Understanding the Antisymmetry of the Maxwell Tensor

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In summary, Malcolm Ludvigsen introduces the Maxwell tensor and states that it is "clearly" antisymmetric. He then asks for someone to explain how to see the antisymmetry of the Maxwell tensor easily.
  • #1
M. Kohlhaas
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i am reading a book written by malcolm ludvigsen and i have difficulty in understanding the following:

he introduces the maxwell tensor via

m[tex]\ddot{x}[/tex] = eF(v)

where v is the four-velocity and [tex]\ddot{x}[/tex] the four-acceleration of a charged partice.

he then states that F(a,b) = aF(b) is "clearly" antisymmetric, i.e. F(a,b)=-F(b,a).

well, i know that it is antisymmetric. but if i wouldn't i then i'd find it quite hard to reach that conclusion having only the given information.

can someone please tell me how to easily see the obvious antisymmetry by only these information?
 
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  • #2
Hard to say what the author had in mind based on the information you've posted.

One way to see that it has to be antisymmetric is that it has to have the right number of degrees of freedom. It should have 6 d.f., corresponding to the components of E and B.

From the equation of motion of the charged particle, you also get unphysical results if, for example, F has only a nonvanishing time-time component and everything else is zero. Then a particle initially at rest would start to change its energy, without changing its three-velocity.

More generally, you want [itex]v^2=1[/itex] always (with a +--- metric). This can only happen if the derivative of this quantity is zero, which means [itex]<v,\dot{v}>=0[/itex]. To ensure that [itex]v^TFv=0[/itex] for all v, I think F has to be antisymmetric. (I could be wrong about this -- my linear algebra is rusty.)
 
  • #3
We have F(a,b)+F(b,a)=F(a+b,a+b) by linearity, which equals (a+b) dot F(a+b) by definition. Since F(v) is orthogonal to v for all v because the acceleration is orthogonal to the velocity, this equals zero. So F(a,b)=-F(b,a).

-Matt
 
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  • #4
thanks @all

dodelson said:
We have F(a,b)+F(b,a)=F(a+b,a+b) by linearity, which equals (a+b) dot F(a+b) by definition. Since F(v) is orthogonal to v for all v because the acceleration is orthogonal to the velocity, this equals zero. So F(a,b)=-F(b,a).
Aha. That's great. Thank you very much. I'm happy now.
 

1. What is the meaning of the antisymmetry property of the Maxwell tensor?

The antisymmetry property of the Maxwell tensor refers to the fact that the tensor is equal to the negative of its transpose. In other words, switching the indices of the tensor results in a negative sign. This property is a fundamental aspect of the tensor and is essential in understanding its physical significance.

2. How does the antisymmetry property affect the Maxwell equations?

The antisymmetry property has a direct impact on the Maxwell equations, as it allows for the equations to remain unchanged under certain transformations. For example, the equations are invariant under a change of coordinates or a Lorentz transformation, which is a result of the tensor's antisymmetry.

3. Can you explain why the Maxwell tensor is only defined in four-dimensional spacetime?

The Maxwell tensor is defined in four-dimensional spacetime because it is derived from the four-potential, which is a four-dimensional vector. The tensor is a combination of the second derivatives of the four-potential with respect to spacetime coordinates, and therefore, it can only exist in four dimensions.

4. What is the significance of the Maxwell tensor in electromagnetism?

The Maxwell tensor plays a critical role in electromagnetism as it encapsulates all of the information about the electromagnetic field. It is used to calculate the electric and magnetic fields at a given point in spacetime, and it also provides a mathematical framework for understanding the propagation of electromagnetic waves.

5. How is the antisymmetry property related to the conservation of energy and momentum in electromagnetism?

The antisymmetry property of the Maxwell tensor is closely related to the conservation of energy and momentum in electromagnetism. The tensor's antisymmetry allows for the conservation of energy and momentum to be derived from the equations, as it leads to the identification of the stress-energy tensor, which is responsible for these conservation laws.

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