Energy Momentum Tensor 

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Here we define the relationship between stress, energy and momentum in relativity. Consider the diagram below showing an infinitesimal element of matter. 

 Define the tensor T by its components T^ab, which are defined as follows; 

T ⁰⁰ º u = energy density 

g^0j º g^j0 º (linear momentum density in the jth direction)/c 

s^jk º Stress 3-tensor = (momentum flux density º amount of j momentum in the k direction)/c². s^xx = x-component of stress/pressure, s^yy = y-component of stress/pressure and s^zz = z-component of stress/pressure. 

In words 

T⁰⁰ º Energy density (i.e. energy per unit volume) 

g^0j º (jth component of momentum per unit volume)/c 

s^jk º (jth component of momentum that flows across a surface whose normal direction is in the k direction)/c².

 Let v^k be the component of the (spatial) velocity in the kth direction and let b^k º v^k/c.  From these definitions and from the diagram above we get, letting r = inertial mass density = uc². 

To summarize: 

Since the expression above for T has components which are the stress 3-tensor, energy density and the momentum density the tensor T has come to be known as the stress-energy-momentum tensor. What remains is to demonstrate that this tensor is conserved. Consider the equation for the local energy conservation (setting c = 1) 

g = rv is the momentum density given that r is the mass density. Written out in component for this expression becomes

In terms of the components of energy-momentum tensor 

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