Invariance of the Lorentz transform

In summary, you proved the invariance of the electromagnetic wave equation by showing that the corresponding differential operator was an invariant.
  • #1
tina21
14
2
Homework Statement
Prove the invariance of the electromagnetic wave equation by showing that the corresponding differential operator is an invariant.
Relevant Equations
d^2/dx^2+d^2/dy^2+d^2/dz^2-1/c^2d^2/dt^2 = d^2/dx^2+d^2/dy^2+d^2/dz^2-1/c^2d^2/dt^2
of course y and z terms are invariant but for the x and t terms I am getting an additional factor of 1/1-v^2/c^2
 
Physics news on Phys.org
  • #2
tina21 said:
Homework Statement: Prove the invariance of the electromagnetic wave equation by showing that the corresponding differential operator is an invariant.
Homework Equations: d^2/dx^2+d^2/dy^2+d^2/dz^2-1/c^2d^2/dt^2 = d^2/dx^2+d^2/dy^2+d^2/dz^2-1/c^2d^2/dt^2

of course y and z terms are invariant but for the x and t terms I am getting an additional factor of 1/1-v^2/c^2
Can you use Latex to show what you got?

https://www.physicsforums.com/help/latexhelp/
 
  • #3
I haven't used latex before. I hope these images are okay
 

Attachments

  • WhatsApp Image 2019-11-24 at 7.23.02 AM.jpeg
    WhatsApp Image 2019-11-24 at 7.23.02 AM.jpeg
    42.6 KB · Views: 108
  • WhatsApp Image 2019-11-24 at 7.23.16 AM.jpeg
    WhatsApp Image 2019-11-24 at 7.23.16 AM.jpeg
    24.8 KB · Views: 128
  • #4
That doesn't look right. I prefer to differentiate a trial function. Imagine we have a function ##f(t', x')##, where ##t' = \gamma(t - vx/c^2), \ x' = \gamma(x - vt)##.

Now, we differentiate ##f## with respect to ##x## using the chain rule:

##\frac{\partial f}{\partial x} = \frac{\partial f}{\partial t'}\frac{\partial t'}{\partial x} + \frac{\partial f}{\partial x'}\frac{\partial x'}{\partial x} = \frac{\partial f}{\partial t'}(-\gamma v/c^2) + \frac{\partial f}{\partial x'}(\gamma)##

Now you need to take the second derivative of ##f## by repeating this process - and remember that you have to differentiate both terms using the chain rule, so you will get cross terms in ##\frac{\partial^2 f}{\partial t' \partial x'}##.

At the end, you can remove the trial function to leave, for example:

##\frac{\partial}{\partial x} = (-\gamma v/c^2)\frac{\partial }{\partial t'} + (\gamma)\frac{\partial}{\partial x'}##

That's what you get if you only wanted the first derivative.

Note that you can do the chain rule on differentials, but I prefer to have a function to differentiate, and then take the function away when I'm finished.
 
Last edited:
  • Like
Likes tina21
  • #5
yes, I tried doing it too. Made it a whole lot easier and I got the answer. Thanks so much !
 

What is the Invariance of the Lorentz transform?

The Invariance of the Lorentz transform is a principle in physics that states that the laws of nature should remain the same for all inertial observers, regardless of their relative motion. This means that the fundamental equations of physics, such as the laws of motion and the laws of electromagnetism, should be the same for all observers moving at a constant velocity.

Why is the Invariance of the Lorentz transform important?

The Invariance of the Lorentz transform is important because it is a cornerstone of Einstein's theory of special relativity. This principle helps us understand the behavior of objects and systems at high speeds, and it has been confirmed through numerous experiments and observations.

How does the Invariance of the Lorentz transform relate to the speed of light?

The Invariance of the Lorentz transform is closely related to the speed of light, as it is one of the key postulates of special relativity. According to this principle, the speed of light in a vacuum is the same for all inertial observers, regardless of their relative motion. This means that the laws of physics must be modified in order to accommodate this constant speed of light.

What are the mathematical equations for the Invariance of the Lorentz transform?

The mathematical equations for the Invariance of the Lorentz transform are the Lorentz transformation equations, which describe how measurements of time and space are different for observers in different inertial frames of reference. These equations involve the speed of light, the relative velocity between observers, and the concept of time dilation and length contraction.

What are some real-world applications of the Invariance of the Lorentz transform?

The Invariance of the Lorentz transform has many real-world applications, particularly in fields such as particle physics, cosmology, and GPS technology. It helps us understand the behavior of particles at high speeds, the structure of the universe, and how to accurately measure time and space for navigation purposes. It also plays a crucial role in modern technologies, such as particle accelerators and satellite navigation systems.

Similar threads

  • Introductory Physics Homework Help
Replies
10
Views
836
  • Introductory Physics Homework Help
Replies
2
Views
899
  • Introductory Physics Homework Help
2
Replies
40
Views
693
  • Introductory Physics Homework Help
2
Replies
35
Views
3K
  • Special and General Relativity
Replies
20
Views
1K
  • Introductory Physics Homework Help
Replies
5
Views
856
  • Introductory Physics Homework Help
Replies
8
Views
1K
  • Special and General Relativity
3
Replies
101
Views
3K
  • Classical Physics
Replies
4
Views
268
  • Introductory Physics Homework Help
Replies
13
Views
1K
Back
Top