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The Possibility of Time Travel through General Relativity

Philosophers strive for conceptual clarity, making the physical possibility of time travel on Einstein’s General Theory of Relativity (hereafter, ‘GR’) a topic of particular philosophical interest (Arntzenius and Maudlin, 2023). There are indeed universes on GR with closed time-like curves (Effingham, 2020). Time travel—along such structures—is therefore a physical possibility according to GR in the kind of spacetime that permits it. This article will explain how time travel is physically possible and, through a philosophical lens, offer a conceptual study of GR and the time travel possibility. To this end, this essay evaluates how one must very carefully address the concept of 'physical possibility'. The analysis starts by providing some brief background on Einstein’s field equations and the locality of GR, thus going on to explain how GR allows for universes with time travel. A major conceptual threat to this is also considered here, namely the Grandfather Paradox (Effingham, 2020). Then, it is finally discussed what physically possible really means according to GR. Via the Global Constraint concept (Lewis, 1976), this feature overall aims to show that the so-called Grandfather Paradox does not pose a serious threat to the physical possibility of time travel permitted by Einstein’s theory of GR.


Einstein’s Theory of General Relativity


Einstein's Theory of GR reflects his conviction that the theoretical physicist must trust simplicity, thus moving steadily into domains even further removed from direct contact with observation and experiment (Norton, 2000). Having published the Special Theory of Relativity in 1905, the physicist sought to incorporate gravity into his new relativistic framework (Matthews, 2013), and notably, via his field equations, subsequently developed GR in 1915 by providing the key mathematical framework on which he fit his physical ideas of gravity (Carmeli, 2008). His proposed idea of a single continuum where space and time are interwoven is what is now known as spacetime (Kennedy, 2014). In GR, spacetime is notably intrinsically curved. For example, energy bends spacetime, and spacetime is not embedded in a higher dimensional space but is instead intrinsic. Hence, there is no extrinsic curvature since there is no exterior space to look at it from – maps of large regions may not be drawn without distortion of the map (Collas, 1997). Broadly speaking, though, GR detects curvature in space-space sheets (i.e., the ordinary two-dimensional slices of three-dimensional space). The summed curvature (of all sheets) is then equal to the total matter-energy density (Hobson, Efstathiou, and Lasenby, 2006). Einstein’s famous gravitational field equations, therefore, relate the geometry of spacetime to the distribution of matter within spacetime. More specifically, the Einstein tensor equals the stress-energy tensor (Norton, 2007), meaning that GR describes spacetime curvature in a manner that is consistent with the conservation of energy and momentum (Lovelock, 1971):


Gμν = Tμν


Importantly, the field equations make up the core of Einstein’s theory and embrace the entirety of gravitational phenomena as well as the geometry of space (Norton, 2007). This explains the locality of GR since Gμν = Tμν can explain every local region (Hobson, Efstathiou, and Lasenby, 2006). Gμν and Tμν are equal then to decide which spacetimes (i.e., which universes) are admissible. If the spacetime curvature and matter density are related appropriately, then the universe is admissible (Norton, 2007). Hence, if a universe is physically possible according to GR, then Gμν = Tμν holds at every region of spacetime in that universe (Norton, 2007), roughly speaking.


Figure 1. NASA's Gravity Probe B corroborates Einstein's prediction of massive objects warping their surrounding spacetime (Wall, 2015)

Time Travel and The Grandfather Paradox


Gμν = Tμν explains all the possible ways a universe can be. Remarkably, in such a universe where space is curved back onto itself, GR discovers that there is a physically possible (admissible) universe where the prospect of time travel arises. This is the Cylinder Universe, which allows for time travel where every single spacetime region is a Minkowski spacetime, that is, with zero curvature and zero-sum curvature, named after German mathematician Hermann Minkowski and allowing for spacetime to curve with the resulting effect of gravity (Norton, 2007). One spatial dimension is wrapped onto itself (hence, ‘cylinder’ universe). What is more, this universe involves only one dimension of time. The cylinder is wrapped onto itself in the direction of time: a closed time-like curve (Weingard, 1979). A time traveller, travelling along the curve, will thus always come back to the same moment in time since the closed time-like curve forms a loop (Weingard, 1979). Every region of spacetime is Minkowski spacetime and is therefore permissible according to GR. As such, time travel is a physical possibility because this universe is a physical possibility (it is important to note that this is all according to GR).


