I How Can Tensors Be Evaluated to Scalars?

[LarryS]

Many descriptions of tensors define them to be these kind of multidimensional arrays of numbers, as generalizations of vectors and matrices, that transform in a certain way. But, mathematically, tensors are functions whose domains are these multidimensional arrays and whose codomains/ranges are the real numbers.

My question: Is there a general rule to evaluate any tensor, to determine the real number that is assigned to it? Perhaps by successive contractions, until it becomes a scalar?

Thanks in advance.

 

[Orodruin]

No. You cannot represent any function as a single real number (unless the domain is a single point or the function is a constant). The number depends on the arguments.

 

[Dale]

the real number that is assigned to it?

I am not sure what you mean by this. There is a real number assigned to $T^{\mu\nu…}a_\mu b_\nu …$ but there is no number assigned to $T^{\mu\nu…}$

 

[LarryS]

I am not sure what you mean by this. There is a real number assigned to $T^{\mu\nu…}a_\mu b_\nu …$ but there is no number assigned to $T^{\mu\nu…}$

I think I see where you are going. I probably need to read/study this subject a little further.

 

[PeterDonis]

Perhaps by successive contractions, until it becomes a scalar?

That is the only way to get a scalar from a tensor, yes: by contracting it, either with itself or with appropriate other tensors, so that there are no free indexes left and you have a scalar.

What are usually called the "components" of a tensor can be viewed as contractions of that tensor with appropriate combinations of basis vectors and covectors of a particular reference frame.