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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 con- nected 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: t v > c v > c x Figure 1. There are no closed timelike curves in special relativity. 663 DOLANSKY: TIME MACHINES Figure 2. An intra-universe wormhole. t 2 1 3 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 l 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 ⇒ W EC ⇒ 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 t Cauchy horizon C 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.
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