{G} = 0 \) ; by Exercise 1.4.7, Ric \( = 0 \) . Conversely, suppose \( 0 = \mathbf{G} = \mathbf{T} + \mathbf{E} \) . Suppose \( x \in M \), and let \( \omega \in {M}_{x}{}^{ * } \) be causal. Then \( \widehat{\mathbf{I}}\left( {\... | No |
Proposition 4.2.2. The mean relative-acceleration of a geodesic reference frame \( Q \) equals \( - \left( {1/3}\right) \operatorname{Ric}\left( {Q, Q}\right) \) . | Proof. For \( \left( {z, Z}\right) \) as above, let \( \left\{ {X}_{i}\right\} \) be an orthonormal basis of \( {M}_{z} \) such that \( {X}_{4} = Z \) ; let \( \left\{ {\omega }^{i}\right\} \) be the dual basis and \( \mathbf{R} \) be the curvature tensor. Then for the rest space we have \( R = \operatorname{span}\left... | Yes |
Proposition 4.2.3. Ric \( = 0 \) iff the mean relative-acceleration of every locally defined geodesic reference frame vanishes. | Proof. The \ | No |
Proposition 4.3.2. \( \alpha \leq - \left( {1/6}\right) \left( {\rho + {3p}}\right) - \left( {1/3}\right) E\left( {Q, Q}\right) < 0 \) ; moreover, at \( z \in \mathcal{U} \), the first inequality becomes an equality iff \( \mathbf{Q}z = \mathbf{Z}z \) . | Proof. Since trace \( \widehat{E} = 0 \) (Proposition 3.7.4), trace \( \widehat{G} = - \rho - p + {4p} = \) \( {3p} - \rho \) . Section 1.0.2 and Exercise 1.4.7 give \( \operatorname{Ric} = \left( {\rho + p}\right) \omega \otimes \omega + \) \( \frac{1}{2}\left( {\rho - p}\right) g + E \), where \( \omega \) is physica... | Yes |
(a) \( Z\left( {\operatorname{div}Z}\right) \leq - \rho /2 - {\left( \operatorname{div}Z\right) }^{2}/3 \) . | Proof. To prove (a) let \( \left\{ {{\mathbf{X}}_{1},{\mathbf{X}}_{2},{\mathbf{X}}_{3}}\right\} \) be vector fields defined on an open set \( \mathcal{U} \) of \( M \) such that \( \left\{ {{\mathbf{X}}_{1},{\mathbf{X}}_{2},{\mathbf{X}}_{3},\mathbf{Z}}\right\} \) is orthonormal in \( \mathcal{U} \) . It suffices to ver... | No |
Theorem 4.3.4. Let \( \gamma : \lbrack 0, a) \rightarrow M \) be an observer in \( Z \) with \( \left( {\operatorname{div}Z}\right) ⓾ = \) \( b \in \left( {-\infty ,0}\right) \) . Then \( a \leq 3/\left| b\right| \) ; moreover, if \( a = 3/\left| b\right| \) , \[ \mathop{\lim }\limits_{{u \rightarrow a}}{\rho \gamma u}... | Proof. Let \( f = \operatorname{div}\mathbf{Z} \circ \gamma \) and let \( h = \rho \circ \gamma \) . Then \( f \) and \( h \) are functions defined on \( \lbrack 0, a) \), with \( h > 0 \) . By Lemma 4.3.3: \[ {f}^{\prime } < - \frac{1}{3}{f}^{2},\;{\left( \ln h\right) }^{\prime } = - f \] where prime denotes different... | Yes |
Define \( \lambda : \left( {0,\infty }\right) \rightarrow M \) by \( {\lambda u} = \left( {0,0,3{u}^{1/5},{u}^{3/5}}\right) \). We claim \( \lambda \) is an in-extendible, freely falling photon. | Proof. \( \lambda \) is \( {C}^{\infty } \) and future pointing. Moreover, \( g\left( {{\lambda }_{ * },{\lambda }_{ * }}\right) = {\left( {u}^{4} \circ \lambda \right) }^{4/3} \times \) \( {\left\lbrack d{u}^{3}\left( {\lambda }_{ * }\right) \right\rbrack }^{2} - {\left\lbrack d{u}^{4}\left( {\lambda }_{ * }\right) \r... | Yes |
Proposition 5.2.3. Suppose \( {u}_{1} \in \mathcal{E} \) is given. There exists an open interval \( \mathcal{F} \subset \mathcal{E} \) containing \( {u}_{1} \) and an open neighborhood \( \mathcal{W} \) of \( \beta {u}_{1} \) such that, \( \forall x \in \mathcal{W} - \beta \mathcal{F} \) the following holds. There exis... | Proof. Let \( \mathcal{U} \) be a simply convex open neighborhood of \( \beta {u}_{1},\left\lbrack {a, c}\right\rbrack \) be an interval in \( \mathcal{E} \) containing \( {u}_{1} \) as an interior point such that \( \beta \left\lbrack {a, c}\right\rbrack \subset \mathcal{U} \) . Regard \( \left( {\mathcal{U}, g}\right... | No |
Proposition 5.4.1. The frequency ratio \( \iota \) for \( \left\lbrack {\left( {\sigma }_{0}\right\rbrack ,{Z}_{a},{Z}_{b}}\right) \) is the inverse of the proper time ratio for \( \sigma \) at \( {\sigma }_{0} \) . | Proof. \( \imath = \left\lbrack {\mathbf{g}\left( {{\sigma }_{0} \cdot a,{\gamma }_{a} \cdot 0}\right) /\mathbf{g}\left( {{\sigma }_{0} \cdot b,{\gamma }_{b} \cdot 0}\right) }\right\rbrack \left\lbrack {\left| {{\gamma }_{b} \cdot 0}\right| /\left| {{\gamma }_{a} \cdot 0}\right| }\right\rbrack \) . The first square bra... | Yes |
Doppler effect in 2-dimensional Minkowski space. When \( \left( {M, g}\right) \) is Minkowski space, distant parallelism makes sense-that is, there exists a global basis consisting of covariant constant vector fields (Exercise 2.3.12). In this case, and really only in this case, can one talk of two instantaneous observ... | Let \( \left( {{\mathbb{R}}^{2}, g}\right) \) be 2-dimensional Minkowski space, \( \lambda : \left\lbrack {0,1}\right\rbrack \rightarrow {\mathbb{R}}^{2} \) be a freely falling photon, \( \left( {╎,{Z}_{0}}\right) \) and \( \left( {╏,{Z}_{1}}\right) \) be instantaneous observers. Without loss of generality, we may supp... | No |
