Time Machines J

WDS'05 Proceedings of Contributed Papers, Part III, 663–670, 2005. ISBN 80-86732-59-2 © MATFYZPRESS

Time Machines J. Dolansk´y Charles University, Faculty of Mathematics and Physics, Prague, Czech Republic.

Abstract. A very short review of the time machine problem in physics is presented. First we consider basic knowledge of a time machine. Then we go through two diﬀerent kinds of non-eternal time machine solutions. Further we explore two fundamental consequences of existence of time machine. And ﬁnally part of my recent work will be outlined.

Introduction Time travelling point-like particle is represented by a closed timelike curve, ds2 < 0, in a spacetime. Look at special relativity situation depicted in figure 1. The special relativity, or Minkowski, spacetime is flat and all null cones look in the same direction. In order to travel in time a particle needs to depart from its light cone and so to exceed speed of light. If there are no tachyons – particles moving faster than photons – closed timelike curves in special relativity are forbidden. The idea of time travel entered physics after general relativity was formulated. Since the Einstein equations determine the spacetime only locally, we can consider extremely curved regions or/with topo- logical ’defects’ (holes) at larger scales. Inside such regions an observer using time machine does not exceed speed of light, and even though her light cone still looks into her local future, she gets into her past cone finally.

Time Machine Solutions There are two kinds of time machine solutions: (i) Eternal-time machine spacetimes which contain time machines always. Examples: Tipler’s rapidly rotating cylinder or G¨odel’s rotating spacetime. (ii) Spacetimes where time machines evolve in a certain moment of their history. Examples: Wormhole- based time machines or cosmic string-based time machines.

Wormhole-based Time Machines Wormhole is a hypothetical physical object which connects two separated regions of multiply connected spacetime, i.e. spacetime with holes. We will consider only intra-universe type of wormholes which connect two regions of the same universe. We can imagine a wormhole as a 2-surface embedded in Euclidean 3-space, ﬁgure 2. Or, we can make some surgery on a spacelike 3-hypersurface to get spacetime idea of it [1; 3]: (i) Remove two open balls from an approximately ﬂat spacelike 3-hypersurface. (ii)Identify the boundaries of the holes. (iii) Smooth out the junction. First Example:The Deutsch-Politzer Space. Let us apply the procedure (mentioned in the previous section) to the two dimensional Minkowski spacetime:

v > c

Figure 1. There are no closed timelike curves in special relativity.

663 DOLANSKY: TIME MACHINES

Figure 2. An intra-universe wormhole.

1 -1 2 x -1 1

Figure 3. The Deutsch-Politzer space containing time machines.

(i) Remove the points (t = ±1, x = ±1) from the plane (t, x). (ii) Make cuts along the two spacelike abscissae t = ±1; −1 < x < 1. (iii) Glue the edges of the cuts: the upper edge of the lower cut to the lower edge of the upper cut and vice versa, and receive a depiction of the Deutsch-Politzer space, figure 3. Resulting spacetime is a combination of a cylinder and a plane with two holes. Since we glued the two pairs of holes together we can not get the four points back into the spacetime. The resulting space preserves its flatness, does not contain any singularity, but closed timelike curves occur here – starting from the lower segment and terminate in the upper, the curve 3 in figure 3. The gray truncated 2-cone which starts in the lower segment and spreads to infinity represents non-globally hyperbolic region in which we can not predict results of experiments uniquely. This region neighbours upon the globally hyperbolic part of spacetime, separated from it by a hypersurface (in this case by lines which start from the edge points of the lower cut, slope by angle π/4 and run on to infinity) called the Cauchy horizon. The particle on the curve 1 (figure 3) is passing from the globally hyperbolic part into the evolved non-globally hyperbolic region and so it can be influenced by a particle emerging from the time machine. Thus we are not able to predict its evolution uniquely. The same holds for particle on the curve 2 which is also continuous since running ”under” the handle of the wormhole. Second Example:The Morris-Thorne-Yurtsever Time Machine. The Morris-Thorne-Yurtsever time machine [8](let us abbreviate it simply as the MTY time machine) is based on the static spherically symmetric wormhole constructed by Morris and Thorne [1]. The metric

2 2 2Φ(r) 2 dr 2 2 2 2 ds = −e dt + b(r) + r (dθ + sin(θ) dφ ) 1 − r represents two asymptotically ﬂat universes (glued together) connected by a wormhole if we specify the functions b = b(r) and Φ = Φ(r) as follows. In ﬁgure 4 we can see meaning of r, l and φ. The function Φ

664 DOLANSKY: TIME MACHINES

l φ

r min

Figure 4. The Morris-Thorne wormhole.

