# Identifying Lorentz Covariant Equations

Statement: $\phi , A^{\mu}, T^{\mu \nu}$ are a Lorentz scalar, vector, and tensor. Which of the following equations are Lorentz covariant.

a. $\phi = A_{0}$

b. $\phi = A^{\mu}A_{\mu}$

c. $\phi = A_{0}A^{0}$

d. $\phi = T_{\mu \nu}T^{\mu \nu}$

e. $T_{\mu \nu} = T^{\nu \mu}$

f. $T_{\mu \nu} = T_{\nu \mu}$

g. $T^{\mu \nu} = A^{\mu}+A^{\nu}$

h. $T_{\mu \nu} = -T_{\mu \nu}$

i. $T_{\mu}^{\nu}=-T_{\nu}^{\mu}$

j. $T_{\mu \nu} = A^{\mu} A^{\nu}$

k. $\phi = det( T^{\mu \nu})$

l. $\phi = det( T_{\mu}^{\nu})$

From my (limited) understanding of Lorentz covariance I would identify b. and d. as Lorentz covariant, but I'm having trouble understanding how I would go about determining in general whether an equation is Lorentz covariant. I would appreciate any recommendations for non-group theory reading materials on this, or just help in general. Thanks in advance.

• It would help if you explained a little what your understanding of Lorentz covariance is, and why you chose the answers you did. – Javier Feb 9 '16 at 3:13
• Perhaps Chap.6 in these notes, and the ones leading up to it, may help: ita.uni-heidelberg.de/~dullemond/lectures/tensor/tensor.pdf – udrv Feb 9 '16 at 7:25

As someone said in the comments, you should explain what your understanding of Lorentz covariance is. I'm going to assume the Wikipedia point of view.

Any equation you (rightfully) write with a scalar on one side is Lorentz covariant. In fact, a scalar is independent of the coordinates used. So, assuming that the equations are right, a, b, c, d, k, l are all covariant.

To extend this to tensors of nonzero rank, an equation is Lorentz covariant if it transforms consistently: if on the left you have a type (h,k) tensor, so you must have on the right side: both sides have to transform the same way. With this definition, f, g, h would all be covariant. For this same reason e, j would be wrong: to transform one side you'd need two direct matrices, on the other side two inverse matrices.

For i, since on both sides are present type (1,1) tensors, it's better to manually transform them to check:

$${𝑇_{\mu}}^{\nu}=−{𝑇_{𝜈}}^{\mu}$$

Now I'm going to use the transformation matrix $$\Lambda$$ and its inverse $$\Lambda^{-1}$$, transforming the left side and checking the effects of the transformation on the right side.

$${𝑇_{𝜇}}^{\nu}\Lambda^{\lambda}_{\nu}{\Lambda^{-1}}^{\mu}_{\sigma}=−{𝑇_{𝜈}}^{\mu}\Lambda^{\lambda}_{\nu}{\Lambda^{-1}}^{\mu}_{\sigma}$$

$${𝑇}_{\sigma}^{\lambda}=−{𝑇_{𝜈}}^{\mu}\Lambda^{\lambda}_{\nu}{\Lambda^{-1}}^{\mu}_{\sigma}$$

$${𝑇}_{\sigma}^{\lambda}g^{\nu\nu}g_{\lambda\lambda}g^{\sigma\sigma}g_{\mu\mu}=−{𝑇_{𝜈}}^{\mu}\Lambda^{\lambda}_{\nu}{\Lambda^{-1}}^{\mu}_{\sigma}g^{\nu\nu}g_{\lambda\lambda}g^{\sigma\sigma}g_{\mu\mu}$$

$${𝑇}_{\lambda}^{\sigma}g^{\nu\nu}g_{\mu\mu}=−{𝑇_{𝜈}}^{\mu}\Lambda^{\nu}_{\lambda}{\Lambda^{-1}}^{\sigma}_{\mu}$$

$${𝑇}_{\lambda}^{\sigma}g^{\nu\nu}g_{\mu\mu}=−{𝑇}_{\lambda}^{\sigma}$$

Hence, since the transformed equation is wrong, the equation isn't covariant.