---
title: Vector Volume in Spacetime
url: https://www.emergentmind.com/topics/vector-volume
type: topic
---

# Vector Volume in Spacetime

Searching arXiv for the primary paper and closely related uses of “vector volume.”
arxiv_search query: "ti:\"The Vector Volume and Black Holes\" OR 1310.1935"
Vector volume is an invariant way to associate a finite, physically meaningful volume with a spacetime region by measuring how its invariant \(D\)-volume grows when the region is flowed along a divergence-free vector field \(v^\alpha\). In the formulation introduced in "The Vector Volume and Black Holes" [1310.1935], it is simultaneously a growth rate, a conserved flux, and, in black-hole applications, a framework that recovers Parikh’s stationary volume, the geometric volume used in extended black hole thermodynamics, and Hayward’s Kodama-based volume in spherical symmetry. The same construction also yields a natural null-generator volume and a local relation \(\kappa=\mathcal V_{\mathcal C}/\mathcal V_{\mathcal N}\) between canonical stationary volume, null-generator volume, and surface gravity [1310.1935].

## 1. Definition through growth and flux

Let \((M,g_{\alpha\beta})\) be a \(D\)-dimensional spacetime and let \(R\subset M\) be an oriented region with piecewise-smooth boundary \(\partial R\). Its invariant \(D\)-volume is
\[
\mathcal V(R)\equiv \int_R \sqrt{|g|}\,\mathrm d^D x,
\]
with \(g=\det(g_{\alpha\beta})\). The key hypothesis is that the vector field satisfy
\[
\nabla_\alpha v^\alpha=0.
\]
All Killing vectors satisfy this, and in spherical symmetry the Kodama vector does as well.

The vector volume of \(R\) with respect to \(v^\alpha\) is defined by flowing a hypersurface \(\Gamma\) along the congruence of integral curves of \(v^\alpha\), parameterized by \(\mu\) so that
\[
v^\alpha\partial_\alpha=\frac{\mathrm d}{\mathrm d\mu}.
\]
If \(R(\mu)\) is the subregion reached after parameter distance at most \(\mu\), then
\[
\mathcal V_v(R)\equiv \frac{\mathrm d\mathcal V(R(\mu))}{\mathrm d\mu}.
\]
In adapted coordinates with \(v^\alpha=\delta^\alpha{}_0\) and \(\Gamma:x^0=0\), one obtains
\[
\mathcal V(R(\mu))=\mu\int_{x^i\in\Sigma}\sqrt{|g|}\,\mathrm d^{D-1}x,
\qquad
\mathcal V_v(R)=\int_{x^i\in\Sigma}\sqrt{|g|}\,\mathrm d^{D-1}x.
\]

An equivalent definition is the flux formula
\[
\mathcal V_v(R)=\int_\Sigma v^\alpha\,\mathrm d\Sigma_\alpha,
\]
where \(\Sigma=\Gamma\cap R\) intersects each integral curve of \(v\) exactly once and \(\mathrm d\Sigma_\alpha=n_\alpha\,\mathrm d\Sigma\) is the directed surface element. If \(\Sigma_1\) and \(\Sigma_2\) bound a slab \(Q\), then
\[
\int_{\Sigma_2} v^\alpha \mathrm d\Sigma_\alpha-\int_{\Sigma_1} v^\alpha \mathrm d\Sigma_\alpha
=\int_Q \nabla_\alpha v^\alpha\,\sqrt{|g|}\,\mathrm d^D x=0,
\]
so the flux is independent of the choice of \(\Sigma\). A differential-forms version uses the \((D-1)\)-form \(i_v\epsilon\), with \(\epsilon\) the volume \(D\)-form [1310.1935].

## 2. Structural properties and conditions of validity

The vector volume is linear with respect to the choice of vector \(v^\alpha\). If \(v^\alpha\) and \(w^\alpha\) are divergence-free and \(a,b\in\mathbb R\), then
\[
\mathcal V_{av+bw}(R)=a\,\mathcal V_v(R)+b\,\mathcal V_w(R),
\]
provided the boundary of \(R\) has normal everywhere orthogonal to both fields and the chosen hypersurface intersects both congruences once. Scaling gives
\[
\mathcal V_{cv}=c\,\mathcal V_v,
\]
up to orientation reversal for \(c<0\). The normalization of the flow parameter is therefore fixed by the normalization of \(v\), not by \(\mu\) alone.

The construction depends on several geometric hypotheses. The field \(v^\alpha\) is assumed smooth, at least \(C^1\), and divergence-free. The region \(R\) must have boundary whose normal is everywhere orthogonal to \(v^\alpha\), so that there is no net flux through \(\partial R\). The hypersurface \(\Sigma\) must intersect each integral curve of \(v\) exactly once, excluding caustics or foldings across \(\Sigma\). If \(\nabla_\alpha v^\alpha\neq 0\), the flux depends on the choice of \(\Sigma\). If \(v\) crosses the boundary, boundary flux contributions must be included or the region must be restricted. Non-smooth regions can be handled piecewise but require the usual measure-theoretic caution [1310.1935].