is called the red-shift function, it is ﬁnite everywhere which implies that the component gtt never vanishes and so there are no event horizons. The function b is called the shape function and it controls wormhole’s shape. The radial coordinate r = r(l) gives the circumference 2πr of a circle centred on the throat. The throat is deﬁned as such a location where proper radial distance is equal to zero: r = r(l = 0) (l < 0 on one side of the throat and l > 0 on the other side). Moreover, the coordinate r has its global minimum equal to the shape function

r(l = 0) = rmin = b(rmin) ≡ b0 at the throat. The metric contains a coordinate singularity, grr diverges at the throat, but the radial proper distance

r dr l(r) = ± Z b r b0 1 − ( ) q r remains finite everywhere. For r → ±∞ the metric components gαβ tend to the components of the Minkowski metric. In order to compute the stress-energy tensor we have to choose convenient functions Φ and b and plug them into Einstein equations. It has been shown in [1] and [5] that resulting stress-energy tensor violates all known energy conditions.1 That is why we call the matter – which a wormhole consists of – the exotic matter. Thus macroscopic wormholes are forbidden at classical level. However, there are quantum effects – with locally negative energy densities and flux – which violate locally the energy conditions and could, possibly, maintain a macroscopic traversable wormhole. Let us describe the process of time machine evolution on figure 5. In order to produce a time machine we have to induce time shift between the mouths of the wormhole (by their rectilinear, or circular motions) first and then bring them together. The two cylinders represent time evolution of wormhole mouths with world lines parametrized by a time coordinate τ. The extremely curved interior of the cylinders is surrounded by a nearly flat spacetime with the Lorentz coordinates t, x, y, z

ds2 ≃ −dt2 + dx2 + dy2 + dz2.

After creating the wormhole both mouths remain at rest near each other. Then the mouth R start to move to the right along the z axis with nearly speed of light to the distance, say, units of light years. As R draws near it slows down and ﬁnally stops, and then starts to move to the original location with the

1The energy conditions which express the conjecture that our universe satisﬁes the condition of positivity of the energy density. The energy conditions are hierarchically ordered according to the strength of the requirement on the positivity of energy. They form a chain of implications: DEC ⇒ SEC ⇒ WEC ⇒ NEC ⇒ ANEC. Look at their overview: α β α αβ i) DEC (Dominant): Tαβw w ≥ 0 for all timelike vectors w ,and the vector T wβ is causal, 1 α β α ii) SEC (Strong): (Tαβ − 2 T gαβ)w w ≥ 0 for all timelike vectors w , α β α iii) WEC (Weak): Tαβ w w ≥ 0 for all timelike vectors w , α β α iv) NEC (Null): Tαβ k k ≥ 0 for all null vectors k , α β α v) ANEC (Averaged Null): dλTαβ k k ≥ 0 for all null vectors k , along every inextendible null geodesics with aﬃne parameter λ with correspondingR tangents kα.

665 DOLANSKY: TIME MACHINES

Cauchy horizon

t t = 0 L = 0 R z

Figure 5. The evolution of the MTY time machine.

same velocity thereby producing the ’twin paradox’ (which means that an external observer sees that the mouth L aged more then that of R). Since both mouths are again in the same location, one can crawl from R to L, and so return in time. Problem: The MTY time machine could be destroyed by electromagnetic waves passing through the wormhole. Look at the figure 5. The Cauchy horizon is generated by exactly one closed curve C which runs along the z axis from the mouth L to the mouth R and back again. The rest of the Cauchy horizon is made of null geodesics (the dotted lines in figure 5) which converge in the center of the left mouth and form its future light cone. The electromagnetic wave passing through the wormhole before creation of time machine gets Doppler shifted each passage. As it asymptotes (the dashed lines in figure 5) to the curve C it traverses infinitely many times through the wormhole in finite time. This fact could base possible instability of the Cauchy horizon, since such wave would receive infinite energy as the wormhole would get near the instance of creation of time machine. However, in the case of the MTY time machine this process is compensated by the divergence property of the Morris-Thorne wormhole (it works as a diverging lens) caused by the repulsion of the exotic matter. Problem: In the instance of creation of time machine it could be destroyed by electromagnetic vacuum fluctuations circulating through wormhole.

