As an abstract mathematical entity, is a tensor different from a matrix?
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As an abstract mathematical entity, is a tensor different from a matrix?
They are similar. A matrix has a single value in each position, but a tensor has a function at each position.
(That pretty much exhausts my understanding of tensors!)
I'm sure Markus could say more ...
Just for the sake of curiosity I started a thread on tensors last Feb. It grew like topsy between two "big hitters"
Tensors - Analysis and Calculus - Science Forums
Not very illuminating but it seems to be (as is often the case) a convoluted subject.
Markus does have stuff on his own website but it is too advanced for me to benefit from.
I know though that it is important (central?) to the maths of GR....
Yes, they are not the same. The situation is that every tensor can be represented by ( i.e. written as ) a matrix of appropriate rank and dimension, but not every matrix is automatically a tensor ! The difference is in how these objects behave under changes in coordinate basis - such a change will leave a tensor invariant, but not in general an arbitrary matrix.
Many of the important tensors used in GR are rank-2 tensors in four dimensions ( i.e. tensors written with two indices ), and you can represent such objects as a 4x4 matrix, meaning they have 16 components. Just remember that the opposite is not true - not every 4x4 matrix is automatically a tensor.
The defining characteristic of a tensor is how it transforms under changes in coordinate basis - the individual components can be either constants, or functions. For example, the components of the Minkowski metric tensor are constants, whereas the components of (e.g.) the Ricci tensor are usually functional expressions.
Yes, it is important precisely because of the way a tensor is defined - it's a mathematical object that remains invariant under changes in coordinate basis. Physically, a change in coordinate basis means you pick a different observer, so, if you write a law of physics in terms of tensors, then that means that this law will have the same form for all observers. This is obviously very convenient, since it highlights the actual underlying physics principle, as opposed to obfuscating it with math formalisms that look different for every observer.Quote:
I know though that it is important (central?) to the maths of GR....
Note that you don't have to use tensors, they are not absolutely required - but they do give you the simplest, most concise, most clear formalism.
Do you mean that the tensor components are not invariant, Guitarist? When might a tensor not be invariant under a coordinate transformation?
Well, I guess in order to be more technically accurate, we would need to say that the tensors we are talking about here are invariant under diffeomorphisms, i.e. smooth mappings. So, the coordinate transformations need to be differentiable, they need to be 1-on-1, and they need to be invertible ( and the inverse must also be differentiable ), which is of course a more stringent requirement than just any old change in basis - you are right in that I should have been more accurate with this.
Physically speaking, the central idea in this context is that - when written in terms of tensors - the laws of physics take the same form for all observers.
Same applies - tensors and tensors fields are NOT invariant under any mapping from one coordinate set to anotherMakes no difference - tensors and their fields are not invariant under any mapping from one coordinate basis to another.Quote:
So, the coordinate transformations need to be differentiable, they need to be 1-on-1, and they need to be invertible ( and the inverse must also be differentiable ),
Which is an entirely different statement, an altogether different claim.Quote:
the laws of physics take the same form for all observers.
I'm lost, then. Is it a set of special cases that prevent the general statement from being true? Or is the term "invariant" a bit more subtle than I understand?
Perhaps you can provide for us the context then, that connects the notion of "tensor" to this principle ? While it may be an altogether different claim, in practical terms it is precisely those laws which are written with tensors, that take the same mathematical form for all observers. General covariance is the key concept here.Quote:
Which is an entirely different statement, an altogether different claim.
But do remember that the audience are lay people, not academics...explaining something at a level appropriate to the target audience is a special kind of skill ;-)
P.S. I am just realising that my choice of terminology is probably the problem - it may be necessary to say that tensors are covariant/kontravariant, rather than invariant, and hence that they obey transformation laws that ensure that this is the case. In many physics texts these terms seem to be used interchangeably, but they really don't mean the same thing. I can see that this distinction actually makes a pretty big difference - tensors do vary under changes in coordinates, but they do so in a way that leaves their overall form ( but not their components ) unchanged, so that the form of laws written in terms of them is the same for all observers. That's not the same as "invariance", which would mean that the components of the tensor remain unchanged under such operations, which is obviously not the case. In my defence I have to say though that many physics texts don't always make a clean distinction here - the statement that "tensors are invariant under coordinate transformations" is one that I have come across very, very often in the literature.
Yes, and I am "just realizing" I was starting to sound testy. Sorry for that.
Look, all - the terminology in this subject area is a minefield. To sufficiently confuse you all.......
1. Suppose we denote a vector, tangent to some manifold, as where the are basis vectors, the are called scalar vector components, and the are coordinate functions.
Then for any coordinate transformation , we have two choices; fix the basis vectors and allow the scalar components to vary, or fix the scalar components and allow the basis vectors to vary. Somewhat surprisingly (given the coordinate transformation above, it is common in physics (and elsewhere to be fair) to fix the basis and require that the scalar components to be invariant.
This (perhaps) excuses the physicist's lamentable use of for a vector, for what Markus refers to as a "contravariant" tensor. But note this: If I appear smaller relative to my surroundings, I have either shrunk, or my surroundings have enlarged - Relativity says there is no objective way to distinguish between them. That is why the terms "covariant" and "contravariant" have fallen into disuse.
2. The principle of "general covariance", which for a field theory means that, whatever choice of coordinates at each and every point at which the field is defined, then the theory remains the same. This condition is given (in part) by something called the "covariant derivative" - this allows a form of differentiation of coordinates at a point with respect to those at another point (which is of course not in general possible for "ordinary" differentiation) This was at one time called the "invariant derivative".
Note that this is entirely unrelated to the outdated classification of tensors as "covariant" (for completeness, these are the tensor, or outer, product of vectors dual to elements in a vector space
I wonder if this was the source of crossed wires between me and Markus?
No need, Guitarist ! You were absolutely right in calling me out on this. I actually feel a little stupid for such an elementary mistake. Really just laziness. But the thing is, if a certain choice of terminology - like tensors being invariant under changes in coordinates - is commonly used, you tend to pick it up and not think about it anymore. Even if it's wrong.Quote:
Yes, and I am "just realizing" I was starting to sound testy. Sorry for that.
I think I so readily adopted the terminology, because unconsciously I think of invariance with respect to the form of a mathematical expression. So if I take a tensor equation in one frame, and I perform a change in coordinate basis, then the overall mathematical form of that equation will remain unchanged; so the form of the expression is "invariant" in that sense.Quote:
I wonder if this was the source of crossed wires
I just want to thank everyone for their contributions, I didn't expect this goldmine, sweet!
And a big "Welcome Back!" to our long lost Guitarist.
Yes ,this toing and froing between the experts is very interesting even if there may be no hope of really understanding in "real time" the points being made. It is a bit like breaking a watch to see how it works (for the spectators ,that is).
It is also interesting to realize that the terminology can be as confusing and detailed at times for them as us (although that should be obvious)- and it also allows a little light into terrain that will possibly never be explored directly.
Lol, there is really only one expert on this thread...and it's not me ;)Quote:
Yes ,this toing and froing between the experts is very interesting
I'm just an average Joe doing this as a hobby.