Checking validity of simple tensor equations

1. Apr 19, 2009

trv

Can someone help me sort out the LaTeX firstly?

Hoping someone can check that my solutions are correct/not. Also do comment on any impreciseness in any of my reasons.

1. The problem statement, all variables and given/known data
Determine which of the following equations is a valid tensor equation. Describe the errors in the other equations.

2. Relevant equations

$$g=A^{cc}B_{dd}$$

Not valid. Since summing over two raised indices or two lower indices, does not result in scalars.

$$P^{ab}=Q^{ab}-Q^{ba}$$

Valid. (Almost a guess here.)

$$R^a_b=S^cT^a_cU^cV_cb$$

Not valid. Four "c"s on the right term. Therefore unclear on how to sum over the repeated "c"s.

$$D^a_{cb}=Q^a_c+M^a_b$$

Not valid. Firstly, can't subtract a rank 2 tensor from another rank 2 tensor to get a rank 3 tensor. Also, can't add the two tensors on the right, since although they are of similar rank, their indices are different.

$$f=G^a_bK^d_aH^a_cL^a_d$$

Not valid. 4"a"s on right term so summing over indices ill defined. Also, b and c can't be summed over, so we wouldn't get a scalar.

$$P_a=A_a^bB_b+U_cV^dW_d$$

Not valid. The two terms on the right reduce to covariant vectors, but the resultants would have a and d indices. These being different, the two resultant terms can't be added to get a single term.

$$X_{ab}=Q^c_{bca}+U_bW_a$$

Valid.

$$h=\delta^aV^a-\delta^b\delta_cZ^c_b$$

Not valid. The first term on the right can't be reduced to a scalar, since the two repeated indices are both raised.

$$A_{ab}=B_a+C_b+D_{ab}$$

Not valid. The three terms on the right aren't all of rank 2 as they should be if they are to be added and result in a rank two term(the left term).

$$M^{ab}=G^{ab}+Q^aR^b$$

valid

$$J_b=T^c_aF_{bc}Z^{ad}_d$$

valid.

$$K^c_b=Y_aZ^{ac}_aX^a_b$$

Not valid. The "a" index is repeated 4 times in the right term, while makes summation ill-defined.

3. The attempt at a solution

Below each equation.

Last edited: Apr 19, 2009