This question arises in a somewhat naive form because I am largely unfamiliar with String Theory. I do know that it incorporates higher space dimensions where I shall take the overall dimensionality to be 10 in this question, for concreteness. Now the traditional Hawking-Penrose Singularity results apply to the the General Relativity manifold of 3+1 dimensions; with the 4D Schwarzchild solution providing an example of a Singularity and Black Hole.

So the question is: do singularities (and maybe associated Event Horizons) necessarily form in all 10 dimensions?

Examining this question for myself I see that this paper for mathematicians introduces an $N$ dimensional Schwarzchild metric and in theorem 3.15 an $N$ dimensional Hawking-Penrose singularity theorem. However this cannot answer directly to the intentions of the String theory models. For example it is mathematically possible to extend 4D Schwarzchild to 10D differently by adding a 6D Euclidean metric. So one question is whether this modified 10D Schwarzchild even meets the conditions for the $N$ dimensional Hawking-Penrose theorem. Although such a modification is not likely acceptable as a String Theory extension, it shows that we can consider some cases:

a) All 4D singularities / Event Horizons are actually 10 D ones.

b) Some/all 4D singularities / Event Horizons are "surface phenomena" in String theory - the underlying Bulk Volume is singularity free.

EDIT: Expressed a bit more formally this is saying that the String Theory has a singularity free solution $\Phi$ in 10D, but when $\Phi$ is restricted or reduced to 3+1D it is one of the known singular solutions of GR.

c) Some Singularities in String Theory Bulk (the 6D part) can arise without a corresponding 4D singularity (akin to a "deep earth earthquake" in 10D space-time, perhaps)?


The Schwarzschild metric for $d$ dimensions is the standard form $$ ds^2~=~-e^{2\phi}dt^2~+~e^{2\gamma}dr^2~+~r^2d\Omega^2 $$ These metric terms in the Einstein field equation gives $$ R_{tt}~-~\frac{1}{2}Rg_{tt}~=~G_{tt}~=~ -e^{2\phi}\Big((d~-~1)\frac{e^{2\gamma}}{r} ~+~\frac{(d~+~1)(d~-~2)}{2r^2}(1~-~e^{2\gamma}\Big) $$ A multiplication by $g^{tt}$ removes the $-e^{2\phi}$ and we equate this with a pressureless fluid $T^{tt}~=~\kappa\rho$. So we think of the black hole as composed of “dust.” Some analysis on this is used to compute the $G_r^r$ gives the curvature term $$ G_r^r~=~\Big((d~-~1)(\phi_{,r}~+~\gamma_{,r})\frac{e^{-2\gamma}}{r}~+~\kappa\rho\Big). $$ which tells us $\phi~=~-\gamma$, commensurate with the standard Schwarzschild result, and that $$ e^{-2\gamma}~=~e^{2\phi}~-~\Big(\frac{r_0}{r}\Big)^{d~-~2}. $$ The entropy of the black hole is then computed by writing the density according to these metric elements and computing the Rindler time coordinates $S~=~2\pi(d~-~2)A/\kappa$.

The results more or less follow as with the standard $3~+~1$ spacetime result. The Connection with strings is to work with the entropy of the black hole. The $1~+~1$ string world sheet has $d~-~1$ transverse degrees of freedom which contain the field data. The entropy $S~=~2\pi(d~-~1)T$ may be computed with the string length, which reduces to the holographic results in $d~=~4$ spacetime.

The event horizon is $d~-~2$ dimensional, which for $10$ dimension means the horizon is $8$ dimensional. The singularity is not considered in these calculations. The factors $e^{-2\gamma}~=~e^{2\phi}$ become extremely large. The metric approximates $$ ds^2~\simeq~\Big(\frac{r_0}{r}\Big)^{d~-~2}dt^2~+~r^2d\Omega^2 $$ which is a $d~-~1$ dimensional surface where the Weyl curvature diverges for $r~\rightarrow~0$. For $d~=~4$ this has properties similar to an anti-deSitter space.

The theory of black holes essentially follows in arbitrary dimensions. It is interesting to speculate on what the singularity is from a stringy perspective. The event horizon contains the quantum field information which composes the black hole. This may then have some type of correspondence with the interior singularity, with one dimension larger. For a black hole that is very small $\sim~10^3$ Planck units, the horizon is a quantum fluctuating region, as is the singularity, and the QFT data on the two may have some form of equivalency.