Cosmic String-based Time Machine The couple of cosmic strings is also an exact solution of Einstein equations [6]. The two strings are straight, do not intersect, have infinite length with finite length mass density, and so infinite mass. Despite they do not attract mutually since negative pressure inside of them compensates their gravitational attraction. So the strings are static to each other. We can imagine a cosmic string solution as result of the following spacetime surgery: (i) Cut a wedge from a spacelike 3-hypersurface (represented by the plane (x, y) in figure 6) of Minkowski spacetime. (ii) Join the borders of the cuts together, as in figure 6. (iii) Take the 3-hypersurface y = 0 (which has a zero extrinsic and intrinsic curvature) and make a mirror image y ≤ 0 along it. (iv) Glue both hypersurfaces, y ≤ 0 and y ≥ 0, together back to back. Now we got two strings located at the opposite parts of spacetime with respect to the hypersurface y = 0 and both perpendicular to the plane (x, y) of the figure. The two places A and B in such spacetime are connected by three geodesics: one is going in between the strings G1, the two around them G2 and G3. Due to such a special topology of this spacetime it is possible to beat a light ray radiated from A to B along the G1 if we move close to speed of light along the G2. Moreover, an observer moving also close to speed of light but along to the G1, would see the events of our departure from A and our arrival at B as spacelike separated. How to construct time machine? Take the ’upper’ part y ≥ 0 of the spacetime and move it along the hypersurface y = 0 with such velocity that our departure and arrival are simultaneous in the G1 observer’s frame. The ’lower’ part y ≤ 0 moves with the same velocity but in the opposite direction.

666 DOLANSKY: TIME MACHINES

Identify

y > 0

G2 G2 B String 1 A y = 0 G1 String 2 G3 G3

y < 0

Identify

Figure 6. The static cosmic string solution in (x, y) plane.

t String 1 String 2

Here is time

Here is time travelling allowed.

travelling forbidden.

Moving strings

Figure 7. The Cauchy horizon divide the spacetime into two diﬀerent parts.

Since the hypersuface y = 0 is perfectly flat, both parts match each other and we obtained spacetime with two cosmic strings that move in the opposite directions with a certain velocity. This spacetime is equivalent to the Minkowski spacetime without the cut wedges where the instantaneous travel in the string rest frames between the identified borders is possible. In accord with assumption on the velocity of the strings is our departure from A simultaneous (for the G1 observer) with our arrival at B which is simultaneous with our arrival at A again. Thus we can meet us at A after our arrival before our younger self depart to the B, i.e. the timelike curve is closed. The Cauchy horizon of the two moving strings divide whole spacetime into two parts and only in the outer part is time travelling enabled, figure 7. Note: The cosmic string-based time machine, unlike the wormhole-based time machine, does not imply multiple connectedness. Note: The cosmic string-based time machine, unlike the wormhole-based time machine, does not violate the WEC.

Consequences of Existence of Time Machines In this section we show some problems related to the existence of time machine.

Violation of Global Hyperbolicity 2 As a consequence of existence of time machine we have to consider the loss of the property of global hyperbolicity. We saw in the Deutsch-Politzer section that by some surgery we can transform originally globally hyperbolic spacetime into its opposite. In this section an example of a spacetime loosing its global hyperbolicity in a certain moment of its history will be shown. Such a loss has very

2A spacetime is globally hyperbolic if it has a subset (Cauchy surface) crossed just once by every inextendible causal curve.

667 DOLANSKY: TIME MACHINES serious consequences: If we lived in a non-globally hyperbolic spacetime, we would not be able to predict results of any experiments uniquely. In order to preserve predictability we should get a global hyperbolic spacetime as a solution of Einstein equations under specific initial conditions, and with the requirement of maximality 3 However, there are spacetimes which can lose its global hyperbolicity in the course of their evolution - after a wormhole evolves in them - and still stay solutions of Einstein equations [3]. This can be demonstrated with the help of figure 5: Let us now define the function τ = τ(p) on the boundaries of the cylinders BL,R which stands for the length of the longest timelike curve from a point p ∈ BL,R to some hypersurface t =const. in its past along BL,R. Then is τ(p) = t(p) only on the non-deformed cylinder. Remove the interiors of both cylinders and identify their boundaries according to

pL = pR ⇒ τ(pL) = τ(pR), where pL ∈ BL and pR ∈ BR. Thus, after smoothing out the junction of the two boundaries, we have constructed the Morris-Thorne wormhole spacetime W . We will show that it is non-globally hyperbolic. On the hypersurface t = const. the identified points have the same time coordinate: t(pL) = t(pR). This changes into an inequality t(pR) > t(pL) only a moment later. As coordinate time goes on, the inequality [t(pR) − t(pL)] > [x(pR) − x(pL)] becomes true, and the identified points become causally connected in the initial Minkowski spacetime. Thus, we got MTY time machine which evolved in the future of globally hyperbolic spacetime H = W \ Q, where Q is the light cone-region in which are the closed timelike curves confined. This process can be observed in figure 5 where the dashed lines transform into circles by identification of BL and BR. Along the time coordinate the dashed lines get larger and larger slope, and finally, that one which lies on the boundary of the cone becomes null. Thus the first closed causal curve is enabled. So, we have to choose between the maximal spacetime W which is non-globally hyperbolic and the globally hyperbolic spacetime H which is not maximal. Thus our former requirement stays unsatisfied and we have to give up to predict anything at all in spacetime given only by Einstein equations, initial conditions, and even supposing its global hyperbolicity.