Define the Planck function \( P : \left( {0,\infty }\right) \rightarrow \left( {0,\infty }\right) \) by \( {Pu} = \) \( 2{h}^{-3}{\left\lbrack \left( \exp u\right) - 1\right\rbrack }^{-1} \), where \( h \) is Planck’s constant. Let \( k \) be the universal constant called Boltzmann’s constant; in our units \( k \cong {... | Let \( {F}_{z} \) be Planck with temperature T. The graph shows its specific intensity \( I = {\mathrm{e}}^{3}P\left( {\mathrm{e}/k\mathrm{\;T}}\right) \) as a function of \( \mathrm{e}.{4\pi } \) times the enclosed area is the energy density (Section 5.5.3a)-that is,\n\n\[ \mathrm{u} = {\int }_{{P}_{Z}}\mathrm{e}{F}_{... | Yes |
Proposition 5.6.1. The 3-form \( {\Lambda }_{0} = {\iota }^{ * }\left\lbrack {\left( {1/{u}^{4}}\right) d{u}^{1} \land d{u}^{2} \land d{u}^{3}}\right\rbrack \) on \( {\mathcal{L}}_{0}^{ + } \) can be intrinsically characterized as follows. \( \forall 3 \) -form \( \chi \) over \( \iota \) such that \( \left( {\omega \c... | Proof. There exists at least one such \( \chi \) ; for example, the 3-form\n\n\[ \chi = \left\lbrack {\left( {1/{u}^{4}}\right) d{u}^{1} \land d{u}^{2} \land d{u}^{3}}\right\rbrack \circ \iota \]\n\nworks by our above expressions for \( \omega ,{\Lambda }_{0} \), and \( \Omega \) . We must show uniqueness of \( {\Lambd... | Yes |
Corollary 5.6.2. Suppose \( x \in M \) . Then there is a unique volume element \( {\Lambda }_{x} \) on \( {\mathcal{L}}_{x}^{ + } \) such that, \( \forall \) instantaneous observer \( \left( {x, X}\right) \) at \( x,{\pi }_{X}{}^{ * }{A}_{x} = {\mathrm{e}}^{-1}{\pi }_{X} \) . | Proof. Let \( \left\{ {{X}_{1},\ldots ,{X}_{4}}\right\} \) be an orthonormal basis of \( {M}_{x} \), and let \( \tau : {M}_{x} \rightarrow {\mathbb{R}}^{4} \) be the diffeomorphism given by \( \tau \left( {\mathop{\sum }\limits_{i}{a}^{i}{X}_{i}}\right) = \left( {{a}^{1},\ldots ,{a}^{4}}\right) \) . Then \( \tau {\math... | Yes |
Proposition 5.6.3. Suppose \( f \) keeps sign-that is, \( f \) is either nonnegative or nonpositive-then \( {\int }_{N}{fA} \) exists. | Proof. We may assume \( f \) nonnegative. Let \( \left\{ {\mathcal{K}}_{i}\right\} \) be an exhaustive sequence. Then \( {\int }_{{\mathcal{K}}_{1}}{fA},{\int }_{{\mathcal{K}}_{2}}{fA},\ldots \) is a nondecreasing sequence of nonnegative numbers and hence must approach a limit, say \( a \in \left\lbrack {0,\infty }\rig... | Yes |
Let \( \left( {x, X}\right) \) be an instantaneous observer and define \( f : {\mathcal{L}}_{x}^{ + } \rightarrow \left( {0,\infty }\right) \) by \( f\left( Y\right) = \exp \left( {-u}\right) /u \), where \( u = - g\left( {X, Y}\right) \). We claim \( f \) is of rapid decay. | Note that \( f \circ {\pi }_{x} = {\mathrm{e}}^{-1}\exp \left( {-\mathrm{e}}\right) \). To estimate integrals involving \( {\left( \widetilde{\omega }\right) }^{N}f \) we need bounds on \( \widetilde{\omega } \). \( \forall Y \in {\mathcal{L}}_{x}^{ + } \) we have \( Y = \mathrm{e}\left( {X - U}\right) \), \( \mathrm{e... | Yes |
Proposition 5.6.5. The future lightcone \( {\mathcal{L}}^{ + } \) in TM is a 7-dimensional imbedded submanifold. | Proof. As usual, let \( K : {TM} \rightarrow \mathbb{R} \) be defined by \( K\left( {x, X}\right) = g\left( {X, X}\right) \) . Let \( T{M}^{\prime } = {TM} - \{ \left( {x,0}\right) \in {TM}\} .T{M}^{\prime } \) is an open submanifold of \( {TM} \) and \( {dK} \) is nowhere zero on \( T{M}^{\prime } \) (Exercise 0.0.10)... | No |
Proposition 5.7.3. If \( \mathrm{F} \) is spatially isotropic for \( \left( {z, Z}\right) \) the stress-energy tensor of \( \mathrm{F} \) is spatially isotropic for \( \left( {z, Z}\right) \) . | Proof. Suppose \( \sigma \in {\mathcal{O}}^{3} \) and denote the extension of \( \sigma \) to \( \left( {r, s}\right) \) tensors by \( {\sigma }_{s}{}^{r} : {T}_{s}{}^{r}\left( {M}_{z}\right) \rightarrow {T}_{s}{}^{r}\left( {M}_{z}\right) \) (Exercise 0.0.14). Then \( {\sigma }_{2}{}^{0}\left( {gz}\right) = {gz} \) and... | Yes |
Corollary 5.7.4. Suppose there exists a reference frame \( Z \) such that \( \mathrm{F} \) is spatially isotropic for \( \left( {x,{Zx}}\right) \forall x \in M \) . Then the stress-energy tensor of \( \mathbf{F} \) is \( \widehat{\mathbf{T}} = \left( {\rho /3}\right) \left( {\widehat{\mathbf{g}} + 4\mathbf{Z} \otimes \... | Proof. If \( \mathrm{F} \) is spatially isotropic for \( \left( {x,{Zx}}\right) \forall x \in M \), the same then holds for \( \widehat{\mathbf{T}} \) . So we have \( \widehat{\mathbf{T}} = \rho \mathbf{Z} \otimes \mathbf{Z} + p\left( {\widehat{\mathbf{g}} + \mathbf{Z} \otimes \mathbf{Z}}\right) \) with \( p \) a funct... | No |
