Uniform Black Strings and Branes

A uniform black string or brane is an exact product of a lower-dimensional Ricci-flat black hole with flat spatial directions. The product does not change the transverse local curvature, but it changes horizon topology, asymptotics, and the meaning of conserved charges. A noncompact worldvolume carries mass density rather than finite total mass; a compact one introduces moduli and a global length scale absent from the local metric.

Required background. The Einstein Equation supplies the product Ricci tensor and vacuum equation.

Helpful background. The Riemann Curvature Tensor supplies product-curvature checks; Exact Solutions of Einstein’s Equations places the family; Global Structure develops horizon topology; and Killing Fields and Isometry Groups supplies translational symmetries.

Product construction

Let $(\mathcal B^d,g^{(d)})$ be a $d$-dimensional Ricci-flat spacetime and append $p$ Cartesian directions $z^i$:

$$ds_D^2=ds_d^2+\sum_{i=1}^p(dz^i)^2, \qquad D=d+p.$$

For a direct product, the Levi–Civita connection has no mixed components. Therefore

$$R_{\mu\nu}^{(D)}=R_{\mu\nu}^{(d)}, \qquad R_{ij}^{(D)}=0, \qquad R_{\mu i}^{(D)}=0.$$

If the seed is Ricci-flat, the product is Ricci-flat. Taking the $d$-dimensional Schwarzschild–Tangherlini seed gives

$$ds_D^2=-f(r)dt^2+\frac{dr^2}{f(r)} +r^2d\Omega_{d-2}^2+\sum_{i=1}^p(dz^i)^2,$$ $$f(r)=1-\left(\frac{r_0}{r}\right)^{d-3}.$$

The horizon at $r=r_0$ has cross-section $S^{d-2}\times\mathbb R^p$ if every $z^i$ is noncompact. If $z^i\sim z^i+L_i$, the corresponding factor is a torus $T^p$, and the horizon topology is $S^{d-2}\times T^p$.

The full Kretschmann scalar equals that of the seed because every curvature component with a flat index vanishes. Thus the product has the same transverse curvature singularity at $r=0$ and a regular horizon at $r=r_0$. This equality is an independent local check, not a statement that the global spacetimes are identical.

Mass density and asymptotics

When the worldvolume is noncompact, the ADM mass is infinite. Let

$$V_p=\prod_{i=1}^pL_i$$

for a compact regulator or actual torus. Dimensional reduction of the gravitational coupling gives $G_d=G_D/V_p$. The seed mass formula then yields

$$\frac{M}{V_p} =\frac{(d-2)\Omega_{d-2}}{16\pi G_D} r_0^{d-3}.$$

The left side is the mass density along the flat directions. Sending $L_i\to\infty$ holds this density fixed while the total mass diverges. Conversely, changing a compact $L_i$ changes the total charge without changing the local metric coefficients. This is exactly how global identifications change charges while preserving transverse local geometry.

The product is asymptotic to $\mathbb R^{1,d-1}\times T^p$ for compact directions, not to $D$-dimensional Minkowski space with an $S^{D-2}$ at infinity. Its gravitational field falls as $r^{-(d-3)}$ because flux spreads only through the noncompact transverse dimensions. Applying the $D$-dimensional Tangherlini mass formula would give the wrong falloff and normalization.

There are also tensions conjugate to the compact lengths. For the uniform neutral string they are fixed fractions of the mass density by scaling relations, but their definition requires a specific asymptotic charge prescription. The exact metric alone tells us that varying $L_i$ is a physical change of boundary data, not a coordinate rescaling.

Exactness does not imply stability

The uniform solution is translationally invariant, but sufficiently long neutral black strings admit long-wavelength perturbations that grow—the Gregory–Laflamme instability Gregory and Laflamme 1993, pp. 2837–2840. The threshold depends on the dimensionless ratio of compact length to $r_0$ and must be found from a perturbation problem. It is not visible as a failure of the background Einstein equation.

This provides an important logical separation. The product check proves that the metric is an exact vacuum solution. Regularity proves that its horizon is geometrically sound. Neither proves dynamical stability or thermodynamic preference. Nonuniform strings and localized Kaluza–Klein black holes belong to other branches with the same asymptotic compactification.

Boosting a string along a compact direction adds momentum charge after identification. Reducing along that direction produces a lower-dimensional charged solution with a scalar and gauge field. Those operations alter the interpretation of parameters and fields; they are not the same as the unboosted direct product developed here.

Common pitfalls

Calling the mass finite for a noncompact brane. The natural charge is a density. A total mass requires finite worldvolume periods or a regulator whose volume is shown explicitly.

Using the total dimension in the radial falloff. The exponent is controlled by the transverse seed dimension $d$, because the solution is uniform along $p$ directions.

Exercises

Show that the horizon area density of the uniform brane is $\Omega_{d-2}r_0^{d-2}$.

Solution

At $r=r_0$ and fixed horizon time, the induced metric is the direct product

$$r_0^2d\Omega_{d-2}^2+\sum_{i=1}^p(dz^i)^2.$$

Its volume element factorizes. For compact lengths $L_i$,

$$\mathcal A_H=\Omega_{d-2}r_0^{d-2}V_p.$$

Dividing by $V_p$ and then allowing noncompact directions gives

$$\frac{\mathcal A_H}{V_p}=\Omega_{d-2}r_0^{d-2}.$$

References

• Ruth Gregory and Raymond Laflamme, “Black Strings and $p$-Branes Are Unstable,” Physical Review Letters 70, 2837–2840 (1993). DOI.