These hypotheses are not ancillary. They are precisely what make the rate-of-growth and flux definitions equivalent, and they identify vector volume as a conserved throughput of invariant \(D\)-volume along the flow of \(v^\alpha\).

## 3. Stationary axisymmetry and Kerr–Schild simplifications

In stationary, axisymmetric spacetimes with timelike Killing vector \(t^\alpha\) and axial Killing vector \(\phi^\alpha\), the axial contribution drops out for an axisymmetric region whose boundary normal is orthogonal to both fields. In adapted coordinates with \(t^\alpha=\delta^\alpha{}_0\) and \(\phi^\alpha=\delta^\alpha{}_1\), and with \(x^1\) cyclic,
\[
\int_\Sigma \phi^\alpha \mathrm d\Sigma_\alpha=0,
\]
so
\[
\mathcal V_{t+\Omega\phi}(R)=\mathcal V_t(R)
\quad\text{for any constant }\Omega.
\]
For black holes, the vector volume computed with the stationary field \(t^\alpha\) therefore equals that computed with the horizon generator \(\xi^\alpha=t^\alpha+\Omega_H\phi^\alpha\).

A second structural simplification occurs in Kerr–Schild geometries. If
\[
g_{ab}=\bar g_{ab}+2H\,\ell_a\ell_b,
\]
with \(\ell^a\) null with respect to both \(g_{ab}\) and \(\bar g_{ab}\), then the Matrix Determinant Lemma yields
\[
\det(g)=\det(\bar g),
\qquad
\sqrt{|g|}=\sqrt{|\bar g|}.
\]
Hence the full spacetime’s volume element equals that of the background spacetime. The vector volume computed in the full spacetime equals that computed in the background, in the same coordinates, for any divergence-free \(v^\alpha\). In the flat-background case this reduces many black-hole volume integrals to Euclidean volumes of the corresponding spatial regions [1310.1935].

This Kerr–Schild determinant equality is the main computational shortcut of the formalism. It explains why Schwarzschild, Kerr, and Kerr–(A)dS volumes take elementary closed forms despite their nontrivial spacetime geometry.

## 4. Black-hole interior volume and the null-generator volume

For black holes, one choice is the canonical stationary volume: take \(v^\alpha=t^\alpha\), the timelike Killing field, and let \(R\) be the black-hole interior. In \(D=4\), the resulting expressions are explicit.

| Spacetime | Horizon | Canonical stationary volume |
|---|---:|---:|
| Schwarzschild | \(r=r_+\) | \(\displaystyle \mathcal V_{\mathcal C}=\frac{4\pi}{3}\,r_+^3\) |
| Kerr | \(r=r_+\) | \(\displaystyle \mathcal V_{\mathcal C}=\frac{4\pi}{3}\,r_+\big(r_+^2+a^2\big)\) |
| Kerr–(A)dS | \(r=r_0\) | \(\displaystyle \mathcal V_{\mathcal C}=\frac{4\pi}{3}\,\frac{r_0(r_0^2+a^2)}{\Xi},\quad \Xi=1+\frac{\Lambda a^2}{3}\) |

For Kerr in Boyer–Lindquist coordinates, \(\sqrt{-g}=\rho^2\sin\theta\) with \(\rho^2=r^2+a^2\cos^2\theta\), so
\[
\mathcal V_{\mathcal C}
=\int_0^{r_+}\!\mathrm dr\int_0^\pi\!\mathrm d\theta\int_0^{2\pi}\!\mathrm d\phi\,
(r^2+a^2\cos^2\theta)\sin\theta
=\frac{4\pi}{3}\,r_+\big(r_+^2+a^2\big).
\]
Geometrically, in the flat background, this is the Euclidean volume of an oblate spheroid with semi-axes \(\sqrt{r_+^2+a^2},\sqrt{r_+^2+a^2},r_+\). In Schwarzschild Kerr–Schild coordinates the same logic gives
\[
\mathcal V_T(R_{\text{BH}})
=\int_0^{r_+}\!\mathrm dr\int_0^\pi\!\mathrm d\theta\int_0^{2\pi}\!\mathrm d\phi\,
r^2\sin\theta
=\frac{4\pi}{3}\,r_+^3.
\]

A second black-hole volume is the null-generator volume. Let \(\xi^\alpha=t^\alpha+\Omega_H\phi^\alpha\) be the horizon generator, with surface gravity \(\kappa\) defined by
\[
\xi^\beta\nabla_\beta\xi^\alpha=\kappa\,\xi^\alpha
\quad\text{on the horizon}.
\]
Define \(k^\alpha\equiv \xi^\alpha/\kappa\). The null-generator volume is
\[
\mathcal V_{\mathcal N}\equiv \frac{\mathrm d\mathcal V(R_{\text{BH}})}{\mathrm d\ln\lambda}
=\mathcal V_k(R_{\text{BH}}).
\]
In stationary spacetimes,
\[
\mathcal V_{\mathcal N}=\frac{\mathcal V_{\mathcal C}}{\kappa}.
\]
This yields the local relation
\[
\kappa=\frac{\mathcal V_{\mathcal C}}{\mathcal V_{\mathcal N}}.
\]
A stated consequence is that driving \(\kappa\to 0\) quasi-statically requires \(\mathcal V_{\mathcal N}\to\infty\), that is, infinite advanced time [1310.1935].

