Physical Constraints on Time Travel

David R. Tribble
26 Aug 2010

Abstract

One of the problems with time travel is the possibility of the Grandfather Paradox, caused when interactions between a time traveler and his past history alter the time-line history in ways that lead to existential paradoxes. In order to avoid such paradoxes, all possible interactions between the time traveler and his past history must be made impossible, so that nothing he does in the past could possibly affect the time-line history leading up to his present departure time. To this end, a Spatial-Temporal Interaction Exclusion Principle (STIEP) is postulated, a physical constraint which prevents any matter-energy from traveling to a point in space-time history that could potentially alter the course of the history leading up to its point of departure. This constraint makes it impossible to travel to any point within the light-cone originating at the point of departure (the light-cone extending both forwards and backwards in time), restricting the influence of any matter-energy interactions between the time traveler and the time-line of his past history to only occur outside that light-cone, making it impossible for him to alter his past time-line, and thereby rendering the Grandfather Paradox a physical impossibility. However, travel to any space-time location outside the light-cone is still a possibility for a time traveler, since the light-cone originating at any such destination point does not intersect his departure-centered light-cone, and therefore any matter-energy interactions from such a destination point could not affect any of the history leading up to the departure point.

Keywords

Time travel, Grandfather paradox, light-cone, space-time history, time-line.

Assumptions

It is assumed in this discussion that time travel involves a single space-time history, or time-line. This means that travel backward or forward in time does not involve alternate or parallel universes, and that any interactions a person or object has with other matter-energy affects the history of the single time-line.

Practical considerations about the mechanism for affecting time travel, or the source of vast energies required for such a device, are not considered in this discussion. Likewise, problems concerning the accurate transport through time to locations on a rotating and moving Earth are ignored.

Lastly, it is assumed for the sake of this discussion that time travel in some form is possible. The constraints discussed here do not render time travel completely impossible, but simple place certain specific physical constraints on any time travel mechanism that might be theoretically plausible.

Background

Consider that Terrence, a time traveler, wishes to travel from his current departure point in space-time, back in time to some other arrival point in space-time. Assuming a single time-line model of time travel (as described above), one of the problems is the potential for the Grandfather Paradox to arise. The paradox requires that Terrence travel far enough back in time (perhaps 70 years or so) to meet his grandfather, Godfrey, before Godfrey has fathered Terrence's father, Fred. Terrence kills his grandfather, or in some equally effective way prevents him from fathering Fred. A paradox is then created: since Fred was not born, then Terrence likewise could not be born; and yet Terrence came back in time, having been born to Fred, who had been born to Godfrey. Thus the paradox: how can Terrence exist, having been born and having not been born?

One suggestion as to how to resolve this paradox states that by traveling backward in time, a time traveler actually arrives in a different time-line, an "alternate" or "parallel" time-line. In such a time-line, Terrence could meet Godfrey and kill him (or otherwise prevent his son Fred from being born), but this would have no affect on Terrence's existence. Since the Godfrey in the alternate/parallel time-line is a different Godfrey than his real grandfather back in Terrence's original time-line, anything that happens to this second Godfrey, including being killed, could not possibly influence or affect the history of the first Godfrey back in Terrence's original time-line. For the purposes of this discussion, we do not consider this solution.

A second suggested solution to the Grandfather Paradox involves a single history time-line, but proposes that the laws of physics act somehow in concert to prevent any chain of interactions from altering history in a way that would lead to a paradox. In this scenario, Terrence goes back in time and meets his actual grandfather, but is unable to prevent him from fathering Terrence's father. Perhaps Terrence cannot arrive at any date within Godfrey's lifetime, or perhaps he cannot arrive at a place that is close enough geographically to influence Godfrey's future, or perhaps the laws of nature make it physically unlikely for Terrence to get anywhere close enough in time and space to Godfrey, requiring Terrence to expend larger and larger amounts of energy as he tries to get closer and closer to Godfrey, to the point that it becomes a practical impossibility. Or perhaps nature somehow conspire against him, so that all of the sequence of events necessary for Terrence to kill his grandfather simply cannot occur. Perhaps the gun he uses to kill Godfrey malfunctions at a critical time, or perhaps the bullet is defective, or perhaps the subway he rides breaks down on the way to meeting his grandfather, and so on.

Under the conditions of this resolution, Terrence is able to travel back in time to arrive close enough in space and time to interact with his grandfather, but not with the outcome that he had intended. So while he may be in the right place and time to create the conditions for the Grandfather Paradox, the laws of physics prevent any sequence of events from occurring that could actually do so.

