A collimated laser beam of intensity (power per unit area) travels along the -axis in…

PHYSICS · Multi-part · Fermi University Physics · Electricity, Magnetism & Electromagnetism · Relativity & Gravitation · Relativistic dynamics and fields in flat spacetime

The Geometry of Gravity, Ch The stress-energy tensor and relativistic continua

Problem

A collimated laser beam of intensity (power per unit area)

I = 3.0\times10^{8}\ \mathrm{W\,m^{-2}}

travels along the

x

-axis in vacuum. Someone reasons as follows: The energy density is

T^{00} = I/c = 1.0\ \mathrm{J\,m^{-3}}

. The energy flux through a surface of constant

x

is just the intensity itself, so

T^{0x} = I = 3.0\times10^{8}\ \mathrm{J\,m^{-2}\,s^{-1}}

. The momentum density is

T^{x0} = T^{00} = 1.0\ \mathrm{J\,m^{-3}}

, since for a light beam

T^{00}=T^{xx}=T^{0x}=T^{x0}

should all be equal. But my numbers give

T^{0x} = 3.0\times10^8

while

T^{x0}=1.0

, so this beam's stress-energy tensor is not symmetric. ((a)) Find the error in this reasoning. ((b)) Give the correct common value of

T^{0x}=T^{x0}

(and hence of

T^{00}=T^{xx}

too) for this beam.

Related theory

Relativistic electrodynamics

Browse the problem library