By Derek F. Lawden

Undemanding advent can pay particular awareness to features of tensor calculus and relativity that scholars locate so much tough. Contents contain tensors in curved areas and alertness to common relativity idea; black holes; gravitational waves; program of common relativity rules to cosmology. various workouts. resolution consultant on hand upon request.

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**Extra info for An Introduction to Tensor Calculus, Relativity, and Cosmology**

**Example text**

Their ‘equivalence classes’. Then Rp and Rq are either disjoint (have no point in common) or identical. In other words, two simultaneity classes cannot intersect in a proper subset. Therefore, Einstein’s deﬁnition of simultaneity partitions space-time into mutually disjoint sets, each containing mutually simultaneous events. The converse is also obviously true, namely that any partitioning into mutually disjoint sets deﬁnes an equivalence relation. These conditions therefore should be expected from any workable deﬁnition of simultaneity, though relaxations may be considered, in particular concerning transitivity.

G. a yardstick) resting in K as observed from K . Here and in the rest of this book it is convenient to plot ct (rather than just t) on the vertical axis, where c is the vacuum speed of light. This endows the unit on the time axis with the physical dimension of a length, like that on the spatial axes. We further agree that these units should be the same, so that world lines of light rays are depicted at an inclination of 45 degrees. In Fig. 2 the world lines of the leading and trailing end of the rod are depicted by solid lines, whereas the world line m of its midpoint is dotted.

Then C2 meets C2 at the event B and C2 reads 12:30. This last reading is also arbitrarily set by choosing the common rate of all clocks appropriately. This being done, all other readings are determined. The reading of C2 at B is 12:20, corresponding to a factor of 1/γ = 2/3 by which the moving clock lags behind. Now we take the position of an observer resting in K . He compares his clocks C1 and C2 with a single clock at rest in K , say C1 . Event A again denotes the meet of C1 and C2 where both clocks read 12:00.