Constraints on Time-Travel

Taking the Grandfather Paradox to its logical extreme, it is clear that any possible interaction that Terrence has with the environment after his arrival in the past might possibly alter history from that point forward, changing the time-line of events leading up to the time of his departure into the past. In fact, the simple acts of breathing oxygen, walking, touching objects, disturbing the air, making sound, etc., would have a non-negligible effect on the matter-energy particles of the universe within Terrence's immediate vicinity. Considering all of the matter-energy particles, quantum effects, and the like comprising Terrence's body, clothes, and anything else he brought with him on his journey, interacting with the particles and energy in the environment of the past into which he arrived, it is clear that the time-line history would be altered from the very point of his arrival, so that it would no longer be identical to the time-line as it (presumably) existed prior to his arrival.¹

To prevent any paradox from occurring, there must be some kind of physical constraint upon the interactions that his presence in the past could possibly produce. Since any interaction at all could affect the history leading up to the time of his departure from the future, this constraint must eliminate all possible interactions within the past light-cone originating at the space-time point of his arrival in the past. Such a postulated constraint would place all physical interactions caused by Terrence, from the point of his arrival in the past up to the point of his departure, far enough away from all possible matter-energy interactions within the past time-line leading up to his departure location in space-time. This constraint therefore must restrict his arrival point in the past so that it is located outside the past light-cone centered at the point of his departure. Only matter-energy interactions inside this light-cone can have an effect on the time-line history leading up to his point of departure, so by restricting all possible interactions to occur at space-time locations outside the light-cone, such interactions cannot happen.

Such a constraint is dubbed the Spatial-Temporal Interaction Exclusion Principle, or STIEP for short.²

Figure 1 – Light-cones for two points in space-time

Figure 1 illustrates this graphically. A time-traveler departs from the time-line at space-time location P_d (at location s_d and time t_d) and arrives at space-time location P_a (at location s_a and time t_a) in the past. If his arrival location is anywhere within the past light-cone −C_b originating at P_d, the matter-energy comprising him will interact with matter-energy of the past time-line, affecting the history of events leading up to the point of his departure P_d. Any such interaction could cause a paradox, or at least would alter the course of history. Since that is not allowed, his point of arrival in the past must exist at some point P_c outside the past light-cone. Arrival point P_c can be located at a point in time different than t_a, or located at a point in space different than s_a.

Figure 2 – Arrival point P_c located inside light-cone C_d

Figure 2 shows a point of arrival P_c in the past that is located at a different spatial coordinate space-time than P_a but is still located within the past light-cone −C_d. At point P_i the light-cone originating from P_c intersects the spatial coordinates of P_d, illustrating the earliest point in space-time that any interactions originating from P_a could affect any matter-energy located at the same spatial location as P_d. While there are fewer matter and energy particles brought back in time with the time traveler that could affect the time-line leading to P_d than if they had arrived at point P_a, any matter-energy particles that could possibly interact with matter-energy particles within the past light-cone −C_d could potentially affect the time-line leading to P_d, and therefore could create a paradox.

Figure 3 – Arrival point P_c located outside light-cone C_d

Figure 3 illustrates an arrival point P_c in space-time that is located outside the past light-cone −C_d. All matter-energy interations from that point on occur within the future light-cone +C_c originating at P_c, and since +C_c does not intersect the past light-cone −C_d, none of those interactions can affect the time-line leading up to P_d. In fact the earliest possible interactions that could affect the time-line for P_d occur at point P_i, which occurs later in time than P_d. Therefore all possible interactions from the time-traveling particles can only occur after the point of departure P_d, and therefore a paradox cannot occur.

Examples

As an example, consider Terrence the time-traveler departing from Dallas on a Tuesday, traveling back exactly 24 hours earlier, and arriving one day ealier in the past, on the preceding Monday. STIEP prohibits his arrival at the same spatial location of Dallas, however. The closest he could get to his original departure point (ignoring the effects of a rotating Earth moving in its orbit in space) would be 24 light-hours (about 26 billion kilometers) away from Dallas (which would put him at a location roughly six times the distance of the orbit of Neptune, or about three times further from the Sun than the Kuyper Belt). Not a pleasant prospect for Terrence, unfortunately.

Futhermore, once he arrived in the past (assuming he had taken proper precautions about wearing a spacesuit, carrying a big supply of oxygen, water, and food, and in fact taking the trouble of transporting himself back while inside a fully-stocked spaceship), assuming that he could travel the speed of light back towards Earth, any matter-energy affected along the way during the journey back to Dallas would not (could not) affect any matter-energy within the light-cone centered at the time-travel booth at the moment he departed. Therefore, nothing he (or his ship) could interact with could possibly alter the past time-line history leading up to moment and place of his departure.

