Can Lorentz transformations be represented by matrices in EM fields?

In summary, the conversation discusses the partial operator and its use in calculating the gradient on M_{4}. It also mentions the representation of A'^{\alpha}(x') and the argument that both Ls represent the same dimensions. The question of whether L should be represented as an inverse is also raised.
  • #1
stunner5000pt
1,461
2
Show that [tex] \partial'_{\alpha} A'^\alpha (x') = \partial _{mu} A^{\mu}(x') [/tex]

lets focus on the partial operator for now
[tex] \partial'_{\alpha} = \frac{\partial}{\partial x'^{\alpha}} = \frac{\partial}{L_{\nu}^{\alpha} \partial x^{\nu}} [/tex]

Now A represents the Scalar and vector fields of an EM field.

[tex] A'^{\alpha}(x') = L_{\sigma}^{\alpha} A^{\sigma}(x') [/tex]
is that fine?

when i put them together
[tex] \partial'_{\alpha} A'^\alpha (x') = \frac{\partial}{L_{\nu}^{\alpha} \partial x^{\nu}} L_{\sigma}^{\alpha} A^{\sigma}(x') [/tex]
the argumetn is that both the L s represent the same dimensions thus the they are the same thing?

But Since L is a matrix... i can't be int eh denominator... can it? Would it simply be represented as an inverse? The two Ls still turn into idnetity matrix which is simply 1.

your helpsi greatly appreciated!
 
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  • #2
Well, the gradient on [itex] M_{4} [/itex] is a covector, so your first equation should read

[tex] \frac{\partial}{\partial x' ^{\alpha}} =\Lambda_{\alpha}{}^{\mu} \frac{\partial}{\partial x^{\mu}} [/tex].


Daniel.
 

What are Lorentz Transformations?

Lorentz transformations are mathematical equations used in physics to describe the relationship between space and time in different inertial reference frames. They were first developed by Dutch physicist Hendrik Lorentz in the late 19th century and later incorporated into Albert Einstein's theory of special relativity.

Why are Lorentz Transformations important?

Lorentz transformations are important because they allow us to understand how measurements of space and time are affected by the relative motion between two observers. They are essential for reconciling the apparent differences in measurements made by observers in different inertial frames, which is a fundamental principle of special relativity. They have also been used in various applications, including the development of the theory of electromagnetism.

What is the formula for Lorentz Transformations?

The formula for Lorentz transformations depends on the variables being transformed and the relative velocity between the two frames of reference. In general, the transformation equations involve the Lorentz factor (γ), which is equal to 1/√(1 - v²/c²), where v is the relative velocity and c is the speed of light. The specific equations can be found in most physics textbooks or online resources.

How do Lorentz Transformations differ from Galilean Transformations?

Lorentz transformations differ from Galilean transformations in that they take into account the constancy of the speed of light in all inertial frames. Galilean transformations, on the other hand, assume that the speed of light is infinite. This difference is crucial for understanding the effects of relative motion on space and time measurements.

What are some real-world applications of Lorentz Transformations?

Lorentz transformations have been used in various real-world applications, including satellite navigation systems, particle accelerators, and GPS technology. They are also used in the development of relativistic mechanics, which is essential for space exploration and understanding the behavior of objects moving at high speeds.

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