Tensor Densities and Weights

Density weights are not a historical footnote — they are compiled into the software that does modern gravitational physics. Every numerical-relativity code that evolves a binary black-hole merger for LIGO–Virgo templates works with the metric determinant and the densitized lapse and shift of the BSSN or Z4c formulations, where a conformal factor built from $\ln g$ is evolved precisely because the determinant is a weight-$2$ density with especially clean equations. The stress-energy conservation law $\nabla_\mu T^{\mu\nu}=0$ is implemented in its density form $\partial_\mu(\sqrt{-g}\,T^{\mu\nu})=\ldots$, and the ADM mass and total charge of a spacetime are surface integrals of weight $+1$ densities, well-defined only because the Jacobian factors cancel. The same structure runs through lattice field theory, where the path-integral measure must be a density, and through the anomaly calculations of quantum field theory in curved space, where the failure of a classically weight-zero current to stay weight-zero under regularization is the trace anomaly. Wherever an integral has to mean the same thing to two observers using different coordinates — from a cosmological simulation's comoving volume to the phase-space measure of statistical mechanics — the weight-$+1$ density $\sqrt{-g}$ is doing the silent work.

Picture a coordinate chart as graph paper laid over the manifold. Each little cell $\mathrm{d}^nx$ is one unit of coordinate volume — but coordinate volume is a fiction of the labelling, not a geometric fact. Re-label the manifold and the same physical patch is covered by cells of a different size; the Jacobian determinant $\mathcal{J}=\det(\partial x'/\partial x)$ is precisely the factor by which each cell shrinks or stretches under the re-labelling. An ordinary tensor is blind to this stretching: its components rotate and mix but do not scale with cell size. A density of weight $w$ is not blind — it responds to the stretching with a factor $\mathcal{J}^{-w}$. The weight is a counter of how many powers of coordinate volume the object secretly carries.

The metric is what supplies the true, geometric size of a coordinate cell. Written out, the invariant proper volume is $\sqrt{-g}\,\mathrm{d}^nx$: the factor $\sqrt{-g}$ carries exactly one power of coordinate volume, cancels the one power the cell $\mathrm{d}^nx$ picks up under a chart change, and leaves an invariant. That single power is its weight, $+1$. The Levi-Civita symbol is the cleanest illustration of the opposite bookkeeping. We insist that its components stay $\pm1$ and $0$ in every chart, because that is what makes it a useful counting device for antisymmetrization. No nonzero genuine tensor can have chart-independent components — but a weight $-1$ density can, because its non-trivial transformation is entirely spent absorbing the Jacobian, leaving the numbers fixed. Constancy of components is bought with a nonzero weight. Finally, weight behaves like a conserved charge under the algebra of tensor operations. Contracting a pair of indices touches the index structure but never the weight. Forming a tensor product adds the weights. And $\sqrt{-g}$, with its clean weight $+1$, acts as a unit of weight you can slide in or out at will: multiply a weight $-1$ object by it to reach weight zero, divide any density by an appropriate power of it to strip the weight entirely. Every density is, up to a power of $\sqrt{-g}$, a tensor in disguise.

Everything starts from one definition: a density carries the ordinary tensor factors and a power of the Jacobian determinant. We fix the convention once — the Jacobian is $\mathcal{J}=\det(\partial x'/\partial x)$, all chart transitions here are taken to be orientation-preserving ($\mathcal{J}>0$, so $\mathcal{J}$ and $|\mathcal{J}|$ agree throughout), and the extra factor is $\mathcal{J}^{-w}$ — and never change it, because the signs of every weight below are tied to that single choice.

where: $w$ — the weight, an integer (or real number); $w=0$ recovers an ordinary tensor. $\mathcal{J}$ — Jacobian determinant of the change $x\to x'$; the volume element scales as $\mathrm{d}^nx'=|\mathcal{J}|\,\mathrm{d}^nx$. $(p,q)$ — tensorial type: $p$ upper, $q$ lower indices. $\mathfrak{T}$ — the density components in the old chart. Physical reading: the ordinary index factors handle rotation and mixing; the lone $\mathcal{J}^{-w}$ handles scaling with coordinate volume. A weight $+1$ density transforms as $\mathcal{J}^{-1}$ — inversely to the cell $\mathrm{d}^nx$ — which is exactly what makes its integral survive a chart change. The metric determinant is the doorway to the whole subject: because $g_{\mu\nu}$ is an ordinary $(0,2)$ tensor, taking its determinant pulls out two factors of the Jacobian, so $g$ is a scalar density of weight $+2$ and its square root has weight $+1$.

