Energy condition

In the general theory of relativity  (ART) the mass and energy distribution is described with an energy-momentum tensor . In the context of this theory, energy conditions are inequalities for contractions of this tensor. They are applied in the singularity theorems , of which different versions exist and which differ in the strength of the applied energy condition.

A strong condition results in causal singularities that are easy to prove, but there may be forms of matter in the universe that contradict a strong energy condition and only obey weaker conditions. The weakest (light-like) energy conditions are very likely fulfilled by all matter, but only light-like singularities follow from this.

The strong energy condition

The strong energy condition says that the energy-momentum tensor only causes attractive gravitation , and is therefore a very clear condition that corresponds to intuitive observation.

In the formulation of the ART, curvatures of space describe the gravitational effects and are represented mathematically by the Ricci tensor . The strong energy condition now states that the double contraction of the Ricci tensor with any time-like vector field must be greater than 0: ${\displaystyle R_{ab}}$ ${\displaystyle X}$

${\displaystyle R_{ab}X^{a}X^{b}\geq 0\quad \forall \,X^{a}{\text{ zeitartig}}}$

Such a vector field corresponds to e.g. B. the tangential vector to the world line of an observer and is thus the time axis of his local Lorentz system .

Using Einstein's field equations , this condition can also be translated as required for the energy-momentum tensor : ${\displaystyle T_{ab}}$

${\displaystyle (T_{ab}-{\frac {1}{2}}g_{ab}T)X^{a}X^{b}\geq 0\quad \forall \,X^{a}{\text{ zeitartig}},}$

by taking the trace inverse :

${\displaystyle R_{ab}-{\frac {1}{2}}g_{ab}R=\kappa T_{ab}\quad /\cdot g^{ab}\quad \Rightarrow \quad R-2R=\kappa T\quad \Rightarrow \quad R_{ab}=\kappa \left(T_{ab}-{\frac {1}{2}}g_{ab}T\right)}$

With

• the curvature scalar ${\displaystyle R}$
• the trace of the energy-momentum tensor ${\displaystyle T}$
• a geometric and gravitational constant . ${\displaystyle \kappa }$

The weak energy condition

The weak energy condition also has an intuitive equivalent and requires that all observers, i.e. systems with time-like world lines, see a positive energy density from the observed energy distribution:

${\displaystyle T_{ab}X^{a}X^{b}\geq 0\quad \forall \,X^{a}{\text{ zeitartig}}}$

Energy condition for light-like vectors

This condition is also often called the zero energy condition, as it relates to light-like vectors (also called zero vectors ) whose scalar product is zero by definition. This condition is much weaker than the previous two, and in each of them as a special case in Limes contain high speeds:

${\displaystyle T_{ab}Y^{a}Y^{b}\geq 0\quad \forall \,Y^{a}{\text{ lichtartig}}}$

literature

• Hawking, Stephen; and Ellis, GFR: The Large Scale Structure of Space-Time . Cambridge University Press, Cambridge 1973, ISBN 0-521-09906-4 .