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## Main Question or Discussion Point

Hey,

I have not done any proper differential geometry before starting general relativity (from Sean Carroll's book: space time and geometry), so excuse me if this is a stupid question.

The metric tensor can be written as

$$ g = g_{\mu\nu} dx^{\mu} \otimes dx^{\nu}$$

and its also written as

$$ds^2 = g_{\mu\nu} dx^{\mu}dx^{\nu} $$

now lets say you have some metric ##ds^2 = A dt^2 + B dx^2##, in Sean Carroll, and various other places on the internet, they then manipulate this to obtain certain derivatives. For example, if ##ds^2 = 0##, they say that

$$A dt^2 + B dx^2 = 0 , \rightarrow \left(\frac{dx}{dt}\right)^2 = -\frac{A}{B} $$

How is this manipulation of the tensor justified? doesnt ##dx^2## actually mean ##dx \otimes dx## ? why are the elements just treated as numbers?

Thanks

I have not done any proper differential geometry before starting general relativity (from Sean Carroll's book: space time and geometry), so excuse me if this is a stupid question.

The metric tensor can be written as

$$ g = g_{\mu\nu} dx^{\mu} \otimes dx^{\nu}$$

and its also written as

$$ds^2 = g_{\mu\nu} dx^{\mu}dx^{\nu} $$

now lets say you have some metric ##ds^2 = A dt^2 + B dx^2##, in Sean Carroll, and various other places on the internet, they then manipulate this to obtain certain derivatives. For example, if ##ds^2 = 0##, they say that

$$A dt^2 + B dx^2 = 0 , \rightarrow \left(\frac{dx}{dt}\right)^2 = -\frac{A}{B} $$

How is this manipulation of the tensor justified? doesnt ##dx^2## actually mean ##dx \otimes dx## ? why are the elements just treated as numbers?

Thanks

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