General Relativity/Rigorous Definition of Tensors
We have seen that a 1-form ("covariant vector") can be thought of an operator with one slot in which we insert a vector ("contravariant vector") and get the scalar . Similarly, a vector can be thought of as an operator with one slot in which we can insert a 1-form to obtain the scalar . As operators, they are linear, i.e., .
A tensor of rank n is an operator with n slots for inserting vectors or 1-forms, which, when all n slots are filled, returns a scalar. In order for such an operator to be a tensor, it must be linear in each slot and obey certain transformation rules (more on this later). An example of a rank 2 tensor is . The symbol (pronounced "tensor") tells you which slot each index acts on. This tensor is said to be of type because it has one contravariant slot and one covariant slot. Since acts on the first slot and acts on the second slot, we must insert a 1-form in the first slot and a vector in the second slot (remember, 1-forms act on vectors and vice-versa). Filling both of these slots, say with and , will return the scalar . We can use linearity (remember, the tensor is linear in each slot) to evaluate this number:
We don't have to fill all of the slots. This will of course not produce a scalar, but it will lower the rank of the tensor. For example, if we fill the second slot of , but not the first, we get a rank 1 tensor of type (which is a contravariant vector):
For another example, consider the rank 5 tensor . This is a tensor of type . We can fill all of its slots to get a scalar:
Filling only the 3rd and 4th slots, we get a rank 3 tensor of type :
As a final note, it should be mentioned that in General Relativity we will always have a special tensor called the "metric tensor" which will allow us to convert contravariant indices to covariant indices and vice-versa. This way, we can change the tensor type and be able to insert either 1-forms or vectors into any slot of a given tensor.