Lemma 6.0.10. If \( \lambda \) is the standard photon and \( a \in \left( {0,\infty }\right) \) then \( \imath \left( {{\lambda u},{\lambda a}}\right) = \) \( {\left( {u}^{4}\lambda a/{u}^{4}\lambda u\right) }^{2/3}\forall u \in \left( {0, a}\right) \) . Furthermore, for \( u \in \left( {0, a}\right), u \rightarrow \ma... | Proof of Lemma 6.0.10. \( \imath \left( {{\lambda u},{\lambda a}}\right) = {\left( {u}^{4}\lambda a/{u}^{4}\lambda u\right) }^{2/3} \) by Example 5.4.3. Now \( u \rightarrow {u}^{4}{\lambda u} \) is monotonically increasing (Section 6.6.6b) so \( u \rightarrow \) \( \imath \left( {{\lambda u},{\lambda a}}\right) \) is ... | Yes |
Proposition 6.0.13. Suppose \( x, z \in M \) . Then \( x \) causally precedes \( z \) iff \( \delta \left( {x, z}\right) \leq \) \( 3{\left( {u}^{4}z\right) }^{1/3} - 3{\left( {u}^{4}x\right) }^{1/3} \) . | Proof. Define the function \( u = 3{\left( {u}^{4}\right) }^{1/3} \) on \( M \) . Then \( {du} = {\left( {u}^{4}\right) }^{-2/3}d{u}^{4} \) , so \( g = {\left( {u}^{4}\right) }^{4/3}\{ - {du} \otimes {du} + \mathop{\sum }\limits_{{\mu = 1}}^{3}d{u}^{\mu } \otimes d{u}^{\mu }\} \) . In view of our comments on conformal ... | Yes |
Proposition 6.2.2. Suppose \( \left( {M,\mathcal{M}, F}\right) \) is a finite superposition of relativistic models and each component is either a particle flow, or a perfect fluid, or a photon gas. Suppose at least one of the following generality conditions holds for the superposition: (a) One component is a perfect fl... | Proof. We will prove \( \widehat{T} \) is normal. The proof that \( \widehat{T} \) obeys the timelike convergence condition is similar, easier and will be left as an exercise (Exercise 6.2.12a).\n\nIn the notation of Section 3.0.3, we have \( \widetilde{T} = {\widetilde{T}}_{1} + \cdots + {\widetilde{T}}_{N} \), where ... | No |
Lemma 6.2.6. Let \( \left( {M, g}\right) \) be a simple cosmological spacetime. (a) The metric volume element is \( \Omega = {R}^{3}\left( {u}^{4}\right) d{u}^{1} \land d{u}^{2} \land d{u}^{3} \land d{u}^{4} \) . (b) The Einstein tensor is \( G = {\rho }_{0}d{u}^{4} \otimes d{u}^{4} + {p}_{0}\left( {g + d{u}^{4} \otime... | Here (a) follows directly from Section 3.0.1b since \( \left( {{Rd}{u}^{1},{Rd}{u}^{2},{Rd}{u}^{3}, d{u}^{4}}\right) \) is a (consistently oriented) orthonormal basis. The proof of (b) follows the proof of Proposition 1.4.4 almost verbatim and is left to the reader as per an earlier agreement. To prove (c) note that \(... | No |
Proposition 6.2.7. Let \( \left( {M,\mathcal{M}, F}\right) \) be a relativistic model, \( \left( {z, Z}\right) \) be an instantaneous observer on \( M \) . Suppose: (a) The nonquantum general relativistic cosmology assumptions Section 6.2.3 hold and \( \mathbf{F} = 0 \) ; (b) \( M \) is a simple cosmological spacetime ... | Proof of Proposition 6.2.7. Since \( \mathbf{F} = 0 \), the Einstein field equation \( \mathbf{G} = \) \( T + E \) becomes \( {\rho }_{0}d{u}^{4} \otimes d{u}^{4} + {p}_{0}\left( {g + d{u}^{4} \otimes d{u}^{4}}\right) = {\rho \omega } \otimes \omega \), where \( \omega \) is the 1 -form physically equivalent to \( Z \)... | Yes |
Proposition 6.3.4. Let \( \mathfrak{t} \) be the predicted Hubble time, and \( d \) be as above. Then: (a) \( d = \mathrm{d}\left( i\right) \), where \( \mathrm{d} : \left( {1,\infty }\right) \rightarrow \left( {0,2\mathrm{t}}\right) \) is the increasing function \( \mathrm{d}\left( u\right) = \) 2t \( \left( {1 - {u}^... | Proof. By Section 6.0.3, we have \( d = {\left( {u}^{4}z\right) }^{2/3}\delta \left( {x, z}\right) \), where we use the notation of Section 6.0.12. Now for any isometry \( \psi \) we know that \( {u}^{4} \circ \psi = {u}^{4} \) , \( \delta \left( {{\psi x},{\psi z}}\right) = \delta \left( {x, z}\right) \), and \( \imat... | Yes |
Proposition 6.3.7. \( \Omega /A = {\left( \imath /\mathrm{d}\right) }^{2} \) -that is, the area-distance is \( {d}_{A} = \mathrm{d}/\imath \) . | Proof. \( A = {4\pi }{\left( {u}^{4}x\right) }^{4/3}{\delta }^{2}\left( {x, z}\right) \) . Using the results in the proof of Proposition 6.3.4, we have:\n\n\[ \nA = {4\pi }{r}^{-2}{\left( \frac{2t}{3}\right) }^{4/3} \cdot 9{\left\lbrack {\left( \frac{2t}{3}\right) }^{1/3} - {r}^{-1/2}{\left( \frac{2t}{3}\right) }^{1/3}... | Yes |
Proposition 6.3.8. (a) The luminosity distance is \( {d}_{L} = \mathfrak{{id}} \) . (b) Regarded as a function of \( r,{d}_{L} \) is an increasing onto function: \( \left( {1,\infty }\right) \rightarrow \left( {0,\infty }\right) \) . | Remarks. We shall postpone the (long winded) proof of (a) to the Appendix of this chapter (Theorem 6.7.5). However, note that the much more general result \( {d}_{L} = {\imath }^{2}{d}_{A} \) quoted (without proof) in Section 6.1.6 gives (a) directly from Proposition 6.3.7. Given (a), (b) is immediate from Proposition ... | No |
Proposition 6.3.13. Suppose \( {d}_{0} \in \left( {0,\infty }\right) \) and let \( {i}_{0} \) be the unique cosmological frequency ratio such that \( {\imath }_{0}\mathrm{\;d}\left( {\imath }_{0}\right) = {d}_{0} \) . (Notation as in Proposition 6.3.4). Then \( N = \left( {{4\pi }/3}\right) n\left( {{u}^{4}z}\right) {\... | Proof. Since \( \operatorname{div}\left( {\eta \mathbf{P}}\right) = 0, N = - {\int }_{\mathcal{B}}{\eta i}\left( \mathbf{P}\right) \mathbf{\Omega } \) by Stokes’ theorem. By Exercise 3.2.5, \( \eta = \left( {1/m}\right) n\left( {{u}^{4}z}\right) {\left( {u}^{4}z\right) }^{2}{\left( {u}^{4}\right) }^{-2} \) . Now \( i\l... | Yes |
