SI unit and coordinate conventions of the classical gravitational framework

This item fixes the unit and coordinate readings of the explicitly defined Lorentzian objects, smooth-curve proper time and four-velocity of Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model, the tensor fields of Smooth covariant 2-tensor fields on a smooth manifold, and the constructed connection, curvature, Ricci and scalar fields of Metric compatibility, torsion freedom and the metric-selected connection class. It states conventions, not a physical law or a measurement.

(a) Constants and tensor slot. Adopt $c>0$ with SI dimension $\mathrm{m}{\mathrm{s}}^{-1}$, $G_N>0$ with dimension ${\mathrm{m}}^3{\mathrm{k}\mathrm{g}}^{-1}{\mathrm{s}}^{-2}$, and a signed parameter $\mathrm{\Lambda }$ with dimension ${\mathrm{m}}^{-2}$. No numerical value is asserted. Define $\kappa :=8\pi G_N\mathrm{/}{c}^4$, of dimension ${\mathrm{m}}^{-1}{\mathrm{k}\mathrm{g}}^{-1}{\mathrm{s}}^2$. For a smooth symmetric covariant two-tensor $T$ adopt the convention that its components $T_{AB}=T(e_A,e_B)$ in an ordered $g$-orthonormal basis are energy per volume, $\mathrm{J}{\mathrm{m}}^{-3}=\mathrm{k}\mathrm{g}{\mathrm{m}}^{-1}{\mathrm{s}}^{-2}$. This is a tensor slot and unit convention; interpreting a particular $T$ as matter is a subsequent definition. Therefore $\kappa T_{AB}$ has dimension ${\mathrm{m}}^{-2}$. No positivity, conservation or matter realization is assumed.

(b) Coordinates and dimensions. In a length-coordinate chart, $x^0=ct$ and all four coordinates have units $\mathrm{m}$; $g_{ab}$ are dimensionless and the quadratic line element $d{s}^2=g_{ab}d{x}^{a}d{x}^b$ has units ${\mathrm{m}}^2$. The Christoffel symbols defined by ${\mathrm{\nabla }}_{{\mathrm{\partial }}_b}^g{\mathrm{\partial }}_c={\mathrm{\Gamma }}^a{}_{bc}{\mathrm{\partial }}_a$ (Christoffel symbols of an affine connection) have units ${\mathrm{m}}^{-1}$: the constructed formula differentiates dimensionless $g_{ab}$ once with respect to a length coordinate. Curvature components have units ${\mathrm{m}}^{-2}$ since the curvature operator differentiates the connection once more and includes products of two connection coefficients; so do ${\mathrm{Ric}}_{ab}$ and scalar curvature $S$ in these coordinates. In a time-coordinate chart $x^0=t$ has units $\mathrm{s}$, and $g_{00}$ has units ${\mathrm{m}}^2{\mathrm{s}}^{-2}$; mixed, spatial, angular or other chart components have the dimensions obtained by the coordinate Jacobians. A single component has no universal dimension in a general chart. Orthonormal-frame readings use the normalized geometric basis, with its length normalization.

(c) Curves and invariant readings. For the smooth future timelike curve and positive scaling parameter of Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model (e), read $c$ as in (a) and $\tau (s)={\tau }_0+c^{-1}{\int }_{s_0}^s\sqrt{-g({\gamma }^{\mathrm{\prime }},{\gamma }^{\mathrm{\prime }})}dr$ as a proper-time parameter in seconds. Its four-velocity $U=d\gamma \mathrm{/}d\tau$ has length-coordinate and normalized orthonormal components in $\mathrm{m}{\mathrm{s}}^{-1}$; in a time-coordinate chart $U^0=dt\mathrm{/}d\tau$ is dimensionless and other component dimensions follow the coordinates. Lorentzian length $c\mathrm{\Delta }\tau$ is in metres. The tensors $R^g$, $\mathrm{Ric}$ and the scalar $S$ have the orthonormal readings ${\mathrm{m}}^{-2}$, whereas the lowered curvature components in a mixed chart have the units required by the metric and coordinate slots.

(d) Future observer readings and SI restoration. Let $n$ be a future unit timelike vector, $g(n,n)=-1$ and $g(n,N)<0$. Define ${\rho }_n:=T(n,n)$, $j_n:=-\mathrm{♯}(T(n,\cdot ))$ and its spatial part ${\varphi }_n:=j_n-{\rho }_{n}n$; indeed $g(j_n,n)=-{\rho }_n$, so $g({\varphi }_n,n)=0$. In a $g$-orthonormal basis with $e_0=n$, set ${\varphi }^i=-T(e_0,e_i)=T^{0i}$ for $i=1,2,3$. The SI energy flux vector and SI momentum density covector are, respectively,

$q_n:=c{\varphi }_n,p_n:={\varphi }_n^{\mathrm{♭}}\mathrm{/}c,q_n^i=-c{T}_{0i},(p_n{)}_i=-T_{0i}\mathrm{/}c\mathrm{.}$

Thus ${\rho }_n$ and each ${\varphi }^i$ have units $\mathrm{J}{\mathrm{m}}^{-3}$; $q_n^i$ has units $\mathrm{W}{\mathrm{m}}^{-2}$ and $(p_n{)}_i$ has units $\mathrm{k}\mathrm{g}{\mathrm{m}}^{-2}{\mathrm{s}}^{-1}$. Stress components $T_{ij}$ have units $\mathrm{J}{\mathrm{m}}^{-3}=\mathrm{P}\mathrm{a}$. Tensor components retain the common energy-density units; the factors of $c$ convert their mixed readings to different SI quantities. Future-directedness and the signs here are part of the convention, not positivity assumptions on $T$.

(e) Natural units. Setting $c=1$, or additionally $G_N=1$, abbreviates these dimensionful conventions. Restore the factors displayed above before comparison with SI quantities; in particular the same spatial tensor component corresponds to energy flux multiplied by $c$ and momentum density divided by $c$. In a chart whose coordinates $x^0,x^1,x^2,x^3$ are all lengths and whose metric components are $\mathrm{diag}(-1,1,1,1)$, a nonzero null tangent vector $v=(v^0,\mathbf{v})$, null as in clause (b) of Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model, satisfies $\mathrm{\parallel }\mathbf{v}{\mathrm{\parallel }}_2:=\sqrt{(v^1{)}^2+(v^2{)}^2+(v^3{)}^2}=\mathrm{\mid }{v}^0\mathrm{\mid }$. Since $v\mathrm{\ne }0$, this implies $v^0\mathrm{\ne }0$, so $\mathrm{\parallel }\mathbf{v}{\mathrm{\parallel }}_2\mathrm{/}\mathrm{\mid }{v}^0\mathrm{\mid }=1$. This is a coordinate identity for the null condition, not a postulate about light propagation, and it does not change the physical constant $c$.