where: $g=\det(g_{\mu\nu})$ — the metric determinant (negative for Lorentzian signature, so $-g>0$); weight $+2$. $\sqrt{-g}$ — the canonical scalar density of weight $+1$, the volume factor. Physical reading: $\sqrt{-g}$ scales as $\mathcal{J}^{-1}$, so it shrinks precisely when coordinate cells grow; it is the metric's answer to "how much proper volume is one coordinate cell worth here?" The Levi-Civita symbol is defined by fixed components, and that very rigidity forces it to be a density. Its components are the totally antisymmetric array; keeping them constant in all charts costs it a weight of $-1$ (lower indices).

where: $\epsilon_{\mu_1\cdots\mu_n}$ — the alternating symbol, identical in every chart; a density of weight $-1$. $n$ — manifold dimension ($4$ in spacetime). Physical reading: because its numbers are frozen, all of its transformation is spent cancelling a Jacobian; the genuine antisymmetric tensor is recovered only after restoring the missing weight with $\sqrt{-g}$. Restoring that weight is a one-line operation, and it is how one builds the object that actually integrates invariantly — the Levi-Civita tensor.

where: $\varepsilon$ — the true antisymmetric tensor (weight $0$); $\sqrt{-g}$ (weight $+1$) absorbs the symbol's weight $-1$. Upper indices flip the sign: $\epsilon^{\mu_1\cdots\mu_n}$ has weight $+1$, so it is divided by $\sqrt{-g}$ (weight $+1$) with $\mathrm{sgn}(g)=-1$ for Lorentzian signature. Physical reading: the tensor $\varepsilon$ is the geometric volume form; the symbol $\epsilon$ is its numerical skeleton. The trace of the Christoffel connection is not an independent quantity — it is the logarithmic derivative of the volume density. This single identity is the hinge on which the calculus of densities turns.

where: $\Gamma^{\lambda}{}_{\lambda\mu}$ — contraction of the Levi-Civita connection over its upper and first lower index. $g^{\alpha\beta}$ — inverse metric. Physical reading: the connection trace measures the fractional rate at which proper volume changes as you step in the $\mu$ direction — literally the divergence of the coordinate basis. That identity means the covariant derivative of a density needs exactly one correction term beyond the usual $\Gamma$ terms, proportional to the weight.

where: $w$ — weight of $\mathfrak{T}$; the final term is the density correction. Physical reading: the correction is fixed by demanding $\nabla_\mu\sqrt{-g}=0$ — with $w=+1$ it gives $\partial_\mu\sqrt{-g}-\partial_\mu\ln\sqrt{-g}\cdot\sqrt{-g}=0$ exactly, the consistency check that pins the sign. The payoff is that the covariant divergence of a vector collapses into an ordinary partial derivative — the practical engine behind every conservation law in curved spacetime.

where: $V^{\mu}$ — a contravariant vector (weight $0$); $\sqrt{-g}\,V^{\mu}$ — a vector density of weight $+1$, whose plain divergence is automatically a scalar density. Physical reading: no Christoffel symbols survive; the connection has been fully absorbed into the volume factor, which is why this formula, not the raw $\nabla_\mu V^\mu=\partial_\mu V^\mu+\Gamma^\mu{}_{\mu\lambda}V^\lambda$, is what you compute with.

The whole family is organized by the single integer $w$; the physically important members are just a handful of special values.

1 · The volume factor and Levi-Civita tensor in spherical coordinates (numeric)

Flat three-dimensional space in spherical coordinates has line element $\mathrm{d}s^2=\mathrm{d}r^2+r^2\,\mathrm{d}\theta^2+r^2\sin^2\theta\,\mathrm{d}\phi^2$. Find $\sqrt{g}$, verify it equals the Cartesian-to-spherical Jacobian (confirming weight $+1$), and use it to compute the volume of a ball of radius $R=2$.

