metric $(dx) ^a+(dt)^a$

Question

Could a metric defined on a 2-dimensional space $(x,t)$ by

$(ds) ^a=(dx) ^a+(c(dt))^a $

where $a=a(x,t)>1$ and $c=$ speed of light

have any sense in physics? $\left(\text{for example } a=2-\frac{\sin x}{1000}\right)$

Answer 1

The exponent really does need to be $2$. We want to be able to raise and lower indices with metric tensors viz. $V_a=g_{ab}V^b,\,V^b=g^{bc}V_c$ so $g_{ab}g^{bc}=\frac{\partial V_a}{\partial V_c}=\delta_a^c$. Then $ds^2=g_{ab}dx^adx^b$. An alternative coordinate system with $ds^2=g_{AB}dy^Ady^B$ implies $g_{AB}=g_{ab}\frac{\partial x^a}{\partial y^A}\frac{\partial x^b}{\partial y^B}$ by the chain rule.