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Over the last couple of weeks, I have been learning about the relativistic description of electromagnetism through Leonard Susskind's Theoretical Minimum lectures, and although I have managed tofollowit, there are some parts which I am becoming increasingly confused by, not helped by the fact Susskind uses the (-+++) convention rather than the seemingly more popular (+---).

Is the definition of the electromagnetic tensor different depending on the convention? I am aware that the matrix changes (the contravariant and covariant forms are multiplied by -1 when moving between conventions), but does the definition of each component in terms of the vector potential change?

The reason I ask this is because when deriving the relativistic Lorentz force from the Lagrangian:

L=-m√(-dX_{ϑ}/dt dX^{ϑ}/dt)-qA_{ϑ}dX^{ϑ}/dt

I always seem to get the following answer (using the (-+++) convention):

Which is the negative of what I would expect from the definition of the Electromagnetic Tensor.

Any help would be appreciated, and thanks in advance.

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# Electromagnetic Tensor with (-+++) convention

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