Einstein Tensors and Energy-Momentum Tensors as Operators

[Alain De Vos]

Can these tensor be seen as operators on two elements.
So given two elements of something they produce something, for instance a scalar ?

 

[andrewkirk]

Yes, they are both functions that take two four-dimensional vectors as input and give a scalar as an output.
Although AFAIK there are no strict rules about this, it would be unconventional to call them 'operators'. From my observation, 'Operator' seems to usually be used to mean a function with one input that returns a result of the same type as the input (eg vector in and vector out, or scalar in and scalar out).

 

[Alain De Vos]

I wander, which four vectors do you have to choose for the tensors to operate upon in order for the resulting scalar to have an physical meaning.
Does it work on any two covariant vectors, only two and the same displacement vectors, the 4-speed vector ?

 

[andrewkirk]

For the energy-momentum tensor $T$, any two vectors can be used. $T(\vec v_1,\vec v_2)$ gives you the rate of flux, in direction $\vec v_1$, of the energy-momentum item defined by $\vec v_2$.
For the Einstein tensor it will also be any two vectors, but somebody else will need to tell you what the physical significance is.