The String Coffee Table

Introduction

It has long been known that general relativity seems to allow interesting solutions such as wormholes [1] and warp drives [2]. This is perhaps not terribly surprising in that we can take any metric on a spacetime, apply Einstein’s field equation and declare, by fiat, that $G_{\mu \nu }=8\pi T_{\mu \nu }$. Thus, in order to restrict the possible solutions, we must supplement Einstein’s field equation with a restriction on the types of stress-energy allowed. These are termed energy conditions. It can be shown that traversable wormholes [1,3,4], warp drives and, indeed, any method of faster-than-light travel [5,6], must violate an energy condition, usually the averaged null energy condition.

Nonetheless, we are not out of the woods. There are many quantum systems which are known to violate various energy conditions. The Casimir effect serves as the most common example. Other examples include the theory of inflation which depends on the violation of the strong energy condition. There are also known to be exotic fluctuations around the horizon of an evaporating black hole that allow the violation of Hawking’s area increase theorem. There has been research in constraining the types of energy condition violation allowed by quantum mechanics [7] but the situation remains uncertain. To my knowledge, while very much doubted, neither the existence of wormholes nor of warp drives has been ruled out.

Susskind [8] has recently proposed two gedankenexperiments that purport to show how wormholes, if they exist, lead to some unexpected weirdness irrespective of the energy conditions. In the first scenario, he uses the fact that a charge traversing a wormhole adds a unit of flux along the wormhole. Thus, if we take two systems in some correlated charge superposition state and pass one through the wormhole, it becomes correlated with the flux quantum number of the wormhole. If we then attempt to measure the correlation between the two original systems, taking one the long way around, we will not measure any correlation because of the entanglement with the flux.

The subject of this paper is the second scenario described by Susskind. There, he uses the fact that the wormhole seems to exactly measure energy to show that the uncertainty in time is infinite, as follows from the energy/time uncertainty relation. While there are many subtleties associated with this uncertainty relation, they will be irrelevant for the purposes of this paper. In section two, we will review Susskind’s argument in more detail. In section three, we will resolve the apparent paradox in 3+1 dimensions and show how the energy/time uncertainty relation is always satisfied. We demonstrate how this (generally quite small) time uncertainty can be seen to follow from the usual uncertainty relation for position and momentum. Interestingly, this argument does not seem to apply to the 2+1 dimensional case, leaving an intriguing problem. As is usual in such arguments involving the uncertainty relations, we will assume that they are saturated. We will also neglect leading constants and only keep track of the dependence on the various parameters of the problem.

Susskind’s Wormhole

To model Susskind’s wormhole, we can begin with flat ${ℝ}^{\mathrm{2,1}}$. In the two dimensional space, select two circles (cylinders in spacetime) extremely far apart. If we identify these circles, we have created a wormhole. While in higher dimensions, the wormhole mouths will generally warp the spacetime far away, in 2+1 dimensions they only create an angle deficit. Thus, we can measure their masses. Call the two mouths A and B.

Now, let us place two identical synchronized clocks near mouth A. These clocks must have some mass, so as to be able to measure time with any accuracy. Now take one of the clocks to mouth B, going the long way around, ie, not through the wormhole. Pass the other clock through the wormhole. The question is whether the clocks are still correlated. For example, if we were to use our knowledge of general relativity to compute the time change for the clock that took the long way, would that be the difference in times shown on the clocks? Or, if we were to repeat the experiment with any number of clocks, would the differences in time have a highly peaked distribution or be relatively flat?

An interesting fact about wormholes is that they change in mass as an object passes through them. To see this, imagine a wormhole connecting two distinct asymptotically flat spacetimes. In each spacetime, the ADM mass is conserved. Thus, if we pass an object of mass $m$ through the wormhole, from A to B, the ADM mass on either side cannot change. This means that the mass of the mouth A increases by $m$ and the mass of mouth B decreases by $m$. In other words, the wormhole has measured the mass of the object. Note that this argument applies in any dimension in which the ADM mass makes sense.

Now we come to the crux of Susskind’s argument. Since the wormhole has measured the mass of the clock, we have $\Delta E=0$. By the usual energy/time uncertainty relation,

$$\Delta E\Delta t\gtrsim \hslash ,$$

we have $\Delta t=\mathrm{\infty }$. In other words, the times on the clocks are completely uncorrelated. If you were to enter the wormhole, you would have no idea when you would come out. Performing the experiment repeatedly, we would observe a distribution with infinite standard deviation for the time differences of the clocks. If not a stark paradox, this makes wormholes significantly weirder than they might appear at first glance.