Step 1. The metric is diagonal, $g_{\mu\nu}=\mathrm{diag}(1,\,r^2,\,r^2\sin^2\theta)$, so $g=\det g_{\mu\nu}=1\cdot r^2\cdot r^2\sin^2\theta=r^4\sin^2\theta$, giving $\sqrt{g}=r^2\sin\theta$. So the proper volume element is $\mathrm{d}V=\sqrt{g}\,\mathrm{d}r\,\mathrm{d}\theta\,\mathrm{d}\phi=r^2\sin\theta\,\mathrm{d}r\,\mathrm{d}\theta\,\mathrm{d}\phi$.

Step 2. Check the weight directly. The Cartesian metric is $\delta_{ij}$ with $\sqrt{g_{\text{cart}}}=1$, and the Jacobian of $(x,y,z)\to(r,\theta,\phi)$ is the familiar $\mathcal{J}=\det\big(\partial(x,y,z)/\partial(r,\theta,\phi)\big)=r^2\sin\theta$. A weight $+1$ density satisfies $\sqrt{g_{\text{sph}}}=\mathcal{J}\cdot\sqrt{g_{\text{cart}}}=r^2\sin\theta\cdot1$. The volume factor picks up exactly one power of the Jacobian — weight $+1$ confirmed.

Step 3. Integrate over the ball: $V=\displaystyle\int_0^{2\pi}\!\!\int_0^{\pi}\!\!\int_0^R r^2\sin\theta\,\mathrm{d}r\,\mathrm{d}\theta\,\mathrm{d}\phi=2\pi\cdot 2\cdot\frac{R^3}{3}=\frac{4}{3}\pi R^3$. For $R=2$, $V=\frac{32\pi}{3}=33.51$ — the correct Euclidean volume, obtained entirely from the density $\sqrt{g}$ with no appeal to Cartesian geometry. The Levi-Civita tensor here is $\varepsilon_{r\theta\phi}=\sqrt{g}\,\epsilon_{r\theta\phi}=r^2\sin\theta$, the oriented volume 3-form.

2 · The volume element of the Schwarzschild spacetime

The Schwarzschild metric is $g_{\mu\nu}=\mathrm{diag}\!\big(-(1-2M/r),\,(1-2M/r)^{-1},\,r^2,\,r^2\sin^2\theta\big)$ in coordinates $(t,r,\theta,\phi)$. Find $\sqrt{-g}$.

Step 1. Multiply the diagonal entries: $g=-(1-2M/r)\cdot(1-2M/r)^{-1}\cdot r^2\cdot r^2\sin^2\theta$. The first two factors are reciprocals, so their product is $-1$. The gravitational potential cancels out entirely: $g=-r^4\sin^2\theta$.

Step 2. Therefore $\sqrt{-g}=r^2\sin\theta$ — identical to flat space in spherical coordinates. The invariant 4-volume element of Schwarzschild is $\sqrt{-g}\,\mathrm{d}^4x=r^2\sin\theta\,\mathrm{d}t\,\mathrm{d}r\,\mathrm{d}\theta\,\mathrm{d}\phi$.

Step 3. The physical lesson: curvature is hidden in the individual metric components, but the coordinate volume measured by $\sqrt{-g}$ is unchanged from flat space because the $g_{tt}$ time-dilation and $g_{rr}$ radial-stretch factors are exact reciprocals in these coordinates. This is special to Schwarzschild coordinates; in isotropic or Kruskal coordinates $\sqrt{-g}$ looks entirely different, yet $\sqrt{-g}\,\mathrm{d}^4x$ integrates to the same invariant, exactly as the weight $+1$ bookkeeping guarantees.

3 · The covariant divergence reproduces the vector-calculus divergence

Use $\nabla_\mu V^{\mu}=\tfrac{1}{\sqrt{g}}\partial_\mu(\sqrt{g}\,V^{\mu})$ with $\sqrt{g}=r^2\sin\theta$ to recover the divergence in spherical coordinates, then evaluate it for the radial field $V^r=r$, $V^\theta=V^\phi=0$.

