I've been trying to grasp the concept of 'tensors' mathematically and their role in the EFE for a long time, and kept thinking I was missing some part of the explanation. I but I think it just clicked for me.

 

Does each tensor boil down to a 4th dimensional derivative? And when put together gives a value for velocity at a given point which can be plotted out consecutively to determine geodesics?

Edited January 31, 20169 yr by metacogitans

Not quite, tensors are used to show linear, or geometric relations such as vectors or scalar relationships in an arbitrary coordinate system.

 

This isn't necessarily velocity. Though is involved in the EFE. As well as geodesic mapping.

 

There is for example scalar tensor theories for gravity. Brans-Dicke gravity.

 

The Riemann curvature tensor for example represents the tidal force experienced by a rigid body.

 

https://en.m.wikipedia.org/wiki/Riemann_curvature_tensor

 

the metric tensor may loosely be thought of as a generalization of the gravitational potential familiar from Newtonian gravitation. The metric captures all the geometric and causal structure of spacetime, being used to define notions such as distance, volume, curvature, angle, future and past.

 

https://en.m.wikipedia.org/wiki/Metric_tensor

Edited January 31, 20169 yr by Mordred

 

metacogitans

Does each tensor boil down to a 4th dimensional derivative?

 

The multidimensional equivalent of the derivative is called the Jacobian and usually appears in Matrix form.

 

https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant

Edited January 31, 20169 yr by studiot

By tensor in physics one usually means the components of a tensor field. The important thing is that theses components have a nice transformation law when changing coordinates on the space-time manifold. You can look this up on wikipedia easily enough.

 

The key thing is that any equations between (the components of) tensors holds true in any coordinate system you choice. In particular setting a tensor to zero is well-defined.