# What kind of tensor is the electromagnetic field tensor?

1. Mar 14, 2012

### center o bass

The covariant form of the Lorentz force can be written as

$$m \ddot x^\mu =q F^{\mu \nu} \dot x_\nu$$

and such a relation should prove by the quotient rule that F is indeed a tensor.
But what kind of tensor is it? One can show that it transforms from an unprimed
to a primed system like

$$F'^{\mu \nu} = \Lambda^\mu_{\ \alpha} \Lambda^{\ \nu}_{\beta} F^{\alpha \beta} = \frac{\partial x'^\mu}{\partial x^\alpha} \frac{\partial x^\nu}{\partial x'^\beta} F^{\alpha \beta}$$,

where $$\Lambda^{\ \nu}_{\beta}$$ is the inverse Lorentz transformation matrix. But what kind of tensors transforms like this? Does it have a name? I know about covariant, contravariant and mixed tensors. The closest I get is a mixed tensor but it would transform like

$$T^\mu_\nu = \frac{\partial x'^\mu}{\partial x^\rho }\frac{\partial x^\sigma}{\partial x'^\nu} T^\rho_\sigma$$.

Last edited: Mar 14, 2012
2. Mar 14, 2012

### tiny-tim

3. Mar 14, 2012

### center o bass

Hi Tim!

But then it should transform like

$$F'^{\mu\nu} = \frac{\partial x'^\mu}{\partial x^\alpha} \frac{\partial x'^\nu}{\partial x^\beta} F^{\alpha \beta}$$

and not as

$$F'^{\mu \nu} = \frac{\partial x'^\mu}{\partial x^\alpha} \frac{\partial x^\nu}{\partial x'^\beta} F^{\alpha \beta}$$

which arises from the fact that the inverse lorentz transformation gives the equality

$$\frac{\partial x^\mu}{\partial x'^\nu} = \Lambda_\nu^{\ \ \mu}$$

and that I have shown that

$$F'^{\mu \nu} = \Lambda^\mu_{\ \ \alpha} \Lambda^{\ \ \nu}_{\beta} F^{\alpha \beta}$$

from transforming the lorentz force law in covariant form.

Last edited: Mar 14, 2012
4. Mar 14, 2012

### center o bass

Maybe there is a flaw in my proof? It goes as follows: Ignoring the constants m and q

$$\ddot x'^\mu = F'^{\mu \nu} \dot x'_\nu =\Lambda^\mu_{\ \alpha} (\dot x^\alpha) = \Lambda^\mu_{\ \alpha} (F^{\alpha \beta} \dot x_{\beta}) = \Lambda^\mu_{\ \alpha} F^{\alpha \beta} \Lambda_\beta^{\ \ \nu} \dot x'_\nu$$,
from which it should follow that

$$F'^{\mu \nu} = \Lambda^\mu_{\ \alpha} \Lambda_\beta^{\ \ \nu} F^{\alpha \beta} = \frac{\partial x'^\mu}{\partial x^\alpha} \frac{\partial x^\nu}{\partial x'^\beta} F^{\alpha \beta}$$.

Here I have used the lorentz transformation and the inverse transformation
$$x'^\mu = \Lambda^\mu_{\ \ \nu} x^\nu$$

$$x'_\mu = \Lambda_\mu^{\ \ \nu} x_\nu$$

from which it follows that

$$\frac{\partial x^\mu}{\partial x'^\nu} = \Lambda_\nu^{\ \ \mu}$$
and

$$\frac{\partial x'^\mu}{\partial x^\nu} = \Lambda^\nu_{\ \ \mu}$$.