Question

I'm a bit puzzled about the concept of an observer in 3+1 or ADM decomposition in General Relativity (GR).

The decomposition is typically described as starting with a scalar field t, whose spacelike level surfaces Σt form a foliation. Then take a vector field t with t(t) = 1 or dt(t) = 1. Denote the unit future timelike normal of Σt by n. There is then a decomposition t = Nn + β, where N and β are the lapse and shift vectors, respectively.

I see people take n as the four-velocity of the Eulerian observer. But since spatial coordinates (x₁, x₂, x₃) are constant along the integral curve of t, I would think t - |t|²√ as the four-velocity of the observer in this setting. But then it comes with an issue that t is non-timelike when
N² ≤ |β|².

Any clarification or suggestion in thinking about observers here? Thanks in advance!

Answer 1

In the ADM decomposition in General Relativity (GR), the observer can be described using either the scalar field t or t - |t|²√ as their four-velocity. The choice depends on the interpretation and context of the observation.

Step-by-step explanation:

In the ADM decomposition in General Relativity (GR), the observer is usually described as starting with a scalar field t and a vector field t, with a unit future timelike normal denoted as n. The decomposition t = Nn + β is then used, where N and β are the lapse and shift vectors respectively. The four-velocity of the observer can be taken as either t or t - |t|²√, depending on the interpretation.

When taken as t, the observer is known as an Eulerian observer, as the spatial coordinates are constant along the integral curve of t. However, when taken as t - |t|²√, there is an issue when N² ≤ |β|² as t becomes non-timelike. Both interpretations have their uses and can provide different insights, so it's important to consider the context and requirements of the observation.