Check invariance under rotation group in spacetime

In summary, the conversation discusses the use of Lorentz transformations to show the covariance of an equation involving ##dp'^{\mu}/ds## and its transformation into ##dp^{\mu}/ds##. The properties of ##C##, the transformation, are used to show that the equation is manifestly covariant under Lorentz transformations. There is also a discussion on the transformation properties of various quantities involved in the equation.
  • #1
mcas
24
5
Homework Statement
Check invariance under rotation group in spacetime of a relativistic Newton equation of a charged particle in e-m field with Lorentz force.
Relevant Equations
##\frac{dp^\mu}{ds}=\frac{e}{c}F^{\mu \nu} u_\nu##
I started by inserting ##ds=\sqrt{dx'^{\mu} dx'_{\mu}}## and ##p'^{\mu}=mc \frac{dx'^{\mu}}{ds}##.
So we have:
$$\frac{dp'^{\mu}}{ds}=mc \frac{d}{dx'^{\mu}} \frac{d}{dx'_{\mu}} (x'^{\mu})$$
Now I know that
##dx'^{\mu}=C_\beta \ ^\mu dx^\beta##
and
##dx'_{\mu}=C^\gamma \ _\mu dx_\gamma##
where ##C## is the transformation and ##C_\beta \ ^\mu C^\gamma \ _\mu = \delta^\gamma _\beta##.
Taking this, we have
$$\frac{dp'^{\mu}}{ds}=mc \frac{d}{C_\beta \ ^\mu dx^\beta C^\gamma \ _\mu dx_\gamma} (x'^{\mu})=mc \frac{d}{dx^\beta dx_\beta} (x'^{\mu})=mc \frac{d}{ds} (x'^{\mu})$$

And now if I were to write ##x'^{\mu}=C_\delta \ ^\mu dx^\delta##, this equation wouldn't be an invariant but I think it should be.
 
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  • #2
I'm a bit puzzled, what is really asked? You have an equation which is manifestly covariant under Lorentz transformations. Since the rotations form a subgroup of the Lorentz group, it's also covariant under rotations.

NB: I guess "invariance" here should read "covariance" or "invariance" is in fact "form invariance" which is synonymous with "covariance" ;-))
 
  • #3
vanhees71 said:
I'm a bit puzzled, what is really asked? You have an equation which is manifestly covariant under Lorentz transformations. Since the rotations form a subgroup of the Lorentz group, it's also covariant under rotations.

NB: I guess "invariance" here should read "covariance" or "invariance" is in fact "form invariance" which is synonymous with "covariance" ;-))

I'm sorry, English is not my first language so the translation might be a little bit clanky and I assume you must be right.
So let me try rephrasing it - I have to show that this equation is covariant under Lorentz transformations using the following properties:
## A_\alpha=C_\alpha \ ^\beta A'_\beta##
##A^\alpha=C^\alpha \ _\beta A'^\beta##
##A'_\alpha=C^\beta \ _\alpha A_\beta##
##A'^\alpha=C_\beta \ ^\alpha A^\beta##
##C^\gamma \ _\alpha C_\gamma \ ^\beta=\delta_\alpha ^\beta##
##C_\alpha \ ^\gamma C^\beta \ _\gamma=\delta_\alpha ^\beta##
##\frac{\partial}{\partial x_\alpha}=C^\alpha \ _\beta \frac{\partial}{\partial x'_\beta}##
##\frac{\partial}{\partial x'_\alpha}=C _\beta \ ^\alpha \frac{\partial}{\partial x_\beta}##
(I don't have to use all of them, obviously.)

I've tried to go from ##\frac{dp'^{\mu}}{ds}## to ##\frac{dp^{\mu}}{ds}## but that didn't work as seen in my attempt above.
 
  • #4
Well, for the left-hand side you first need to deduce, how ##p^{\mu}## and ##s## transform under Lorentz transformations, and on the right-hand side ##q## and ##F^{\mu \nu}## as well as ##u_{\nu}##. So just think about the definition of the various quantities and their transformation properties under Lorentz transformations.
 

Related to Check invariance under rotation group in spacetime

1. What is the rotation group in spacetime?

The rotation group in spacetime refers to the set of all possible rotations that can be applied to a system in four-dimensional spacetime. This group includes rotations in three-dimensional space as well as rotations in time.

2. Why is it important to check for invariance under the rotation group in spacetime?

Checking for invariance under the rotation group in spacetime is important because it ensures that the laws of physics remain the same regardless of the orientation or direction of motion in spacetime. This is a fundamental principle of relativity.

3. How is invariance under the rotation group in spacetime tested?

Invariance under the rotation group in spacetime is tested by applying different rotations to a system and observing if the physical laws and equations remain the same. This can also be done mathematically by using transformation matrices to represent the rotations.

4. What are the consequences if invariance under the rotation group in spacetime is not satisfied?

If invariance under the rotation group in spacetime is not satisfied, it would mean that the laws of physics are not the same in all frames of reference. This would violate the principle of relativity and could lead to contradictory or inconsistent results in experiments.

5. Are there any exceptions to invariance under the rotation group in spacetime?

There are some cases where invariance under the rotation group in spacetime may not hold, such as in extreme conditions like near a black hole or at the quantum level. However, in most cases, the laws of physics are still expected to be invariant under the rotation group in spacetime.

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