Numerical Relativity: Components of the Lapse Function?

[Wallace]

I have a technical question about numerical relativity, hopefully someone can help.

In the usual 3 + 1 decomposition in NR, the four gauge freedoms are expressed via the lapse, $$\alpha$$ and three shift components $$\beta^i$$. In a finite element numerical scheme, each grid point will have a value for these 4 components.

Now, looking at evolution equations on the other hand, I see terms that want to take the covariant derivative of the lapse function, which I can't understand since it is a scalar, and hence the covariant derivative reduces to the regular derivative.

To be explicit, take for instance the ADM evolution equation for the extrinsic curvature. The textbook I have ("Elements of Numerical Relativity" by Carles Bona and Carlos Palenzuela-Luque), write this down as

$$(\partial_t - L_{\beta} ) K_{ij} = -\alpha_{j;i} + \alpha [ ...]$$

I've left the rest of the equation out for simplicity (note that $$L_{\beta}$$ is the Lie Derivative, I couldn't work out how to make the nice curly L with the tex tags). See that both $$\alpha$$ and $$\alpha^i$$ appear which I don't understand. It would make sense if this book was using $$\alpha^i$$ to denote the shift vector, but as you can see from the LHS (and it made clear in the book) it uses $$\beta^i$$ for this.

Any ideas? I'm just replacing the Covariant derivative with the regular one for $$\alpha$$ but maybe the equation is actually telling me something very different?

 

[Wallace]

I think I found the answer, comparing to other literature it looks like they were meaning

$$\alpha_i \equiv \partial_i \alpha$$

that makes sense, because the covariant derivative of that would not be 'trivial', i.e. reduce simply to the regular derivative.

 

[Entropia]

Thank you for your question about numerical relativity and the components of the lapse function. The lapse function, \alpha, is a scalar quantity that represents the rate at which time is measured in a particular coordinate system. In the 3+1 decomposition used in numerical relativity, it is one of the four gauge freedoms that are expressed through the lapse and three shift components, \beta^i. These components are necessary for the numerical scheme to accurately represent the evolution of the spacetime.

In the evolution equations, it is common to see terms that involve the covariant derivative of the lapse function. This may seem confusing at first since the lapse is a scalar quantity and the covariant derivative of a scalar reduces to the regular derivative. However, the use of the covariant derivative in these equations is not meant to represent a literal derivative of the lapse function, but rather it is used as a mathematical tool to simplify the equations and make them easier to solve numerically.

To better understand this, let's take a look at the ADM evolution equation for the extrinsic curvature that you mentioned. The term \alpha_{j;i} represents the covariant derivative of the lapse function with respect to the spatial coordinates, which may seem redundant since the lapse is a scalar. However, in this equation, the covariant derivative is being used to represent the time derivative of the lapse function, which is necessary for the numerical scheme to accurately evolve the spacetime.

In summary, the use of the covariant derivative in the evolution equations for the lapse function is a mathematical tool that simplifies the equations and allows for accurate numerical solutions. I hope this helps clarify your question.