Lemma 6.4.3. Suppose \( M \) is a simple cosmological spacetime with \( \dot{R} > 0 \) . Let \( \mathcal{M} = \left( {\rho, p,{\partial }_{4}}\right) \) be a perfect fluid on \( M \) such that \( {\partial }_{\mu }\rho = 0 = {\partial }_{\mu }p \) for \( \mu = \) \( 1,2,3 \), and that the stress-energy tensor \( \wideh... | Proof of Lemma 6.4.3. \( \operatorname{div}\widehat{T} = 0 \) implies \( \operatorname{div}\left( {\rho {\partial }_{4}}\right) = - p\operatorname{div}{\partial }_{4} \) (Proposition 3.15.3), which is equivalent to \( {\partial }_{4}\rho = - \left( {\rho + p}\right) \operatorname{div}{\partial }_{4} \) . Now \( \operat... | Yes |
Proposition 6.4.5. Let \( \left( {M,\mathcal{M}, z}\right) \) be a simple cosmological model such that: (a) \( \mathcal{M} \) is a superposition of a dust \( {\mathcal{M}}_{1} = \left( {{\rho }_{1},{\partial }_{4}}\right) \) and a rest-mass zero perfect fluid \( {\mathcal{M}}_{2} = \left( {{\rho }_{2},{\rho }_{2}/3,{\p... | Proof of Proposition 6.4.5. Let \( \left( {M, z}\right) \) be a simple cosmological model such that Assumptions 6.4.5a-c hold. The Einstein field equation \( G = {T}_{1} + {T}_{2} \) implies div \( {\widehat{T}}_{1} = 0 \), because of (b) and div \( \widehat{G} = 0 \) ; it also implies \( {\partial }_{\mu }{\rho }_{1} ... | Yes |
Proposition 6.5.1. Suppose there exist constants \( a \in \mathcal{E} \) and \( {\mathrm{T}}_{a} \in \left( {0,\infty }\right) \) such that \( \mathrm{F} \) is Planck with temperature \( {\mathrm{T}}_{a} \) for \( \left( {x,{\partial }_{4}x}\right) \) whenever \( x \in M \) and \( {u}^{4}x = a \) . Then \( \mathrm{F} \... | Proof. Suppose \( y \in {\mathbb{R}}^{3} \times \mathcal{E} \) and \( Y \in {\mathcal{L}}_{y}{}^{ + } \) . Let \( \lambda : \mathcal{F} \rightarrow M \) be the in-extendible freely falling photon for which \( ╎ = y \) and \( {\lambda }_{ * }0 = Y \) . Then there is a unique \( b \in \mathbb{R} \) such that \( {u}^{4}{\... | No |
Proposition 6.7.3. Let \( x,\gamma ,\mathcal{U} \) , \( r \) and \( T \) be as above, let \( L = \left( {a,\infty }\right) \rightarrow \left( {0,\infty }\right) \) be a \( {C}^{\infty } \) function, and suppose \( \mathrm{e} \in \left( {0,\infty }\right) \) . On \( \mathcal{U} \), define the function \( \eta = \) \( \l... | Proof. From the definition of \( r,{dr} = {r}^{-1}\mathop{\sum }\limits_{{\rho = 1}}^{3}{u}^{\rho }d{u}^{\rho } \) . Using the form of \( g \) we find that the vector field physically equivalent to \( {dr} \) is \( {\left( {u}^{4}\right) }^{-2/3}\partial \) , where \( r\partial = {\left( {u}^{4}\right) }^{-2/3}\mathop{... | Yes |
Theorem 6.7.5. \( \ell = \left( {{L}_{0}/{4\pi }}\right) {\left\lbrack \imath \mathrm{d}\left( \imath \right) \right\rbrack }^{-2} \) . | Proof. Let \( y \) be the point on \( \gamma \) such that there is a light signal from \( y \) to \( z \) . Then \( \imath = {\left( {u}^{4}z/{u}^{4}y\right) }^{2/3} \) and the function \( r\left( z\right) \) of Proposition \( {6.7.3} = \delta \left( {y, z}\right) = \) \( \left. {3\left\lbrack {{\left( {u}^{4}z\right) ... | Yes |
Proposition 7.2.1. \( \imath = \left| {\mathbf{X}{\lambda b}}\right| /\left| {\mathbf{X}{\lambda a}}\right| \) . | Proof.\n\n\( \imath = g\left( {{\lambda }_{ * }a,{Z\lambda a}}\right) /g\left( {{\lambda }_{ * }b,{Z\lambda b}}\right) = \left( {\left| {X\lambda b}\right| /\left| {X\lambda a}\right| }\right) \cdot \left\lbrack {g\left( {{\lambda }_{ * }a,{X\lambda a}}\right) /g\left( {{\lambda }_{ * }b,{X\lambda b}}\right) }\right\rb... | Yes |
Proposition 7.3.2. \( \widehat{R} = \left( {{4\mu }/{r}^{3}}\right) \lbrack 2\left( {{dr} \land {dt}}\right) \otimes \left( {{dr} \land {dt}}\right) - {r}^{2}T + 2{r}^{4}\left( {{P}^{ * }\zeta }\right) \otimes \left( {{P}^{ * }\zeta }\right) \rbrack \) . | Proof. Since the reader has worked through each step of Proposition 1.4.4 an outline will suffice. Let \( \left( {{\chi }^{1},{\chi }^{2}}\right) \) be an orthonormal basis of 1- forms on a nonempty open subset of \( {\mathcal{S}}^{2} \) . Then locally on \( N,{\omega }^{A} = r{P}^{ * }{\chi }^{A} \) , \( {\omega }^{3}... | No |
Corollary 7.3.3. Both \( N \) and \( B \) are vacuum but not flat. | Proof. \( \widehat{R}\left( {\partial /\partial t,\;\partial /\partial r,\;\partial /\partial t,\;\partial /\partial r}\right) = \left( {\mu /{r}^{3}}\right) {\left\lbrack dr\left( \partial /\partial r\right) dt\left( \partial /\partial t\right) \right\rbrack }^{2} = \mu /{r}^{3} \neq 0 \) by Proposition 7.3.2. Thus \(... | No |
