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The universe is often described as being 4-dimensional (having 3 spatial dimensions + the dimension of time).

Some theories (such as some string theories) consider the universe as having even more dimensions. In some cases, some of the dimensions are viewed as "curled up".

For example: according to many versions of string theory, the universe could have 10 dimensions. Assuming that these theories are supported by relevant mathematics, it should be possible to explain the conclusion of the amount of dimensions mathematically. The equations used for this explanation therefore would use some notation containing the amount of dimensions.

What notation is used for the dimensions in physics equations? Are they indicated as exponents, and if so, are there cases when they are not present as exponents?

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  • $\begingroup$ Comments to the question (v1): This notational question seems off-topic/non-constructive, cf. this meta post. That said, here are some examples appearing in the literature: 1. The dimension is sometimes indicated with a superscript, e.g. ${\cal M}^{10}=M^4\times K^6$. 2. Whether we mean the 3-dimensional spatial metric tensor $g^{(3)}_{ij}$ or the 4-dimensional spacetime metric tensor $g^{(4)}_{\mu\nu}$ is sometimes indicated with a superscript in parenthesis. $\endgroup$
    – Qmechanic ♦
    Commented Dec 10, 2014 at 14:17

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Typically we use a superscript for the dimension in contravariant objects and a subscript for covariant objects. Traditionally greek super/subscripts are used to indicate all the dimensions and latin super/subscripts if we are considering only the spatial dimensions. So for example if $\bf x$ is a four vector we would write its components as $x^\alpha$, or if we were only considering the spatial components we would write $x^a$. Alternatively if $\bf g$ is a oneform we would write $g_\alpha$ or $g_a$.

Introductory books will warn students not to get mixed up with superscripts meaning components and superscripts meaning exponents, but the context is usually obvious so it's hard to get seriously confused.

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