Time Travel Paradoxes The travel paradoxes represent rather logical aspect of existence of closed timelike curves. The most famous of them is the so called grandfather paradox: Grandfather ask his grandson to go back to the past and kill him (grandfather) in his infancy. But if grandson succeeds he can not be born, so he can not succeed in killing his grandfather in his infancy, etc. This paradox can be reinterpreted as a badly formulated problem, since the initial conditions are affected by the outcome of the experiment. This can be resolved so that the grandson always fails in killing the baby. A little different is the case when time machine is created after initial conditions have been fixed: Wife asks her husband to create a time machine, enter it and kill his younger self. There are two possibilities: (i) If the man is dead when he meets his younger self, and so he can not kill him, why is he dead? (ii) If the man is alive and kills his younger self, so why the victim survived? Since in that case the initial conditions do not depend on the result of the experiment the paradox can not be resolved as above. Mathematically, such a situation means that in a non-globally hyperbolic spacetime, a simple Cauchy problem need not to have any solution. This means that world lines of initial particles can not be continued in agreement with the formulated laws of interaction. Or, that these initial conditions does not determine evolution of particles – no evolution is consistent with initial data [4]. The twisted Deutsch-Politzer space, figure 8, comes up from the Deutsch-Politzer space if we rotate one of the cut by π. This space is especially convenient for showing these paradoxes because the world lines even of free falling particles have self-intersections. Suppose the simplest initial data – no particles at all initially. Even under such conditions there is no way to determine evolutions of the particles, i.e. there is no unique solution of the uniquely formulated Cauchy problem. The spacetime may always remain empty, or some particles may appear there later as well – emerging from a hole.

Sketch of my Work Time machines would enable us to travel to our past and so we could meet our younger selves. The problem of consistency of such meetings is the main interest of my work. A demonstration program

3A solution is maximal if it has no further extension.

668 DOLANSKY: TIME MACHINES

t 1

-1

x -1 1

Figure 8. The twisted DP space with world line of a particle.

T = t wormhole enter e tip L con α ψ β γ L σ a σ ρ ρ r xit φ

le e b mho urin wor u ‘ v u

t ∆ v ‘ urout T = t -

Figure 9. The self-collision in the conical, eternal time travel spacetime.

representing self-collisions of either a point-like particle or a billiard ball with ﬁnite diameter will be made. The work should enlighten the question if the Chauchy problem is well posed for a spacetime with wormhole-based time machine. The Chauchy problem is well posed if – roughly speaking – the given initial data and evolution equations yield just one solution. The question arises if there are so called ”dangerous” trajectories for which we assume no collision (in initial conditions). An observer moving along such curves enters the wormhole, exits it before she entered and then runs into collisions with herself (see the time travel paradoxes section), i.e. such Cauchy problem is ill posed.

Geometrical Model for a Point-like Particle We assume the eternal-time machine spacetime made of cone with some time shift △t (ﬁgure 9). As initial data we take the impact parameter ρ > 0 and the magnitude of initial velocity of the particle u > 0. Additional problem parameters are represented by the cone angle 0 < γ < π and the time delay △t > 0. We look for the location of the self-collision, given by the radial coordinate r and the angle σ, from which we are able, with respect to the symmetries of the problem, to determine the components of all velocities u, u′, v, v′, where u′ is the velocity of the younger self after the collision, v of the older self before the collision and v′ of the older self after the collision. Thus we should be able to decide on existence and uniqueness of solutions under such simpliﬁed assumptions.

669 DOLANSKY: TIME MACHINES References [1] M. S. Morris, K. S. Thorne, Am. J. Phys. 56, 395 (1988) [2] S. Krasnikov, A traversable wormhole, gr-qc/9909016 (1999) [3] S. Krasnikov, The wormhole hazard, gr-qc/0302107 (2003) [4] S. Krasnikov, Time machine (1988-2001), gr-qc/0305070 (2003) [5] M. S. Morris, K. S. Thorne, U. Yurtsever, Phys. Rev. Lett. 61, 1446 (1988) [6] J. R. Gott III., Phys. Rev. Lett. 66, 1126 (1991) [7] F. Echeverria, G. Klinkhammer, K. S. Thorne, Phys. Rev. D 44, 1077 (1991) [8] T. A. Roman, Phys. Rev. D 47, 1370 (1993)

670