The resolution

Energy and time uncertainties

The problem in the above is the idea that we can measure the mass of the mouth of the wormhole exactly. The see the problem, let us recall the definition of the ADM mass. We will hence forth work in 3+1 dimensions¹. I will be schematic; for more rigor, see [9] for example. In an asymptotically flat spacetime, we know that the metric approaches a flat metric as some coordinate, $r$, approaches infinity. The ADM mass is then defined by the integral

$$M\cong \frac{1}{16\pi G}\underset{r\to \mathrm{\infty }}{\lim }\int \sum _{\mu ,\nu =1}^3(g_{\mu \nu ,\mu }-g_{\mu \mu ,\nu })N^{\nu }\mathrm{dS}.$$

Here, we integrate over a surface at infinity where $N^{\nu }$ is the unit normal to this surface.

One immediately sees a problem. In our wormhole spacetime, there is only one asymptotic infinity and, thus, only one ADM mass. While we might consider this as invalidating the scenario, this does not accord with our intuition that, when the wormhole mouths are sufficiently separate, we should not have to worry about the other mouth. Thus, we ought to be able to measure some sort of mass for each mouth in this limit. To see how we might do this, note that, sufficiently far away from the mouth, we can recast the metric in the following form

The $M$ that appears here will serve as a definition of the mass of the wormhole mouth. The $V$s are just velocities that we can set to zero.

Now we can see the difficulty with Susskind’s scenario. In fact, we cannot precisely put the metric in the form (1). As we can superimpose the perturbations, the true metric will be of the form

$$g_{\mu \nu }={\eta }_{\mu \nu }(1+2G(\frac{M_A}{r_A}+\frac{M_B}{r_B}))={\eta }_{\mu \nu }+{\eta }_{\mu \nu }\frac{2G}{r_B}(M_B+\frac{M_A{r}_B}{r_A}).$$

Assuming we are close to mouth B, we can approximate $r_A\sim d$. Comparing with (1), this gives the measured `mass’ of mouth B as

$$M\sim M_B+\frac{M_{𝔸}{r}_B}{d}$$

Thus, we have

$$\Delta M\sim \frac{{\mathrm{GM}}_A{M}_B}{d}.$$

Note that, in accordance with our intuition, as we separate the wormhole mouths, we decrease the uncertainty. Now, if we want to transit an object of mass $m$ from mouth A to mouth B, the value of $M_B$ must be at least $m$ so as to not give the mouth a negative mass when the object emerges². This implies that the best accuracy we can expect is ${\mathrm{GM}}_{A}m/d$.

Now, we plug this into the energy/time uncertainty relation to obtain:

This is generally a quite small number. However, we can certainly imagine an enormous universe where we can set $d$ as large as we wish, giving arbitrarily large time uncertainties. Thus, we have apparently not eliminated the weirdness from our world.

Momentum and position uncertainties

In fact, there is more uncertainty in the problem. At the beginning, each box has to reside in the region of the wormhole’s mouth. Let’s say it begins at a radius $q$. Thus, the uncertainty in its position is certainly less than $q$. This means that the usual uncertainty relation, $\Delta p\Delta x\gtrsim \hslash$, implies an uncertainty in the momentum

From the standard fact, $d=\mathrm{vt}=\mathrm{pt}/m$, we have

where $\mathrm{KE}$ is the kinetic energy of our object, and we have plugged in (3).

Now, in order to escape the gravitational well of wormhole A, assuming no other energies in the problem, we must have an initial $\mathrm{KE}$ of at least ${\mathrm{GM}}_{A}m/q$. Plugging this into (4), we obtain

$$\Delta t\sim \frac{\hslash d}{G{M}_{A}m}$$

which is exactly the relation from equation (2) that we derived from the energy/time uncertainty relation. One might ask what would happen if we were to decrease $M_A$. In fact, because we passed one clock through the wormhole, $M_A$ has a minimum value of $m$ and cannot be decreased to zero.

There is still a problem here, however. We have shown that we can account for the uncertainty in one scenario, but it seems as if we can increase KE to lower the uncertainty in the time of transit. Must think more….

Acknowledgments

I would like to thank Jacques Distler, Lenny Susskind and Uday Varadarajan for useful conversations.

¹As noted above, in 2+1 dimensions, the ADM mass corresponds to a deficit angle. As no matter implies flatness in 2+1 D, we can measure the mass of an object unambiguously without going to infinity.

² To my knowledge, this statement is unproven. Normally one would expect that the ADM mass must be positive from the positive mass theorem. However, this theorem depends on the dominant energy condition which will be violated in a wormhole spacetime. Nonetheless, the prospects of an unbounded from below mass are disturbing enough that this seems like a reasonable assumption.

Bibliography

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[3] M. Visser, Lorentzian Wormholes: From Einstein to Hawking.

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[5] R. J. Low, “Speed limits in general relativity,” Class. Quant. Grav. 16, 543 (1999).

[6] K. D. Olum, “Superluminal travel requires negative energies,” Phys. Rev. Lett. 81, 3567 (1998).

[7] L. H. Ford and T. A. Roman, “Quantum Field Theory Constrains Traversable Wormhole Geometries,” Phys. Rev. D 53, 5496 (1996), and many more by the same authors and by others.

[8] L. Susskind, “Wormholes and time travel? Not likely”.

[9] R. M. Wald, General Relativity.