Corollary 7.3.4. \( f = {144}{\mu }^{2}/{r}^{6} \) . | Proof. Let \( \widetilde{\mathbf{T}} \) be the \( \left( {4,0}\right) \) -tensor field physically equivalent on \( U \) to \( \mathbf{T} \) in Proposition 7.3.2; let \( A \) be the \( \left( {2,0}\right) \) -tensor field physically equivalent to \( {P}^{ * }\zeta \) . Using the Lorentzian metric \( g \) we find that th... | Yes |
Corollary 7.3.6. \( Z \) is a static, absolute reference frame on \( N;\widehat{t} : N \rightarrow \mathbb{R} \) is a time function for \( \mathbf{Z} \) such that \( \mathbf{Z}\widehat{t} \rightarrow 1 \) as \( r \rightarrow \infty \) iff \( d\widehat{t} = {\left. dt\right| }_{N} \) ; an observer \( \gamma \) in \( N \... | Proof. Chase down the definitions. | No |
Proposition 7.3.7. There exist unique numbers \( \mathrm{E} \in \mathbb{R}, J \in \lbrack 0,\infty ),{J}_{x} \in \left\lbrack {-J, J}\right\rbrack \) such that:\n\n(a) \( g\left( {{\gamma }_{ * },\partial }\right) = - \mathrm{E} \) ;\n\n(b) \( {\rho }^{4}h\left( {{\delta }_{ * },{\delta }_{ * }}\right) = {J}^{2} \) ;\n... | Proof. Since \( \partial /\partial t \) is Killing,(a) follows from Section 3.6.3e. Similarly, the vector field \( V : U \rightarrow {TU} \) determined by \( {P}_{ * }V = {K}_{x} \) and \( {Q}_{ * }V = 0 \) is Killing and \( {\rho }^{2}h\left( {{\delta }_{ * },{K}_{x}}\right) = g\left( {V,{\gamma }_{ * }}\right) \), so... | No |
Proposition 7.4.5. fobeys the ordinary differential equation \( {\left( {f}^{\prime }\right) }^{2} = \mathrm{b} + 2\mathrm{a}f - {f}^{2} + 2{f}^{3} \) . | Proof. If we write \( {\gamma }_{ * } = {a}_{1}\partial + {a}_{2}\left( {\partial /\partial r}\right) \circ \gamma + {\delta }_{ * } \), where each \( {a}_{i} \in \mathbb{R} \) and \( \partial = \left( {\partial /\partial t}\right) \circ \gamma \) as in Proposition 7.3.7 and \( {Q}_{ * }{\delta }_{ * } = 0 \), then \( ... | Yes |
Proposition 7.5.2. \( \\left( {{M}_{\\mathrm{I}},{\\left. g\\right| }_{{M}_{\\mathrm{I}}}}\\right) \) is isometric to the normal Schwarzschild spacetime of active mass \( {8\\pi \\mu } \) . | Proof. Let \( t = {\\left. 2\\mu \\left\\lbrack \\ln u - \\ln \\left( -v\\right) \\right\\rbrack \\right| }_{{M}_{1}} \) . Then \( t \) is \( {C}^{\\infty } \) and maps \( {M}_{\\mathrm{I}} \) onto \( \\left( {-\\infty ,\\infty }\\right) \) . Let \( \\psi : {M}_{1} \\rightarrow \\left( {-\\infty ,\\infty }\\right) \\ti... | Yes |
Proposition 7.5.3. \( \left( {{M}_{11},{\left. g\right| }_{{M}_{11}}}\right) \) is isometric to the Schwarzschild black hole of active mass \( {8\pi \mu } \) . | Proof. Define \( t = {2\mu }\left( {\ln u - \ln v}\right) \) on \( {M}_{\mathrm{{II}}} \) . The calculation now duplicates that of Proposition 7.5.2 except for the change in domains. | No |
Proposition 7.5.5. \( \gamma \mathcal{E} \subset {M}_{\mathrm{{II}}} \) . | Proof. Let \( V \) be the vector field physically equivalent to \( - {dv} \) . Then \( g\left( {V, V}\right) = 0 \) and \( g\left( {U, V}\right) < 0 \), so \( V \) is future pointing, lightlike, and nowhere parallel to \( \mathbf{U} \) . We compute \( \left( {d/{ds}}\right) \left( {r \circ \gamma }\right) \) : On \( {M... | Yes |
Proposition 7.6.1. \( \left( {{\mathbb{R}}^{4}, g}\right) \) is vacuum but not flat. | Proof. Let \( \left\{ {\omega }^{i}\right\} \) be the basis dual to \( \left\{ {\mathbf{X}}_{i}\right\} \) above. By algebra, \( {\omega }^{1} = \mathbf{d}{u}^{1} \) , \( {\omega }^{2} = d{u}^{2},{\omega }^{3} = \frac{1}{2}\left( {d{u}^{4} + d{u}^{3}}\right) - {Fd\phi },{\omega }^{4} = {d\phi } \) . To compute the conn... | Yes |
Proposition 7.6.3. A vector field \( \mathbf{X} \) is parallel iff \( \mathbf{X} = a\mathbf{Y} \) for some \( a \in \mathbb{R} \) . | Proof. Suppose \( \mathbf{X} = a\mathbf{Y} \) . Then \( \forall \left( {z, Z}\right) \in {TM} \) ,\n\n\[ \n{D}_{Z}\mathbf{X} = a\left\{ {Z\left\lbrack {{\omega }^{i}\left( \mathbf{Y}\right) }\right\rbrack + {\omega }_{j}{}^{i}\left( Z\right) {\omega }^{j}\left( \mathbf{Y}\right) }\right\} {\mathbf{X}}_{i} = a{\omega }_... | Yes |
Proposition 1. The intersection of any family of subspaces of \( V \) is again a subspace of \( V \) . | Proof. Put \( M = \cap \left( {{M}_{i} : i \in I}\right) \) . Since \( {0}_{V} \in {M}_{i} \) for each \( i, M \) is not empty. If \( a, b \in M \) and \( x \in F \), then \( a + b \in {M}_{i} \) and \( {xa} \in {M}_{i} \) for each \( i \) ; thus \( a + b \in M \) and \( {xa} \in M \) . | Yes |
Proposition 2. (1) If \( {a}_{1},\ldots ,{a}_{r} \) where \( r \geq 2 \) are linearly dependent, then one of the \( {a}_{i} \) is linearly dependent on the rest. | PROOF. (1) We are given that \( {x}_{1}{a}_{1} + \cdots + {x}_{r}{a}_{r} = 0 \) for some \( {x}_{1},\ldots ,{x}_{r} \) in \( F \), not all zero. Suppose that \( {x}_{j} \neq 0 \) ; then we may multiply by \( {x}_{j}{}^{-1} \) and express \( {a}_{j} \) as a linear combination of the remaining \( {a}_{i} \) ’s. | Yes |
