By Malcolm Ludvigsen
ISBN-10: 0511040067
ISBN-13: 9780511040061
ISBN-10: 0521630193
ISBN-13: 9780521630191
Beginning with the belief of an occasion and completing with an outline of the traditional big-bang version of the Universe, this textbook presents a transparent, concise and updated creation to the idea of common relativity, appropriate for final-year undergraduate arithmetic or physics scholars. all through, the emphasis is at the geometric constitution of spacetime, instead of the conventional coordinate-dependent process. this permits the idea to be pared down and awarded in its easiest and so much based shape. issues lined comprise flat spacetime (special relativity), Maxwell fields, the energy-momentum tensor, spacetime curvature and gravity, Schwarzschild and Kerr spacetimes, black holes and singularities, and cosmology. In constructing the idea, all actual assumptions are truly spelled out and the mandatory arithmetic is built besides the physics. workouts are supplied on the finish of every bankruptcy and key principles within the textual content are illustrated with labored examples. suggestions and tricks to chose difficulties also are supplied on the finish of the publication. This textbook will allow the scholar to advance a legitimate realizing of the idea of normal relativity, and all of the priceless mathematical equipment.
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Extra info for General Relativity - a geometric approach
Example text
Throughout this book we will employ the convention that Greek indices take values 0 to 3 and use the Latin letters i, j, k, l, m, and n to indicate indices that take values 1 to 3. Thus, for example, we write (e α ) for (e 0 , e 1 , e 2 , e 3 ) and (e i ) for (e 1 , e 2 , e 3 ). The components w α of a vector w with respect to (e α ) are given by w = w 0e 0 + w 1e 2 + w 2e 2 + w 3e 3 ≡ w α e α . Here and throughout the rest of the book we employ the Einstein summation convention whereby we automatically sum over repeated indices, one upstairs and the other downstairs.
Since E(v ) is strictly positive, p must be future-pointing. Unlike E(v ), which depends on v , the four-momentum vector depends only on the physical system itself and is therefore an intrinsic, observer-independent property of it. By linearity and conservation of energy we see that if p 1 and p 2 are the four-momenta of two isolated systems that are brought together to form a combined isolated system, then the four-momentum of the combined system is p = p 1 + p 2 . If p is timelike, then there will exist a four-velocity vector w and a positive number m such that p = mw.
3. The box spanned by a , b, and c captures the same four-momentum as that spanned by a = α p + a , b = β p + b, and c = γ p + c . p v' v a' a 46 ENERGY write p = Xv + Aa + Bb + Cc = X v + A a + B b + C c , where, by orthogonality, X = p · v and X = p · v . Using the antisymmetry of ε, we thus have ε( p, a , b, c ) = ε( p, a , b , c ) = p · v V = p · v V , which, since p = p , can be written p · v V = p · v V or ρ (v ) · v VV = ρ (v ) · v VV . ✷ Our next theorem shows that ρ (v ) · v defines a quadratic from T, namely the energy–momentum tensor.