Stationary Lorentzian data and tensor conventions

Work in ZFC, including the axiom of choice. Stationary Lorentzian data here consist of a connected smooth four-manifold $M$ without boundary, an orientation, a smooth Lorentzian metric $g$ of signature $(-,+,+,+)$, a time orientation $N$, and a smooth vector field $T$ satisfying ${\mathcal{L}}_{T}g=0$. The orientation is a smooth choice of a positive ray in $\det T_{p}M$ at each point. The time orientation is a smooth field with $g(N,N)<0$; a timelike vector $v$ is future-directed when $g(v,N)<0$.

The metric is a smooth symmetric nondegenerate covariant two-tensor with one negative and three positive directions at each point. The Lie derivative ${\mathcal{L}}_{T}g$ is $\frac{d}{dt}{\mathrm{\mid }}_{t=0}{\mathrm{\Phi }}_t^{\ast }g$, where ${\mathrm{\Phi }}_t$ is the local flow of $T$; a field satisfying its vanishing is called a Killing field. Thus stationary refers here to the specified pair $(g,T)$; it does not assert that $T$ is timelike everywhere or that its flow is complete. For a designated open region called an end, the additional convention that it is a stationary end requires $T$ to be future timelike throughout that region. Further topology or asymptotic conditions on an end must be stated separately.

Let $\mathrm{\nabla }$ be the Levi–Civita connection: the unique torsion-free affine connection compatible with $g$. Its curvature convention is $R(X,Y)Z={\mathrm{\nabla }}_X{\mathrm{\nabla }}_{Y}Z-{\mathrm{\nabla }}_Y{\mathrm{\nabla }}_{X}Z-{\mathrm{\nabla }}_{[X,Y]}Z$. In a coordinate frame write $R({\mathrm{\partial }}_c,{\mathrm{\partial }}_d){\mathrm{\partial }}_b=R^a{}_{bcd}{\mathrm{\partial }}_a$ and ${\mathrm{Ric}}_{bd}=R^a{}_{bad}$. The Ricci tensor is denoted $\mathrm{Ric}(g)$.

Put $T^{\mathrm{♭}}=g(T,\cdot )$ and $f=-g(T,T)$. In coordinates, $T_a=g_{ab}{T}^b$ and $g^{ab}$ denotes the inverse matrix of $g_{ab}$. Repeated upper and lower indices are summed from $0$ to $3$. Raising indices uses $g^{ab}$ and lowering them uses $g_{ab}$; a contraction or raised tensor associated with another metric uses that metric and its own inverse.

Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model

Killing fields of a Lorentzian metric

Metric compatibility, torsion freedom and the metric-selected connection class

Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields

The inverse metric, musical isomorphisms and index raising

The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors

Oriented smooth manifolds and oriented charts

The Axiom of Choice

Given: The data and hypotheses in the Definition.

Goal: Show that the named fields and conventions are well-defined under those hypotheses.

1.1 The metric and time-orientation hypotheses place $(M,g,N)$ in the Lorentzian interface; the chosen orientation is the determinant-line datum in that interface's smooth-manifold setting. The local-flow Lie derivative and the Killing condition therefore have the stated meanings. The end condition is a further pointwise requirement on the designated region, not an assertion that such a region exists. [given, F1, F2, F6]

2.1 Nondegeneracy and smoothness supply the unique metric-compatible torsion-free connection and the inverse metric. The curvature interface then supplies the displayed Ricci contraction, while $T^{\mathrm{♭}}$ and $f$ are pointwise contractions of the supplied smooth fields. Raising and lowering with the inverse and metric of the metric currently specified is consequently unambiguous. This establishes the stated conventions without asserting completeness or an existence theorem for an end. [given, F3, F4, F5, step 1.1] ∎

Carroll supplies the symmetric nondegenerate metric and Lorentzian signature conventions in Chapter 2. The flow definition of the Lie derivative and the Killing condition are recorded in Chapter 5; only their restriction to local flow domains is used here. Time orientation and future direction use the declared Lorentzian interface.