Integral over 4D Metrics: Volume Element?

[Jim Kata]

I'm going to be working in 4D, and I assume my metric is symmetric. So what I am interested in is an integral of the form $$\int {d^{10} \eta \delta (\eta - \bar \eta } )$$. My question is what is the invariant volume element for an integral of this type? My guess is that it is $$\frac{{d^{10} \eta }}{{J^2 }}$$ where $$J$$ is the jacobian, but I don't know. This my logic:
$$d\bar x^\alpha = \frac{{\partial \bar x^\alpha }}{{\partial x^\beta }}dx^\beta$$ and $$d^4 \bar x = \frac{{d^4 x}}{J}$$. So since:$$\bar g^{\alpha \beta } = g^{\mu \tau } \frac{{\partial \bar x^\alpha }}{{\partial x^\mu }}\frac{{\partial \bar x^\beta }}{{\partial x^\tau }}$$ I figured $$d^{10} \bar g = \frac{{d^{10} g}}{{J^2 }}$$. I would also like the requirements that $$g_{00} < 0$$, $$\det [g_{\alpha \beta } ] < 0$$, and $$\det [\gamma _{ij} ] > 0$$ where $$\det [g_{\alpha \beta } ] = g_{00} \det [\gamma _{ij} ]$$. Any suggestions or guesses?

 

[matt grime]

I'm going to be working in 4D, and I assume my metric is symmetric.

A metric, by definition, satisfies d(x,y)=d(y,x).

 

[tacman]

Your guess is correct. The invariant volume element for an integral over 4D metrics would be \frac{d^{10}\eta}{J^2}, where J is the Jacobian of the coordinate transformation. This is because the volume element must be invariant under coordinate transformations.

To see why this is the case, let's break down the components of the integral. First, we have d^{10}\eta, which represents the infinitesimal volume element in the 4D space. This is the same for any coordinate system, as it is a fundamental property of the space itself.

Next, we have \delta(\eta-\bar{\eta}), which is a Dirac delta function that enforces the constraint that the coordinates in the two systems must be equal. This ensures that we are only integrating over the same region in both coordinate systems.

Finally, we have the Jacobian, which takes into account the stretching and shrinking of the coordinate system. As you correctly pointed out, the Jacobian for a general coordinate transformation is given by J = \det\left(\frac{\partial\bar{x}^\alpha}{\partial x^\beta}\right). Since the volume element must be invariant under coordinate transformations, we must divide by the Jacobian to account for the change in coordinates.

In terms of the requirements you mentioned, g_{00} < 0 and \det[g_{\alpha\beta}] < 0 ensure that the metric is Lorentzian, which is necessary for a 4D spacetime. The requirement \det[\gamma_{ij}] > 0 ensures that the spatial part of the metric is positive definite, which is important for defining distances and volumes in space.

In summary, your logic is correct and the invariant volume element for an integral over 4D metrics is \frac{d^{10}\eta}{J^2}.