Gravitation #3.14: Showing dF=0 as Geometric Version of Maxwell's Equations

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In summary, Maxwell's equations can be obtained from the geometric version by performing a Lorentz boost.
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Living_Dog
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How does one show that dF = 0 is the geometric version of Maxwell's equations??
 
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I guess you mean that dF is a 2-form in four dimensions, so it has six independent fields (the electromagnetic fields):

[tex]dF=E_idx^idt+\frac{1}{2}\epsilon_{ijk}B_idx^jdx^k[/tex]

Now the homogeneous Maxwell equation read dF=0. For the other two equations, introduce a 3-form for the 4-current

[tex]J=J_1dx^1dx^2dt+J_2dx^3dx^4dt+J_3dx^1dx^2dt+\rho dx^1dx^2dx^3[/tex]

So the inhomogeneous Maxwell equations are dF = -4 \pi J. Note that, since d^2 = 0, J satisfies the continuity equation dJ = 0.
 
  • #3
Petr Mugver said:
I guess you mean that dF is a 2-form in four dimensions, so it has six independent fields (the electromagnetic fields):

[tex]dF=E_idx^idt+\frac{1}{2}\epsilon_{ijk}B_idx^jdx^k[/tex]

Now the homogeneous Maxwell equation read dF=0. For the other two equations, introduce a 3-form for the 4-current

[tex]J=J_1dx^1dx^2dt+J_2dx^3dx^4dt+J_3dx^1dx^2dt+\rho dx^1dx^2dx^3[/tex]

So the inhomogeneous Maxwell equations are dF = -4 \pi J. Note that, since d^2 = 0, J satisfies the continuity equation dJ = 0.
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...huh? I know that dF=0 is Maxwell's equations. I asked:
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How does one show that dF = 0 is the geometric version of Maxwell's equations??
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E.g. if I wanted to show it was frame-independent, then I would perform a Lorentz boost and show how the same equation appears, but with primes.
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I'm sorry, but I don't know how to ask this question more clearly. I guess it's b/c I don't understand it. But then again, that's why I posted it.
 
  • #4
Petr Mugver said:
I guess you mean that dF is a 2-form in four dimensions, so it has six independent fields (the electromagnetic fields):

[tex]dF=E_idx^idt+\frac{1}{2}\epsilon_{ijk}B_idx^jdx^k[/tex]

Now the homogeneous Maxwell equation read dF=0. For the other two equations, introduce a 3-form for the 4-current

[tex]J=J_1dx^1dx^2dt+J_2dx^3dx^4dt+J_3dx^1dx^2dt+\rho dx^1dx^2dx^3[/tex]

So the inhomogeneous Maxwell equations are dF = -4 \pi J. Note that, since d^2 = 0, J satisfies the continuity equation dJ = 0.

I misunderstood the question. They were probably asking to show that Maxwell's equations can be obtained from this geometric version. I can do that having read section 4.5 of the text.

Thanks for your help.
 

1. What is "Gravitation #3.14"?

"Gravitation #3.14" is a mathematical concept proposed by renowned physicist Albert Einstein, also known as the "Geometric Version of Maxwell's Equations". It is a theory that explains the relationship between gravity and electromagnetism using geometric principles.

2. What does dF=0 represent in this theory?

In this theory, dF=0 represents the absence of electromagnetic fields in a region of space. This means that there are no electric or magnetic fields present, which is a necessary condition for the existence of a gravitational field.

3. How does this theory relate to Maxwell's Equations?

This theory is based on the idea that the equations of electromagnetism, known as Maxwell's Equations, can be reformulated in a geometric way to also describe the principles of gravity. This shows a deep connection between these two fundamental forces of nature.

4. What is the significance of showing dF=0 in this theory?

Showing dF=0 in this theory is significant because it provides a geometric interpretation of the absence of electromagnetic fields and their relationship to the presence of a gravitational field. It also helps to unify the understanding of these two forces in a single framework.

5. How does this theory impact our understanding of the universe?

This theory has a significant impact on our understanding of the universe as it provides a deeper understanding of the fundamental forces that govern the behavior of matter and energy. It also helps to bridge the gap between classical and quantum physics by showing the connection between gravity and electromagnetism.

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