An Introduction to Riemannian Geometry: With Applications to by José Natário, Leonor Godinho
By José Natário, Leonor Godinho
In contrast to many different texts on differential geometry, this textbook additionally bargains fascinating functions to geometric mechanics and normal relativity.
The first half is a concise and self-contained advent to the fundamentals of manifolds, differential kinds, metrics and curvature. the second one half stories purposes to mechanics and relativity together with the proofs of the Hawking and Penrose singularity theorems. it may be independently used for one-semester classes in both of those subjects.
The major principles are illustrated and additional built via a variety of examples and over three hundred workouts. designated ideas are supplied for plenty of of those routines, making An advent to Riemannian Geometry perfect for self-study.
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Extra resources for An Introduction to Riemannian Geometry: With Applications to Mechanics and Relativity (Universitext)
Example text
13) Relativistic Kinematics in a Deformed Minkowski Space 39 • Aberration law: tgθ = sin θ . 14) We want now to provide a comparison between the main kinematical ˜ (in the laws in the usual Minkowski space M and in the deformed one M hypothesis of spatial isotropy), because their different behaviors may help one to understand the peculiar features of leptonic, hadronic (and gravitational) interactions with respect to the electromagnetic one. 1, where the maximal speed u has been expressed in terms of c, in order to emphasize the dependence of the deformed laws on the parameter ratio b/b0 and exhibit their scale invariance.
24) a Let us notice that the directionally separating procedure can be consistently applied only to (special- or general relativistic) metrics which are fully diagonal. This is obviously due to the mixings between different space directions which arise in the case of nondiagonal metrics. Of course, such a procedure gives (in either subcase) the same standard result when applied to SR. In fact: uiSR = (−gii )1/2 = dxi dxi = (g00 )1/2 c = dt dt (g00 )1/2 c=c (−gii )1/2 ∀ i = 1, 2, 3. 25) The vector u is the (spatially) anisotropic generalization of the maximal causal speed derived in the (spatially) isotropic case, Eq.
24) a Let us notice that the directionally separating procedure can be consistently applied only to (special- or general relativistic) metrics which are fully diagonal. This is obviously due to the mixings between different space directions which arise in the case of nondiagonal metrics. Of course, such a procedure gives (in either subcase) the same standard result when applied to SR. In fact: uiSR = (−gii )1/2 = dxi dxi = (g00 )1/2 c = dt dt (g00 )1/2 c=c (−gii )1/2 ∀ i = 1, 2, 3. 25) The vector u is the (spatially) anisotropic generalization of the maximal causal speed derived in the (spatially) isotropic case, Eq.