Step 1. Insert $\sqrt{g}=r^2\sin\theta$ and expand the sum over $\mu\in\{r,\theta,\phi\}$: $\nabla_\mu V^{\mu}=\dfrac{1}{r^2\sin\theta}\Big[\partial_r(r^2\sin\theta\,V^{r})+\partial_\theta(r^2\sin\theta\,V^{\theta})+\partial_\phi(r^2\sin\theta\,V^{\phi})\Big]$.

Step 2. Pull the $r$-independent and $\theta$-independent pieces out of each derivative: $\nabla_\mu V^{\mu}=\dfrac{1}{r^2}\partial_r(r^2V^{r})+\dfrac{1}{\sin\theta}\partial_\theta(\sin\theta\,V^{\theta})+\partial_\phi V^{\phi}$ — precisely the standard spherical-coordinate divergence, delivered by the density formula with no Christoffel symbols computed by hand.

Step 3. For $V^{r}=r$ (a radial field with $V^\theta=V^\phi=0$): $\nabla_\mu V^{\mu}=\dfrac{1}{r^2}\partial_r(r^2\cdot r)=\dfrac{1}{r^2}\partial_r(r^3)=\dfrac{3r^2}{r^2}=\mathbf{3}$. This matches the elementary result $\nabla\!\cdot\vec{r}=3$ in three dimensions — a clean numeric confirmation that the density machinery reproduces ordinary vector calculus.

Tensor densities are inseparable from the two objects that carry a nineteenth-century name into every modern calculation: the alternating symbol and the absolute differential calculus that made index gymnastics into a coherent theory.

Texts. Misner, Thorne & Wheeler, Gravitation (1973), §3.4 and §8.3, exercise 8.3 · Wald, General Relativity (1984), Appendix B and §4.1 · Carroll, Spacetime and Geometry (2004), §2.8–2.9 · Schutz, A First Course in General Relativity (2009), §6.8 · Weinberg, Gravitation and Cosmology (1972), §4.4. The single convention used throughout is $\mathcal{J}=\det(\partial x'/\partial x)$ with density factor $\mathcal{J}^{-w}$. In this convention the integrand of an invariant integral has weight $+1$, so $\sqrt{-g}$ has weight $+1$, $g=\det g_{\mu\nu}$ has weight $+2$, and the covariant Levi-Civita symbol has weight $-1$ — the internally consistent assignment in which $\varepsilon=\sqrt{-g}\,\epsilon$ is a weight-zero tensor. Texts that define the Jacobian as $\det(\partial x/\partial x')$ or attach $\mathcal{J}^{+w}$ flip every sign (for instance calling $\sqrt{-g}$ "weight $-1$"); the ratios among the weights are convention-independent. The key results are: (1) the Leibniz determinant identity establishing the symbol's weight $-1$; (2) $g'=\mathcal{J}^{-2}g\Rightarrow\sqrt{-g'}=|\mathcal{J}|^{-1}\sqrt{-g}$ and the cancellation against $\mathrm{d}^nx'=|\mathcal{J}|\,\mathrm{d}^nx$; (3) Jacobi's formula giving $\Gamma^{\lambda}{}_{\lambda\mu}=\partial_\mu\ln\sqrt{-g}$ and the divergence formula; (4) the spherical checks $\sqrt{g}=r^2\sin\theta$, ball volume $\tfrac{4}{3}\pi R^3$, Schwarzschild $\sqrt{-g}=r^2\sin\theta$, and $\nabla\!\cdot\vec{r}=3$. No numerical constants beyond $\pm1,0$ (Levi-Civita) and the standard $1/16\pi G$ Einstein–Hilbert prefactor appear.

Builds on. The Tensor Transformation Law · General Tensors and the Tensor Product · Atlases, Transition Functions and Differentiable Structure · The Metric Tensor and the Line Element. Leads to. The Invariant Volume Element · Tensor Algebra: Contraction, Symmetrization and Antisymmetrization · Raising and Lowering Indices.

Ask the tutor: Ask me to work through why the metric determinant has weight $+2$, or to show how the $-w\,\Gamma^{\lambda}{}_{\lambda\mu}$ term makes $\nabla_\mu\sqrt{-g}=0$ exactly.