Proposition 3. Any finite dimensional vector space contains a basis. | PROOF. Suppose that \( {a}_{1},\ldots ,{a}_{r} \) span \( V \) . If these vectors are linearly dependent and \( r \geq 2 \), then by Proposition 2 (1) one of these \( {a}_{i} \) is linearly dependent on the rest. If this vector is discarded then the rest still span \( V \) . Continuing in this way we either arrive at a... | No |
THEOREM 2. Let \( M \) and \( N \) be finite dimensional subspaces of a vector space.\n\n(1) If \( M \subset N \), then \( \dim M \leq \dim N \) and \( \dim M = \dim N \) implies \( M = N \) .\n\n(2) Both \( M + N \) and \( M \cap N \) are finite dimensional and\n\n\[ \dim \left( {M + N}\right) + \dim \left( {M \cap N}... | PROOF. Part (1) is an immediate consequence of Theorem 1. To prove part (2), we may suppose, using Theorem 1, Corollary 1, that \( \left\{ {{a}_{1},\ldots ,{a}_{r}}\right\} \) is a basis of \( M \cap N,\left\{ {{a}_{1},\ldots ,{a}_{r},{b}_{1},\ldots ,{b}_{s}}\right\} \) is a basis of \( M \) and that \( \left\{ {{a}_{1... | Yes |
Lemma 2. If \( a + M = b + N \) then \( M = N \) . | PROOF. If \( a + M = b + N \), then \( b = b + 0 \in a + M \) and so \( a + M = \) \( b + M \) by Lemma 1. Thus \( b + M = b + N \), from which the result follows. | Yes |
Proposition 2. Let \( S \) and \( T \) be cosets in \( V \) and \( M, N \) the subspaces belonging to \( S, T \) respectively. Then \( S \cap T \) is empty if, and only if, | \[ \dim \left( {S \cup T}\right) = \dim \left( {M + N}\right) + 1. \] | No |
Lemma 3. \( \left( {a + M}\right) \cup \left( {b + N}\right) = a + \left\lbrack {-a + b}\right\rbrack + M + N \) . | PROOF. \( \left( {a + M}\right) \cup \left( {b + N}\right) \) is a coset containing \( a \) and \( b \) and so is of the form \( a + P = b + P \) for some subspace \( P \) (Lemma 1). Then \( P \) certainly contains \( \left\lbrack {-a + b}\right\rbrack \) (Lemma 1), and also \( M \) and \( N \) . But the join of \( a +... | Yes |
Proposition 3. Let \( A,{A}^{\prime } \) be affine geometries and \( \alpha \) an isomorphism of \( \mathrm{A} \) onto \( {\mathrm{A}}^{\prime } \). If \( {\left( {S}_{i}\right) }_{i \in I} \) is a family of elements of \( A \), then (1) \( \left( {\cap {S}_{i}}\right) \alpha = \cap \left( {{S}_{i}\alpha }\right) \) if... | PROOF. The coset \( \cap {S}_{i} = C \) is the largest coset contained in every \( {S}_{i} \) , \( i \in I \) : this means that \( C \) is the coset uniquely determined by the two conditions (i) \( C \subset {S}_{i} \) for each \( i \) in \( I \), and (ii) if \( T \subset {S}_{i} \) for each \( i \) in \( I \) , then \... | Yes |
Theorem 3. (Desargues’ Theorem for the affine plane.) If \( {ABC},{A}^{\prime }{B}^{\prime }{C}^{\prime } \) are two coplanar triangles in perspective from a point \( P \), and if the pairs of corresponding sides intersect in points \( L, M, N \), then \( L, M \) , \( N \) are distinct and collinear and the line LMN is... | Proof. We may assume that the plane \( S \) of the triangles \( {ABC} \) , \( {A}^{\prime }{B}^{\prime }{C}^{\prime } \) is contained in an affine geometry \( \mathcal{A}\left( V\right) \) but does not contain \( {0}_{V} \) .\n\nWe choose any homogeneous vector \( p \) for \( P \) and use Lemma 5 to find homogeneous ve... | Yes |
Theorem 5. (The Harmonic Construction in the affine plane.) Given two distinct points \( A \) , \( B \) in an affine plane \( S \) , and a point \( G \) on \( {AB} \) . We choose any point \( C \) in \( S \) but not on \( {AB} \) and also any point \( D \) on \( {GC} \) distinct from \( G \) and \( C \) . If \( E = {AD... | PROOF. As before we assume that \( S \) is contained in an affine geometry \( \mathcal{A}\left( V\right) \) and does not contain \( {0}_{V} \) . We then take homogeneous vectors, \( a, b, c, g \) for \( A, B, C, G \) respectively. Since \( {OA},{OB},{OG} \) are coplanar we have \( g = {xa} + {yb} \), for some scalars \... | Yes |
Theorem 9. (Pappus’ Theorem.) If two triads of points, \( A, B, C \) ; \( {A}^{\prime },{B}^{\prime },{C}^{\prime } \) are taken on two distinct coplanar lines and the intersections \( B{C}^{\prime } \cap {B}^{\prime }C, C{A}^{\prime } \cap {C}^{\prime }A, A{B}^{\prime } \cap {A}^{\prime }B \) are points \( L, M, N \),... | The proof follows exactly the same pattern as that of Theorem 4. (Cf. Fig. 4.) | No |
Proposition 1. Two isomorphisms \( f, g \) of \( V \) onto \( {V}^{\prime } \) induce the same projectivity \( \mathcal{P}\left( f\right) = \mathcal{P}\left( g\right) \) if, and only if, there exists a non-zero scalar \( z \) such that \( g = {zf} \) . | PROOF. If \( g = {zf} \) then \( {Mf} = {Mg} \) for any subspace \( M \) of \( V \) and so \( \mathcal{P}\left( f\right) = \mathcal{P}\left( g\right) \n\nConversely, suppose that \( \mathcal{P}\left( f\right) = \mathcal{P}\left( g\right) \) ; then \( {ag} = {z}_{a}\left( {af}\right) \) where \( {z}_{a} \) is a non-zero... | Yes |
