Black Branes

Black branes are exact black objects whose horizons extend along one or more translational directions. “Black brane” names a structural class, not one theory: neutral vacuum products, charged supergravity branes, and planar AdS horizons solve different field equations. Their extensive quantities are densities unless the worldvolume is compactified, and their exactness does not guarantee stability of the uniform phase.

Black-brane geometries

Property	Value
Object class	Exact solution family
Theories	Higher-dimensional general relativity, supergravity, or a stated string-effective action
Dimensions	Family dependent, usually $D\gt4$ or planar AdS
Parameters	horizon scale, energy and charge densities, boosts, worldvolume periods
Physical status	Exact in the specified classical theory; interpretation depends on compactification and boundary data

A neutral baseline begins with a $d$-dimensional Ricci-flat black hole and appends $p$ flat directions:

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

For a Schwarzschild–Tangherlini seed,

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

The product connection has no mixed curvature, so a Ricci-flat seed gives $R_{AB}^{(D)}=0$ exactly. A noncompact horizon section is $S^{d-2}\times\mathbb R^p$; periods $z^i\sim z^i+L_i$ replace the flat factor by $T^p$. The construction and charge-density bookkeeping are developed in the higher-dimensional review by Emparan and Reall 2008, § 3.2.

Densities, horizons, and regularity

Let $V_p=\prod_iL_i$ be a compact regulator or a physical torus volume. The mass and area densities are

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

and the surface gravity is $\kappa=(d-3)/(2r_0)$. For noncompact directions, the total mass and area diverge while these densities remain finite. The radial falloff is $r^{-(d-3)}$, not $r^{-(D-3)}$, because the field is uniform along the brane.

The horizon at $r=r_0$ is regular in ingoing coordinates. Its local curvature equals the seed curvature because every component with a flat index vanishes; the genuine transverse singularity remains at $r=0$. Compactification changes global asymptotics and introduces gravitational tensions conjugate to $L_i$, even though it leaves the local product metric unchanged.

For one compact neutral direction in total dimension $D$, the mechanics includes

$$dM=\frac{\kappa}{8\pi G_D}\,dA_H+\mathcal T\,dL, \qquad \mathcal T=\frac{M}{(D-3)L}.$$

Treating $L$ as a coordinate convention would miss this work term. Likewise, boosting along a compact direction and then reducing produces momentum, gauge, and scalar charges; it is a different lower-dimensional field configuration, not merely the same unboosted brane in new coordinates.

Charged and extremal branes

A representative asymptotically flat charged D$p$-brane in ten-dimensional type-II string frame, for $0\le p\le6$, has

$$\begin{aligned} ds_s^2&=H^{-1/2}(-f\,dt^2+d\mathbf x_p^2) +H^{1/2}(f^{-1}dr^2+r^2d\Omega_{8-p}^2),\\ f&=1-\frac{r_0^{7-p}}{r^{7-p}}, \qquad H=1+\frac{r_0^{7-p}\sinh^2\alpha}{r^{7-p}},\\ e^\Phi&=g_sH^{(3-p)/4}. \end{aligned}$$

It also carries an R–R form field whose flux fixes a charge density proportional to $r_0^{7-p}\sinh\alpha\cosh\alpha$. D7-branes have codimension-two logarithmic behavior, while D8-branes belong to massive IIA and are domain-wall-like, so neither is described by the displayed harmonic function. The action, frame, form normalization, and compact worldvolume must be stated before comparing charges. Peet 2000, §§ 3.1, 3.4, and 4.3 derives these fields and their validity conditions.

The nonextremal horizon is at $r=r_0$, with

$$T=\frac{7-p}{4\pi r_0\cosh\alpha}.$$

The extremal limit takes $r_0\to0$ and $\alpha\to\infty$ while holding the charge scale fixed. It is not obtained by setting $r_0=0$ at fixed $\alpha$, which removes both horizon and charge. A putative “superextremal brane” is theory dependent: beyond a BPS or charge bound the regular horizon may disappear or the classical description may fail, but there is no universal discriminant shared by all brane theories.

Controlled limits and dynamics

Neutral limit. Taking $\alpha\to0$ at fixed $r_0$ gives $H\to1$ and the neutral Schwarzschild brane.

Extremal limit. The fixed-charge scaling above removes nonextremal energy and often produces a throat. Curvature and $e^\Phi$ must remain small for supergravity to be controlled; this fails in parts of some D$p$ throats.

Compactification and decompactification. Finite $L_i$ give Kaluza–Klein asymptotics, total charges, and tensions. Sending $L_i\to\infty$ at fixed densities recovers a noncompact brane while total charges diverge.

Planar AdS limit. A scaling of a large spherical AdS black hole can yield a planar horizon and finite energy density. The boundary topology and charge normalization change, so this is not a mere large-radius approximation at fixed coordinates.

Uniformity does not imply stability. Long neutral strings and branes possess Gregory–Laflamme modes when a sufficiently long wavelength fits along the worldvolume Gregory and Laflamme 1993, complete article, printed pp. 2837–2840. The threshold depends on dimension, charge, and compactification. It separates an exact background check from a dynamical spectral calculation; the nonlinear endpoint may involve nonuniform branes, localized holes, or topology change.

Connections to the detailed theory

The neutral product construction is developed in Uniform Black Strings and Branes. Horizon topology, density, tension, and Kaluza–Klein phases are organized in Black Strings, Branes, and Kaluza–Klein Horizons. Charged fields, frames, and extremal scalings belong to Extremal and Nonextremal Black Branes.

References

• Emparan, Roberto, and Harvey S. Reall. “Black Holes in Higher Dimensions.” Living Reviews in Relativity 11 (2008): 6. DOI.
• Gregory, Ruth, and Raymond Laflamme. “Black Strings and $p$-Branes Are Unstable.” Physical Review Letters 70 (1993): 2837–2840. DOI.
• Peet, Amanda W. “TASI Lectures on Black Holes in String Theory.” 2000. arXiv:hep-th/0008241.