# Tensor Equations

• I
I understand that tensor equations are expressions that give the same answer regardless of the coordinate system when expressing the laws of nature. Does this invariance apply across different reference frames? Another words, do the tensor equations yield the same laws with respect to an inertial frame and with respect to an accelerated frame?

fresh_42
Mentor
2021 Award
You can think of tensors the same way you think about vectors and matrices, which are tensors. E.g. we can talk about a refection, but as soon as we write it as a matrix we will have chosen a coordinate system, a reference frame. If you observe someone standing in front of a mirror, your position will change what you see, but the reflection is still the same. Tensors are not different.

jambaugh
Gold Member
Yes, when working with tensor fields. In General Relativity in particular, there are no longer assuredly global inertial frames, one can only speak of an instantaneous local inertial frame. Thus the formulation must be expressed in a generalized (accelerated) coordinate frame.

Note that one can speak of tensors in a tensor space (and vectors are a form of tensor) or in a more complex setting one can work with tensor fields (which is what happens in say SR and especially GR). The "best" way to think of tensors in my opinion is as specific representations of the group of general linear transformations, but to do that requires you get more familiar with groups and their representations. You start with scalars (transforming trivially) and vectors (with the fundamental defining representation). Thence all the other enumerated representations defines the other ranks of tensors. But if you want to bypass the group theory you can think of tensors as composites of vectors (you can e.g. think of a matrix as a row vector of column vectors or vise versa and hence it is a "bivector" of sorts. A rank 3 vector as a vector of vectors of vectors etc.)

Now with tensor fields you have a tensor space at each point in a manifold (a manifold being like a space without the flatness that lets you add displacements as vectors, for example the "space time continuum").

So in this setting, in a physical application you can describe the dynamic evolution of a system by its motion in the space-time manifold and its transformation (rotation or or other general linear transformation and change in direction of motion) as its position moves. (Think, for example, of spinning asteroids flying through space as time passes.)

And I'll finally mention, what you were getting at with your question, when we are dealing with a flat manifold, we can treat it as a vector space and this allows us to lump position vectors and displacement vectors and force vectors and so on. We then can work in linear coordinates (vectors) which, when working in unified space-time means we are working within an inertial frame. But we may in those cases and are forced to in the case where the manifold is no longer necessarily flat,... <breathe> to work with arbitrary space-time coordinates. We then need to connect vectors and tensors representing e.g. forces, directions of motion, moments of inertia etc, at any one point to those at other points. We have vectors and tensors at each coordinate point and a geometric structure which sets the framework by which we compare these at different points. This provides a very general setting to describe what is happening (as well as describe artifacts of our choice of description which are not physical). Hence we can formulate rules and physical laws in a most general way without the necessity of assuming such things as "inertial frames" are even defined globally much less that we are actually working within one.

Sorry if this answer is overkill.

It is a lot to absorb. It will take me time to digest it, but thanks for the explanation.

Nugatory
Mentor
I understand that tensor equations are expressions that give the same answer regardless of the coordinate system when expressing the laws of nature. Does this invariance apply across different reference frames? Another words, do the tensor equations yield the same laws with respect to an inertial frame and with respect to an accelerated frame?
Yes, and that's why they're so useful.

A reference frame implies a particular convention for assigning coordinates to events, so it is not surprising that tensor equations which work with all coordinate systems would work in all frames.