Can a Physical Law Formulated by One component Tensor ?

[Antonio Lao]

Can a Physical Law Formulated by One component Tensor ?

The number of component of a tensor of any rank is given by

$$c = d^r$$

where c is the number of component, d is the dimension of the tensor, r is the rank of the tensor.

For r=0, the tensors are the scalars. For r=1, the tensors are the vectors.

For r=0
$$0^0 = 1$$
$$1^0 = 1$$
$$2^0 = 1$$
$$3^0 = 1$$
$$4^0 = 1$$
the above show that for scalar tensors, there is only one component for any dimension. And for scalar tensors even the zero dimension has one component.

For r=1
$$0^1 = 0$$
$$1^1 = 1$$
$$2^1 = 2$$
$$3^1 = 3$$
$$4^1 = 4$$
the above show that for vector tensors, the number of component is the same as the dimension.

For r=2
$$0^2 = 0$$
$$1^2 = 1$$
$$2^2 = 4$$
$$3^2 = 9$$
$$4^2 = 16$$

For r=3
$$0^3 = 0$$
$$1^3 = 1$$
$$2^3 = 8$$
$$3^3 = 27$$
$$4^3 = 64$$

For r=4
$$0^4 = 0$$
$$1^4 = 1$$
$$2^4 = 16$$
$$3^4 = 81$$
$$4^4 = 256$$

From these, it can be noted that only in 1D is the number of component equals 1 for any tensor. So when a physical law is formulated in one dimension, it can represent tensor of any rank and no transformation is needed hence a coordinate system is not necessary.

 

[kurious]

Electromagnetism (rank 1) and gravity (rank 2)would be equal in 1D.

 

[Antonio Lao]

That's right. They have the same number of component.