  • $\begingroup$ I am to conclude from this that the underlying string equation is 10D Einstein i.e. $G_{uv}=8\pi T_{uv}$ in 10D? If so then my question is whether all 4D singular solutions (like SC) map onto a unique 10D solution and whether that 10D solution is necessarily singular? In your answer you claim that the 10D Swarzchild is unique, but there is still the general case. $\endgroup$ – Roy Simpson Feb 16 '11 at 15:30
  • $\begingroup$ For a 1+1 dimensional string, the transeverse degrees of freedom can oscillate in all available dimensions of the theory. The form of the black hole is not terribly sensitive to the number of dimensions, though there is a subtle issue with even v. odd dimensions. A 10 dimensional black hole then has an 8 dimensional null horizon. The singularity is a 9 dimensional surface where classical curvature diverges. In holography the horizon. or stretched horizon, contains all the string QFT data as seen by a stationary observer. What happens on the singularity is less well explored. $\endgroup$ – Lawrence B. Crowell Feb 16 '11 at 17:41

Yes, the singularities have to form - under similar assumptions - in spacetimes of any dimension. This is not a problem: a singularity just means a place where the quantum-gravity, Planckian effects become important. In particular, the Penrose-Hawking theorem and its generalizations guarantees that black holes with singularities are formed even in 10D vacua.

The singularities that would be, if formed, problematic from the viewpoint of another Penrose's claim - the Cosmic Censorship Conjecture (the CCC) - are naked singularities, i.e. those that are not dressed in an event horizon. In 3+1 dimensions, naked singularities are most likely not formed (the CCC seems to hold) but I think that it has become pretty much established that in higher-dimensional GR, they may get formed (the CCC fails). They're not a real inconsistency - a consistent theory of quantum gravity may predict what happens even in the presence of naked singularities even if Dr Roger Penrose found it counterintuitive or dangerous.

a) Whether the 4D or 10D singularity theorem is relevant for string theory depends on the situation you consider. If there are 10 large dimensions, you need to use the 10D theorem; if there are only 4 large dimensions, physics is approximately equivalent to GR in 4D and the relevant singularity theorem is the 4D theorem. Obviously, if the compact dimensions are as small as the Planck length, we can't resolve them from a cosmological viewpoint, so the cosmological implications are identical. A singularity in a 3+1-dimensional GR description may secretly be extended in additional dimensions - either all the compactified dimensions or just some of them - but this fact doesn't affect the conclusions about the singularities for much longer distances than the Planck scale.

b) I don't fully understand what "surface phenomena" are but it is true that e.g. in the AdS/CFT correspondence, it is extremely difficult if not impossible to "see" inside the black holes, i.e. to the regions inside the event horizon. That's why the singularities inside the black holes remain largely invisible to the boundary CFT description. In some sense, all of physics may be imagined to reside outside the event horizons and the interior of the black hole contains just some subset of this information that is severely reshuffled: this principle rejecting the independent existence of the interior is known as the "black hole complementarity".

c) There can be singularities localized in the extra dimensions that are extended over the whole large 3+1-dimensional spacetime. In that case, we simply say that the compactified manifold is singular. Conifolds and orbifolds are two classes of simple examples. In those cases, the 4D effective description may remain non-singular. However, whenever you have any singularities whose character strongly depends on the position in the 3+1-dimensional spacetime (in particular, any singularities that are localized in at least one large dimension among the 3+1 dimensions), they will inevitably manifest themselves as singularities in the 4D description by GR as well. It's simply because one can show that the energy density around those singularities blows up and this energy density has to cause a huge curvature, even according to the effective 4D description. So singularities localized in the large dimensions can't hide themselves - not even if they try to localize themselves in the compact dimensions in any way.

  • $\begingroup$ Motl : I will ask a separate question about CCC if I may. However key to understanding this answer for me is understanding the definition of "large dimension" here. Are there three scales for dimensions in String Theory: a)3+1 GR dimensions; b)(sub-)Planck dimensions; c)larger than Planck but not yet observed. Is "Large Dimension" a) and c) or just c)? $\endgroup$ – Roy Simpson Feb 16 '11 at 15:02
  • $\begingroup$ Dear @RoySimpson, by large dimensions, I meant any dimensions that are not compactified, that are large enough for the black holes and other things to be localized in these dimensions. Otherwise the numerical size is a continuous quantity so it can have many values. It is not clear why you think that there are only three real numbers. We don't know what the size is. There exist vacua or at least string-inspired models with pretty much any size of compactified dimensions between the Planck length and microns. I personally think that extra dimensions are at most 1,000 times the Planck length. $\endgroup$ – Luboš Motl Jun 25 '13 at 6:13

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