# Strange super script notation $^{(4)}R$ in the textbook Numerical Relativity

In Numerical Relativity by Thomas W. Baumgarte and Stuart L. Shapiro.

There are bunch of superscript $$(4)$$ over $$T,\Gamma, R$$
i.e. $$^{(4)}\Gamma^a_{bc}$$ _ $$^{(4)}R_{abcd}$$ _ $$^{(4)}R$$ ... (see example in page 5).

I thought it was just indicating it's $$4$$ vectors, where I encountered a sentence in page 8 saying "the superscript $$(3)$$ (on the left instead of on the right) in the above formula indicates the third time derivative..."

I flipped through the book and this notion continuously poped up, i.e. $$^{(2)}ds^2$$ ...

1. what does the superscript on the left mean? i.e. $$^{(4)}R$$

2. when the book said third time derivative(or higher), does it just mean $$^{(3)}I =\frac{d^{(3)} I}{dt^{3}}$$? But why it wasn't indicated in the indice. i.e. page 9 equation 1.50

• “On the right instead of on the right”? – G. Smith Nov 21 '18 at 7:02

In the first paragraph on page 27, it says "Four-dimensional objects associated with $$g_{ab}$$ are denoted with a superscript $${}^{(4)}$$ in front of the symbol,..."
It most likely indicates the number of dimensions. For example, $$^{(4)}R$$ indicates a 4-dimensional Ricci scalar.