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There has been some heated debate as to whether the laws of physics allow for traversable wormholes. Some physicists claim we require exotic matter to construct wormholes, but then others counter the Casimir effect with ordinary matter is sufficient. But these same physicists seldom come up with an explicit solution or state of ordinary matter keeping the throat of a wormhole open via the Casimir effect. Yet others claim with extra dimensions, a Gauss-Bonnet interaction is sufficient to keep the wormhole throat open, but opposing physicists claim such a term can't arise from string theory.

So, my question is, do traversable wormholes exist as solutions to string theory?

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It is known that Lorentzian wormholes can exist only if they respond to matter sources that violate some energy conditions. This is a general phenomena: a lot of what makes gravity special is that it is universally attractive. If you violate energy conditions then gravity can become repulsive, and then it is less distinguished from other forces.

On the other hand, it is believed that negative energy matter does not exist, mostly because this will cause instability (producing more of it will lower the energy indefinitely, so no reason for such production to stop at any point). It has never been observed, which is another reason to doubt it exists.

There are some attempts to involve quantum mechanics in order to circumvent this classical result (for example the Casimir effect can result in negative energies). However, if quantum effects are to violate a classical result, there is no way they are small. Then, you have to understand the full quantum mechanics, you cannot reach any conclusion based on the first quantum corrections. Same comment applies to any other attempt to add effects that are normally small (like higher derivative terms). Either you understand the full picture or you cannot come to any reliable conclusion.

There is also an independent problem, which is that wormhole spacetimes are unstable - small fluctuations tend to be magnified in parts of the geometry and therefore change it drastically. So even if you can write down a metric, it is not clear it is a good approximation to a real physical situation.

Having explained some of the issues, it should be clear that you have to understand quite a bit of quantum gravity to address these questions. I don't think that string theory is currently well-understood enough to resolve the question, or even to change anybody's mind either way. For what it is worth, my feeling is that what we know so far indicates wormholes don't exist.

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It might be best to assume that a string solution for wormholes exist and see what the consequences are. Wormholes are similar to black holes, but where the even horizon is replaced by a membrane of some type of quantum field which causes geodesics to diverge. The Hawking-Penrose energy condition, in particular the weak energy condition, gives geodesics which focus inwards in a spacetime diagram. In order to get a wormhole it requires that inward focusing geodesics be “defocused” near or at the region where the event horizon of a black hole would otherwise exist. This means the geodesics are defocused into some other region of spacetime. The wormhole is then two 3-balls cut out of spacetime and where the boundaries of the two balls have points identified with each other. The special field which violates the Hawking-Penrose energy conditions defines a junction in spacetime where the curvature exhibits an abrupt change.

In string theory the action defines the world sheet area $$ S~=~-\frac{T}{2}\int d^2\sigma\sqrt{h}h^{ab}(\sigma)g_{\mu\nu}\partial_a X^\mu \partial_b X^\nu $$ where $T$ is the string tension, $h_{ab}$ is the metric for the string world sheet with coordinates $\sigma~=~(\tau,~\sigma^1)$, $g_{\mu\nu}$ is the spacetime metric and $X^\mu$ is the coordinates of the string parameterized in its modes $\alpha_n$ ${\tilde\alpha}_n$. We then want to expand this string into terms which correspond to spacetime curvatures and then to look at the momentum-energy curvature.

The string coordinates are functions of the spacetime coordinates $X^\mu~=~X^\mu$ where we consider the string coordinates expanded as $X^\mu ~\rightarrow~X^\mu~+~\delta X^\mu$. The variations in the string $\delta X^\mu~=~Y^\mu(\sigma)$ are small oscillations of the string which occur around the string path $\sigma^1$. Now consider $\partial_aX^\mu$ in the action expanded by these variation. The linear term in $Y^\mu$ will be the covariant derivative $Y^\nu\nabla_\nu(\partial_aX^\mu)$. Plugging this in and using the geodesic equation the action for the string in first order with these oscillations $$ S~\simeq~-\frac{T}{2}\int d^2\sigma\sqrt{h}h^{ab}\big(g_{\mu\nu}~+~R_{\mu\alpha\nu\beta}Y^\alpha Y^\beta\big)\partial_a X^\mu \partial_bX^\nu $$ The term $ R_{\mu\alpha\nu\beta}Y^\alpha Y^\beta~\sim~R_{\mu\nu}Y^2$ is negative, which defocuses geodesics. This Lagrangian determines the momentum energy tensor $$ T_{ab}~=~-\frac{2}{T\sqrt{h}}\frac{\delta S}{\delta h^{ab}} $$ which is traceless and the field equation have $\frac{\delta S}{\delta h^{ab}}~=~0$. This then leads to a form of the action as a formula for the area of the world sheet.

We then have a funny situation here. This negative Ricci curvature defines a momentum energy tensor in spacetime indices where $T^{00}~<~0$. Further, the state space which constructs this $\langle~|T^{00}|~\rangle$ is not bounded below. This means the negative curvature may become arbitrarily large. This suggests a serious contradiction, for this can imply the world sheet area of a string can be negative, and arbitrarily negative. It is unclear exactly what is meant by a negative area for a string world sheet.

The black hole has an event horizon with an area $A$, where entropy is $S~=~k~A/4L_p^2$. Each unit of area contributes a unit of entropy, and is further identified according to units of $G$ with naturalized units $area$. The event horizon in the holographic setting if covered by strings, and the modes of the string define the degenerate set of states of the black hole. Hence we may think of the horizon area as a summation over the string world sheet areas, which are positive and are identified with a positive entropy. The wormhole from a stringy perspective has then a funny appearance, where there are negative areas and negative entropies. If wormholes exist it is not hard to see that one could connect up with the interior of a black hole and reduce the entropy of the system by accessing states in the interior.

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