Proposition 2. Every coset in \( {F}^{n} \) is the set of all solutions of a suitable system of equations (1). | PROOF. By the reduction already given of Theorem 3 to Theorem 4 we need only prove our proposition when the given coset is actually a subspace \( M \) of \( {F}^{n} \) .\n\nWe define \( {M}^{ \bot } \) to be the set of all \( \left( {{y}_{1},\ldots ,{y}_{n}}\right) \) in \( {F}^{n} \) satisfying\n\n\[ \mathop{\sum }\li... | No |
Proposition 5. Let \( V \) be a vector space of dimension \( \geq 3 \) over a field \( F \) and let \( \varphi ,\psi ,\theta \) be the homomorphisms\n\n\[ \n{F}^{ * }\overset{o}{ \rightarrow }\text{ Aut }V\overset{\psi }{ \rightarrow }\text{ Aut }\mathcal{P}\left( V\right) \overset{\theta }{ \rightarrow }\text{ Aut }F ... | Proof. Clearly \( \varphi \) and \( \psi \) are homomorphisms. Moreover (i) is trivial and (ii) is immediate from Proposition 1.\n\nTo see that \( \theta \) is a homomorphism, we choose two projective auto-morphisms and use Theorem 6 to represent them in the form \( \mathcal{P}\left( f\right) \) , \( \mathcal{P}\left( ... | Yes |
Proposition 6. Let \( V \) be a vector space of dimension \( \geq 2 \) over a field \( F \) .\n\n(i) Aut \( \mathcal{A}\left( V\right) \) is the split product of \( \left( {S, T}\right) \) where \( S \) is isomorphic to \( S\left( V\right) \) by means of the isomorphism \( f \rightarrow \mathcal{A}\left( f\right) \) an... | The non-trivial part of this proposition is the proof of condition (1) for a split product in part (i): this follows from Theorem 5. The rest we leave as an exercise for the reader. | No |
Proposition 7. A central collineation \( \pi \) which is distinct from the identity has one and only one center \( A \) (possibly on the hyperplane of \( \pi \) ) and \( A \) is an invariant point for \( \pi \) . | PROOF. Suppose that \( \pi = \mathcal{P}\left( f\right) \) where \( f \) is an element of Aut \( V \) and that \( {P\pi } = P \) for every point \( P \) of the hyperplane \( H \) of \( V \) . By multiplying \( f \) by a suitable non-zero scalar we may assume that the restriction of \( f \) to \( H \) is the identity ma... | Yes |
Proposition 8. In the notation of Proposition 3, \( \alpha \) is a dilatation of A if, and only if, \( \pi \) is a central collineation with hyperplane \( H \) . Further, \( \alpha \) is a translation if, and only if, \( \pi \) has center on \( H \) . (Note that \( \alpha = {1}_{\mathrm{A}} \) if, and only if, \( \pi =... | PROOF. (1) It follows directly from the definition that \( \alpha \) is a dilatation if, and only if, \( S \) and \( {S\alpha } \) have the same hyperplane at infinity for every \( S \) in A. But the construction of \( \pi \) from \( \alpha \) given in the proof of Proposition 3 shows that this is equivalent to the con... | No |
Proposition 2. If \( f \in \mathcal{L}\left( {U, V}\right), g \in \mathcal{L}\left( {V, W}\right) \) and \( \left( {u}_{i}\right) ,\left( {v}_{j}\right) ,\left( {w}_{k}\right) \) are ordered bases of \( U, V, W \), respectively, then\n\n\[ \left( {f;\left( {u}_{i}\right) ,\left( {v}_{j}\right) }\right) \left( {g;\left(... | PROOF. If \( {u}_{i}f = \mathop{\sum }\limits_{{j = 1}}^{n}{a}_{ij}{v}_{j} \) and \( {v}_{j}g = \mathop{\sum }\limits_{{k = 1}}^{p}{b}_{jk}{w}_{k} \), then\n\n\[ {u}_{i}\left( {fg}\right) = \mathop{\sum }\limits_{{k = 1}}^{p}\mathop{\sum }\limits_{{j = 1}}^{n}{a}_{ij}{b}_{jk}{w}_{k} = \mathop{\sum }\limits_{{k = 1}}^{p... | Yes |
Proposition 3. A square matrix \( A \) is invertible if, and only if, \( A \) is the matrix of an isomorphism. | The \ | No |
Proposition 4. If \( f \in \mathcal{L}\left( {V,{V}^{\prime }}\right) \) has matrices \( A, B \) with respect to two pairs of bases of \( V,{V}^{\prime } \), then there exist invertible matrices \( P, Q \) such that \( B = {PAQ} \) . | PROOF. Let \( A = \left( {f;\left( {v}_{i}\right) ,\left( {{v}_{j}{}^{\prime }}\right) }\right) \) and \( B = \left( {f;\left( {a}_{r}\right) ,\left( {{a}_{s}{}^{\prime }}\right) }\right) \) . If we put \( P = \left( {{1}_{V};\left( {a}_{r}\right) ,\left( {v}_{i}\right) }\right) \) and \( Q = \left( {{1}_{{V}^{\prime }... | Yes |
Proposition 5. \( {\left( AB\right) }^{t} = {B}^{t}{A}^{t} \) . | The proof is left as an exercise. | No |
Proposition 2. Let \( \left( {{a}_{1},\ldots ,{a}_{n}}\right) \) be an ordered basis of \( V \) . The bilinear forms \( \sigma ,\tau \) are congruent if, and only if, the matrices \( \left( {\sigma ;\left( {a}_{i}\right) }\right) \) , \( \left( {\tau ;\left( {a}_{i}\right) }\right) \) are congruent. | PROOF. Given two ordered bases \( \left( {{a}_{1},\ldots ,{a}_{n}}\right) ,\left( {{b}_{1},\ldots ,{b}_{n}}\right) \) there is a unique \( f \) in Aut \( V \) such that\n\n\[ \n{b}_{i} = {a}_{i}f\left( {i = 1,\ldots, n}\right) .\n\] \n\n\( \mathrm{{Now}}\left( {{\sigma }^{f};\left( {a}_{i}\right) }\right) = \left( {\si... | Yes |
