macdonald a elementary general relativity

... coordinate system a planar frame. They postulate: The Planar Frame Postulate for a Flat Surface A planar frame can be constructed with any point P as origin and with any orientation. Fig. 1.2: A planar frame. Similarly, ... other speeds are expressed as a fraction of the speed of light. Ordinarily the fractions are very small. 19 Elementary General Relativity Version 3.35 Alan...

Ngày tải lên: 24/04/2014, 17:07

... physical theories and notions is not always as easy as it might sound. It involves a new, less parochial, and less cozy way of looking at the world and, sometimes, new and unfamiliar mathematical ... of spacetime. It is no accident that I use the word “spacetime” rather than “space and time” or “space-time.” It expresses the fact that, at least as far as the fundamental laws of physics are...

Ngày tải lên: 24/04/2014, 17:07

... additional criteria. In particular, we have to say something about general covariance. We consider an accelerated frame of reference locally equivalent to one which is at rest in a gravitational field. ... more than a mathematical curiosity, can now be applied to realistic astrophysical situations like that in our Galactic Center. Clearly, the time has come for general relativists and...

Ngày tải lên: 19/01/2014, 14:03

... not distinguishable before simple follicle formation in Octopoteuthis sicula, Sepia bertheloti and Abraliopsis atlantica. In Argonauta argo and Pteriqioteutnis gemmata, at the stage when the simple follicle nucleoli start decomposing, they acquire a characteristic blot shape ... male cephalopods, stages IV and Vare short and perhaps of little ecological importance since they have to accumul...

Ngày tải lên: 14/03/2014, 16:20

... (rank one tensors) produces a scalar (rank zero tensor) as A µ B µ = constant ≡ A. B (3.22) and it can be shown that A. B as defined here is a scalar (do Problem 3.2). A scalar is a tensor of rank ... let’s define a contravariant vector A µ as anything whose components transform as (compare (3.5)) A µ ≡ ∂x µ ∂x ν A ν (3.12) and a covariant vector A µ (often also called...

Ngày tải lên: 17/03/2014, 13:34

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Ngày tải lên: 17/03/2014, 14:26

... real-valued map ∞: U’E 1 . If U ≠ M, we sometimes call such a scalar field local. If ∞ is a scalar field on M and x is a chart, then we can express ∞ as a smooth function ˙ of the associated parameters ... Manifolds and Scalar Fields 7 3. Tangent Vectors and the Tangent Space 14 4. Contravariant and Covariant Vector Fields 24 5. Tensor Fields 35 6. Riemannian Manifolds 40 7. Locally M...

Ngày tải lên: 17/03/2014, 14:28

... Earth or any other planet. Thus, we can construct an “artificial” gravitational field. Let us consider such an artificial gravitational field more closely. Suppose we want this artificial gravitational ... an invariant tensor: it has the same form in all coordinate grids. Gradients of tensors The gradient of a scalar field φ transforms as a covariant vector. Are gradients of covariant vectors a...

Ngày tải lên: 17/03/2014, 14:28

... combining the gravitational and matter actions and varying the metric. After a little algebra, equation (33) gives the variation of S G ; we must add to it the variation of S M . Equation (35) gives ... m a a a δg µν (x a i (t), t) = dt  1 m a V 0 i − 2 V a V a δg µν (x a (t), t) . (36) 2 V 0 a a a a Variation of the metric naturally gives the normalize...

Ngày tải lên: 17/03/2014, 14:28

... spacetimes, and we will have to define a “covariant derivative” to take the place of the partial derivative. Nevertheless, we can still use the fact that partial derivatives 1 SPECIAL RELATIVITY AND FLAT ... linear maps belong to the realm of linear algebra, and are appropriate whenever we have an abstract vector space at hand. In the case of interest to us we have not just a vector spac...

Ngày tải lên: 17/03/2014, 14:28