By Derek F. Lawden
User-friendly advent will pay targeted cognizance to features of tensor calculus and relativity that scholars locate so much tricky. Contents comprise tensors in curved areas and alertness to basic relativity conception; black holes; gravitational waves; software of normal relativity ideas to cosmology. quite a few workouts. resolution advisor on hand upon request.
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Extra resources for An Introduction to Tensor Calculus, Relativity, and Cosmology
38) The momentum and energy transform into each other under the Lorentz transformation just as space and time do. 34), the invariant square of the 4-momentum must be pμ pμ = –(mc)2 . 36), we obtain the important relativistic energy–momentum relation: E 2 = (mc2 )2 + (pc)2 = m2 c4 + p2 c2 . 36), a particle’s momentum and energy per unit mass can be written directly in terms of derivatives with respect to proper time: p = r˙ , m E = c˙t . 40) These expressions will be useful when we discuss the SR limit of particle energy in general relativity when spacetime is curved.
31). 28) Tensors in special relativity where we have used the time dilation relation dt = γ dτ and vx = dx/dt, etc. 29) which is invariant, as it should be. 4 4-velocity transformation leads to the SR addition rule for 3-velocities With its transverse components suppressed, the 4-velocity of a particle is x˙ μ = γu c u , with u = dx u2 and γu = 1 – 2 dt c –1/2 . 30) As a 4-vector, x˙ μ has the following boost (of velocity v parallel to u) transformation: γu c u = γv 1 –v/c –v/c 1 γu c . 18). 5), has the (mathematical) group property of [L(ψ1 )][L(ψ2 )] = [L(ψ1 + ψ2 )].
Furthermore, in several applications to be discussed below, the infinitesimal form ds 2 = ds2 is directly relevant. The invariant interval and proper time In this new kinematics, time is no longer absolute (no longer an invariant under coordinate transformation). There is, however, a particular combination of space and time intervals13 that is invariant: ds2 ≡ dx2 + dy2 + dz2 – c2 dt 2 . 28) We will demonstrate this by an explicit calculation. 28) yields ds 2 = dx 2 + dy 2 + dz 2 – c2 dt 2 = γ 2 (dx – v dt)2 + dy2 + dz2 – γ 2 c2 dt – = γ2 1 – v2 c2 v dx c2 dx2 – c2 dt2 + dy2 + dz2 = ds2 .
An Introduction to Tensor Calculus, Relativity, and Cosmology by Derek F. Lawden