Proposition 3. The mappings \( \sigma \rightarrow g,\sigma \rightarrow \widetilde{\sigma } \) defined by the equations\n\n\[ \sigma \left( {a, b}\right) = b\left( {ag}\right) \]\n\n\[ \sigma \left( {a, b}\right) = a\left( {b\widetilde{\sigma }}\right) \]\n\n(for all \( a, b \) in \( V \) ), are linear isomorphisms of \... | PROOF. The mapping \( f \rightarrow \sigma \), constructed above, is the inverse of \( \sigma \rightarrow g \) and hence \( \sigma \rightarrow g \) is one-one and onto \( \mathcal{L}\left( {V,{V}^{ * }}\right) \) . The linearity follows immediately from the definitions of \( \sigma + \tau \) and \( {x\sigma } \) .\n\nI... | Yes |
Proposition 4. If \( \sigma \) is a bilinear form on the vector space \( V \) and \( M \) is any subspace of \( V \), then\n\n(1) \( \dim M + \dim {M}^{ \bot } = \dim V + \dim \left( {M \cap {V}^{\top }}\right) \) ,\n\n(2) \( \dim M + \dim {M}^{\top } = \dim V + \dim \left( {M \cap {V}^{ \bot }}\right) \) . | PROOF. We first give a proof in terms of coordinates. To simplify the argument we may choose an ordered basis \( \left( {{a}_{1},\ldots ,{a}_{n}}\right) \) of \( V \) which extends an ordered basis \( \left( {{a}_{1},\ldots ,{a}_{m}}\right) \) of \( M \) . Let \( b \) have coordinate row \( y = \left( {{y}_{1},\ldots ,... | Yes |
Lemma 1. \( {M}^{ \bot \left( \sigma \right) } = {\left( Mg\right) }^{ \circ } \) and \( {M}^{\top \left( \sigma \right) } = {\left( M\widetilde{\sigma }\right) }^{ \circ } \) . | PROOF. The vector \( b \) lies in \( {\left( Mg\right) }^{ \circ } \) if, and only if, \( {bf} = 0 \) for all \( f \) in \( {Mg} \), i.e., for all \( f = {ag} \) with \( a \) in \( M \) . By the definition of \( g \) this is equivalent to \( \sigma \left( {a, b}\right) = 0 \) for all \( a \) in \( M \), i.e., to \( b \... | Yes |
Lemma 2. \( M \subset {M}^{ \bot \top } \) and \( M \subset {M}^{\top \bot } \) for every subspace \( M \) of \( V \) . | PROOF. If \( a \) is any vector in \( M \) and \( b \) is any vector in \( {M}^{ \bot } \), then, by the definition of \( {M}^{ \bot },\sigma \left( {a, b}\right) = 0 \) . But this implies that \( a \) is an element of \( {\left( {M}^{ \bot }\right) }^{\top } \) and thus \( M \subset {M}^{ \bot \top } \) . The other in... | Yes |
Proposition 5. Let \( \\sigma \) be a bilinear form on the vector space \( V \) . If \( M \) is a non-degenerate subspace of \( V \) with respect to \( \\sigma \) then\n\n\[ V = M \\oplus {M}^{ \\bot }\\;\\text{ and }\\;V = M \\oplus {M}^{\\top }.\] | PROOF. We shall confine our attention to \( \\bot \) : the argument for \( \\top \) is exactly the same. By Lemma \( 3, M \\cap {M}^{ \\bot } = 0 \) and so we only have to show that \( M + {M}^{ \\bot } = V \) . For this it suffices to prove \( \\dim M + \\dim {M}^{ \\bot } = \) \( \\dim V \) . Now \( M \\cap {V}^{\\to... | Yes |
Proposition 6. A bilinear form is orthosymmetric if, and only if, it is either symmetric or skew-symmetric. | Suppose that \( \sigma \) is a skew-symmetric bilinear form. Then\n\n\[ \sigma \left( {a, a}\right) = - \sigma \left( {a, a}\right) \]\n\nfor all \( a \) in \( V \) . In other words\n\n\[ {2\sigma }\left( {a, a}\right) = 0 \]\n\nfor all \( a \) in \( V \) .\n\nIf we assume that the ground field is not of characteristic... | Yes |
Corollary 2. There exists an ordered basis with respect to which \( \sigma \) has matrix  where \( n \) is the number of rows and \( s \) is the number of blocks \[ \left( \begin{array}{rr} 0 & 1 \\ - 1 & 0 \end{arra... | It is easily seen that a matrix \( A = \left( {a}_{ij}\right) \) is the matrix of an alternating bilinear form if, and only if, \( {a}_{ij} = - {a}_{ji} \) whenever \( i \neq j \) and \( {a}_{ii} = 0 \) for all \( i \) . We call such a matrix an alternating matrix. Corollary 2 shows that every \( n \times n \) alternat... | No |
Proposition 7. If \( \sigma \) is a symmetric bilinear form on a complex vector space \( V \), then there exists an ordered basis of \( V \) with respect to which \( \sigma \) has matrix \[ \left( \begin{array}{ll} {I}_{r} & 0 \\ 0 & 0 \end{array}\right) \] where \( r \) is the rank of \( \sigma \) . | PROOF. By the corollary to Theorem 3, there exists an ordered basis \( \left( {{a}_{1},\ldots ,{a}_{n}}\right) \) such that \[ \left( {\sigma ;\left( {a}_{i}\right) }\right) = \operatorname{diag}\left( {{x}_{1},\ldots ,{x}_{r},0,\ldots ,0}\right) , \] with \( {x}_{i} \neq 0, i = 1,\ldots, r \) . Let \( {y}_{i}{}^{2} = ... | Yes |
Proposition 8. Let \( \\left( {V,\\sigma }\\right) \) be a real symmetric space. If\n\n\[ V = P \\oplus Q \\oplus {V}^{ \\bot } \]\n\nwhere the restriction of \( \\sigma \) to \( P \) is positive definite and the restriction of \( \\sigma \) to \( Q \) is negative definite, then \( \\dim P \) and \( \\dim Q \) are inva... | PROOF. If \( M \) is a positive definite subspace and \( N \) is a negative semidefinite subspace then clearly \( M \\cap N = 0 \) . Thus\n\n\[ \\dim M + \\dim N \\leq \\dim V.\]\n\nBut in particular \( N = Q \\oplus {V}^{ \\bot } \) satisfies this inequality and so\n\n\[ \\dim M \\leq \\dim V - \\left( {\\dim V - \\di... | Yes |
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