A boost along with speed has rapidity

PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · Special relativity: spacetime and kinematics · Special relativity: spacetime and kinematics

The Geometry of Gravity, Ch Minkowski spacetime and relativistic dynamics, completed

Problem

A boost along

x

with speed

v

has rapidity

\varphi

, where

\tanh\varphi=v/c

, and is represented by the matrix

\Lambda^\mu{}_\nu

with

\Lambda^0{}_0=\Lambda^1{}_1=\cosh\varphi

\Lambda^0{}_1=\Lambda^1{}_0=-\sinh\varphi

(the

y

and

z

rows and columns are the identity). Two observers,

S\to S'\to S''

, are related by two successive boosts along

x

, each at

v=0.90c

. ((a)) Find the rapidity

\varphi

of a single boost at

v=0.90c

. ((b)) Using

\varphi_{S\to S''}=2\varphi

, find the speed of

S''

relative to

S

, as a multiple of

c

. ((c)) Verify directly, by computing

\cosh(2\varphi)

and

\sinh(2\varphi)

and substituting, that the single boost

\Lambda(2\varphi)

satisfies the isometry condition

\Lambda^\alpha{}_\mu\Lambda^\beta{}_\nu\eta_{\alpha\beta}=\eta_{\mu\nu}

for

(\mu,\nu)=(0,0)

(1,1)

and

(0,1)

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