On the other hand, assuming that Terrence could arrive within a kilometer of his departure point, STIEP would limit the closest point in time that he could get to that location in the past. In this case, he could go no more than three microseconds (the time it takes light to travel one kilometer) back in time and still remain within a kilometer of Dallas. Any further back in time and he'd have to be further away spatially. Once he arrived, the light-cone centered at the time-travel booth from which he departed would not (could not) be affected by any of Terrence's matter-energy. Even if he could travel the speed of light back towards the booth, any matter-energy that his body interacted with along the way would be outside the light-cone originating at the booth the moment he left, and thus the history time-line of the past leading up to the moment of his departure in the booth could not be altered.

Applying STIEP to the scenario in which Terrence attempts to kill his grandfather, since Terrence travels backwards in time 70 years, the closest he could get spatially to his grandfather on Earth is no more than 70 light-years. Even assuming that he was already traveling at the speed of light towards Earth the moment he arrived 70 years in the past, he would not be able to affect any of the history on Earth up to the point in space-time at which he departed from the present (70 years in the future from his arrival time). In fact, he would not be able to arrive on Earth in time to meet his grandfather at all, at least not until 70 years of traveling at the speed of light to get back to Earth, at which point he would be temporally located at the time of this departure, long after his grandfather had fathered Fred and Fred had in turn fathered Terrence. Thus Terrence would not be able to kill his grandfather, and therefore there is no possibility that he could create a spatial-temporal paradox.

A possible objection to the scenario just described presents itself. Suppose that Terrence's grandfather Godfrey is already at a location 70 light-years away from Earth, near the spatial-temporal location where Terrence arrives (70 years in his past). Terrence could then kill Godfrey, preventing Godfrey from fathering Fred at any point in the future (prior to Terrence's departure). The problem is that this scenario is impossible; since Godfrey is 70 light-years from Earth, and it was (is) on Earth that Fred was born, and subsequently where Terrence was born, it is physically impossible for Godfrey to be at a location that far removed from Earth. More specifically, it is impossible for Godfrey to be 70 light-years from the point of Terrence's departure 70 years in the future, because it would then be impossible for Godfrey to be Terrence's grandfather. Thus this objection, which was created to make the Grandfather Paradox possible again, itelf contradicts the initial conditions, that of Godfrey being Terrence's grandfather. So this is not a plausible objection.

Other Applications

These physical constraints affect other applications of time travel. Consider Kip Thorne's speculative time-travel machine which utilizes the two ends of a wormhole. In such a device, one end of the wormhole is sent on a long journey at high velocity and then returned to a location close to the other end of the wormhole. Due to the Twin Paradox, the passage of time experienced at the first end is slower than the second (unmoved) end, thus creating a temporal differential between the two ends. The two wormhole ends thus form the mechanism for a time travel device. Time travelers who enter the first end exit the second end at a later point in time, effectively traveling into the future by a time equal to the temporal differential of the two wormhole ends. Likewise, entering the second end and existing the first allows a traveler to travel backwards in time by the same amount.

The STIEP constraints, however, limit the usefulness of such a device, by limiting how close the ends of the wormhole can be brought together. Specifically, each end of the wormhole can only be brought as close to the other end so that its light-cone is not affected by interactions from the other end, and vice versa. So in a situation similar to the example given above, given a wormhole time machine having a time differential of 24 hours, its two ends could be no closer than 24 light-hours apart, or about 26 billion kilometers. Likewise, a wormhole time machine with its two ends located one kilometer apart can have a time displacement of no more than three microseconds.

Conclusion

Matter-energy particles traveling backwards in time and arriving close to the same spatial location as their departure may interact with matter-energy particles in the past, thus altering the past time-line and allowing for the possibility of the Grandfather Paradox to occur. The postulated Spatial-Temporal Interaction Exclusion Principle (STIEP) eliminates the possibility of any such paradox by restricting the possible points of arrival in the past to fall outside the past light-cone originating at the point of departure, making it impossible for any such interactions to occur before the point of departure.

Notes

1. One way out of this difficulty is to simply assert that Terrence's arrival in the past actually forms part of the whole time-line history. That is, his existence in the past was (is) actually already part of the time-line of the past. He was "meant" or "destined" to travel back in time, even before he decided to do so, because he existed as part of history in the past, even before he actually traveled back in time from the future. The trouble with this resolution is that it still allows the Grandfather Paradox to occur. Saying that Terrence's travels in the past make up the history of the past makes it easier, in fact, to allow for the paradox, by simply saying the paradox would be embedded as part of the total history. Since such a solution only makes things worse, logically, than before, it seems reasonable to reject such a solution.

2. "STIEP" is pronounced under regular American and Germanic pronunciation rules, rhyming with "steep".

References

[1] Time travel.: Wikipedia
[2] Grandfather paradox.: Wikipedia
[3] Novikov self-consistency principle.: Wikipedia
[4] Parallel universe.: Wikipedia
[5] Many-worlds interpretation.: Wikipedia
[6] Light cone.: Wikipedia