Metrics of degree higher than 2

Question

Suppose I have a metric (special relativity) as follows:

$$ (ds)^2 = -(dt)^2+(dx)^2+(dy)^2+(dz)^2 \tag{1} $$

Here there are no cross-terms. Let's consider for instance this metric (which could be used in general relativity, and has cross-terms):

$$ (ds)^2=-(dt)^2+(dx)^2+(dy)^2+(dz)^2+(dxdy)+... \tag{2} $$

Both the previous metrics have "degree 2". I am interested in metrics that are of "degree 4". For example:

$$ (ds)^4=(dt)^4+(dx)^4+2(dx)^2(dy)^2+(dx)^3dy+... \tag{3} $$

We note that this metric cannot be written as a polynomial of degree 2.

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How can I work with such a metric using conventional tools?

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For instance, in special relativity (equation 1), I can write the metric as a tensor:

$$ (ds)^2=\pmatrix{dt&dx&dy&dz}\pmatrix{-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1}\pmatrix{dt\\dx\\dy\\dz}\tag{4} $$

How can I write $(ds)^4$ (equation 3) as a metric tensor using a similar form as equation 4, but adapted to a degree 4 metric polynomial?

Answer 1

Instead of writing $ds^2=g_{ab}dx^adx^b$ with $g$ symmetric, you'd want $ds^4=g_{abcd}dx^adx^bdx^cdx^d$ with $g$ fully symmetric. But there are very good theoretical reasons physicists work with degree-$2$ metrics, such as wanting to be able to move indices viz.$$dx_a=g_{ab}dx^b\implies ds^2=dx_adx^a,\,dx^b=g^{bc}dx_c\implies dx_a=g_{ab}g^{bc}dx_c\implies g_{ab}g^{bc}=\delta_a^c.$$

Answer 2

You should keep in mind that the metric in GR is a "semi-definite bi-linear symmetric form on every tangent space that varies smoothly with the base point". In the same manner you can define a symmetric n-linear form on every tangent space ($n \geq 2$ can be whatever you want), but you wouldn't call it a "metric".