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:
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 :
by taking the trace inverse :
With
- the curvature scalar
- the trace of the energy-momentum tensor
- a geometric and gravitational constant .
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:
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:
literature
- Hawking, Stephen; and Ellis, GFR: The Large Scale Structure of Space-Time . Cambridge University Press, Cambridge 1973, ISBN 0-521-09906-4 .