## 5. Relation to thermodynamic volume

In extended black-hole thermodynamics, the cosmological constant is treated as pressure,
\[
P=-\frac{\Lambda}{8\pi},
\]
and the first law takes the form
\[
\mathrm dM=T\,\mathrm dS+\Omega\,\mathrm dJ+\Phi\,\mathrm dQ+V\,\mathrm dP.
\]
The thermodynamic volume is related to the \(\Lambda\)-conjugate \(\Theta\) by
\[
\mathcal V_{\text{th}}=-\frac{16\pi}{D-2}\,\Theta.
\]

Within this literature, Cvetič et al. showed that a natural geometric volume equals Parikh’s stationary volume and obeys
\[
\mathcal V_{\text{geo}}=\frac{r_+\,\mathcal A}{D-1},
\]
with \(\mathcal A\) the horizon area. In rotating spacetimes,
\[
\mathcal V_{\text{th}}-\mathcal V_{\text{geo}}
=\frac{8\pi}{(D-1)(D-2)}\sum_i a_i J_i.
\]
Thus the thermodynamic volume differs from the geometric, vector-volume notion by an angular-momentum-dependent term. In spherical symmetry, where all \(a_i=0\),
\[
\mathcal V_{\text{th}}=\mathcal V_{\text{geo}}=\mathcal V_v
\quad\text{with }v=t.
\]

The distinction is therefore precise. The vector volume coincides with the geometric or Parikh volume, and it matches the thermodynamic volume whenever rotation is absent. With rotation, the thermodynamic volume incorporates global thermodynamic structure beyond the purely geometric flux or growth-rate interpretation carried by \(\mathcal V_v\) [1310.1935].

## 6. Scope of the term and related literatures

The expression “vector volume” also appears in several other mathematical and physical literatures, but with different meanings. On oriented Riemannian surfaces and 3-manifolds, the volume of a unit vector field is the Sasaki volume of its graph in the unit tangent bundle, with
\[
\mathrm{vol}(X)=\int \sqrt{1+A_0^2+A_1^2}\,\mathrm d\mathrm{vol}
\]
in dimension \(2\) and
\[
\mathrm{Vol}(X)=\int_M \sqrt{\det(I+(\nabla X)^t\nabla X)}\,\mathrm d\mathrm{vol}_g
\]
in dimension \(3\). Calibration methods are used to characterize minimal-volume fields, including Hopf vector fields on \(\mathbb S^3(r)\) and meridian-type fields on punctured spheres [2109.01565], [2110.07759], [2207.08761], [1708.01575], [1408.2706], [1101.5259].

For geodesible and Reeb vector fields on closed, oriented odd-dimensional manifolds, a vector-field-determined volume is defined by
\[
\mathrm{Vol}(X)=\int_M \alpha\wedge (d\alpha)^n,
\]
where \(\alpha(X)=1\) and \(\iota_X d\alpha=0\). In that setting the quantity depends only on the vector field, not on the particular choice of characteristic \(1\)-form, and it is tied to basic cohomology, Seifert Euler numbers, and global surfaces of section [2003.06270].

In dynamical systems and geometric numerical integration, “vector volume” refers to Lebesgue measure preserved by divergence-free flows. Liouville’s formula gives
\[
\frac{\mathrm d}{\mathrm dt}\det(D\phi_t(x))
=(\nabla\cdot F)(\phi_t(x))\,\det(D\phi_t(x)),
\]
so \(\nabla\cdot F\equiv 0\) implies exact volume preservation. This is the setting for explicit volume-preserving splitting methods for polynomial divergence-free vector fields and for analyses of Runge–Kutta and exponential integrators that preserve phase-space volume for structured classes of vector fields [1205.1935], [1507.00535], [1805.11713], [2201.12935].

A further terminological reuse appears in echocardiographic flow reconstruction, where a “vector volume” denotes a full \(3\)-dimensional volume populated by \(3\)-component velocity vectors over time. In \(3\)D-iVFM, this is reconstructed from triplane color Doppler by enforcing mass conservation and free-slip boundary conditions within a constrained least-squares problem [2112.03843].

These usages are not equivalent definitions. They indicate that “vector volume” is a polysemous term whose meaning depends on whether the primary object is a spacetime region, a unit vector field, a divergence-free flow, or a volumetric vector-valued dataset. In the black-hole context, however, the term retains a distinctive content: the conserved rate of growth, or equivalently flux, of invariant spacetime volume along a divergence-free vector field [1310.1935].

Source: https://www.emergentmind.com/topics/vector-volume