Gμν = Tμν is the solution, and each region of spacetime (Minkowski spacetime) connects to the next. Remarkably, this is all that is needed for spacetime to count as a solution to Einstein’s gravitational field equations (Norton, 2007). So, a time traveller on this closed time-like curve would—technically, at some point—wrap all the way around the curve and come back to the same moment in time. Put differently, the time traveller would reconnect with their former self (see figure 2 for a diagrammatical representation of this universe). This extraordinary possibility is somewhat complicated, however. The very possibility (and problem) of time travel seems to yield a paradox (Maudlin, 1990). Said logical problem is famously known as the Grandfather Paradox, and conceptually presents a real challenge to the physical possibility of time travel in GR. In a somewhat simple fashion, it says that if a time traveller goes back to a time before their grandfather had children, and kills him, the grandfather’s death would make their own birth impossible. Indeed, the killing of the grandfather also entails no time traveller to go back in time to kill him (Norton, 2007). The closed loop, therefore, appears contradictory – the time traveller both does and does not exist at some and the same time. The time traveller goes back in time if and only if the time traveller does not go back in time (both P and ~ P, the kind of logical contradiction that must be taken seriously in any physical theory). Perhaps, one wonders, time travel is not physically possible after all. But what does ‘physically possible’ mean in the sense of ‘permitted-by-GR’, though?


Figure 2. The 'grandfather paradox' resulting from a closed time-like curve (Norton, 2007).

‘Physically Possible’


As discussed throughout the article, the argument set out for the physical possibility of time travel is as follows (Norton, 2007):


Premise 1. A universe is physically possible according to GR if it satisfies Einstein’s gravitational field equations in every region of spacetime.

Premise 2. There are time travel universes with closed time-like curves (i.e., the cylinder universe) that satisfy Einstein’s field equations and are physically possible as a result.

Premise 3. If a universe that is physically possible can allow for time travel to arise, then time travel is a physical possibility too.

Premise 4. Einstein’s theory of GR describes universes that allow for time travel to arise.


Conclusion. Therefore, time travel is a physical possibility according to GR.


The Grandfather Paradox objects to the above by asserting that time travel is inherently paradoxical (Maudlin, 1990). It is worth noting, however, that there is no contradiction if the bullet is never fired or if the attempt to kill the grandfather fails. Despite stating the obvious, the point to take away is that a global constraint can suffice as a response to the paradox. As this article discusses hereafter, both a global constraint in this (cylinder) universe and an adequate understanding of what it means for time travel to be physically possible—according to GR—may overcome the Grandfather Paradox. Essentially, this is to say that the paradox is only a serious issue when physically possible is conceptually misunderstood and since the paradox assumes that the attempt to kill the grandfather would always be successful. The global constraint says that the grandson cannot actually kill his grandfather (Lewis, 1976). Hence, the global structure of the time travel universe is quite different to our own. In the time travel universe, all processes must conform to these additional ‘global’ constraints so that the distant future and distant past ‘mesh’ when they meet (Norton, 2007). Unlike our own universe, the cosmos, (or any time-travel-free universe for that matter), global constraints ensure that any attempt to kill the grandfather fails. This event would never happen since processes must conform to the global constraints of the universe (Norton, 2007). The past and the future have to connect or ‘mesh’ to make sense in a time-travel universe as such.


Figure 3. Stephen Hawking's 'M theory' offers the possibility of travelling back in time (Kaiser, 2018).

The physical possibility of time travel according to GR is consistent with the said global constraints. This is therefore how physically possible time travel may be understood. The kind of time travel universe that is physically possible according to GR is indeed both physically possible and wholly different: time travel is a physical possibility in a universe that is vastly different to the cosmos. Crucially, this then must play a role in one’s understanding of physical possibility (according to GR). It is a possible universe since Einstein’s field equations are satisfied (Norton, 2007). As for the Grandfather Paradox, one must remember that there are global constraints and other dissimilar structural features. According to GR, the cylinder universe is physically possible thus making time travel achievable. The Grandfather Paradox is problematic when one assigns features of one’s own universe (i.e., the cosmos) to a time-travel possible universe. With a more adequate description of what it means for time travel to be physically possible according to GR (i.e., in an utterly different universe that permits time travel and contains closed time-like curves where a traveller simply by being travels in time), the Grandfather Paradox does not pose nearly as a serious threat.


Conclusion


The very possibility of a time travel universe enables time travel itself to become a physical possibility, as permitted by GR (where every region of spacetime connects to the next and Gμν = Tμν holds) (Maudlin, 1990). After providing some background on the locality of GR—albeit briefly—this article then went on to describe what time travel really refers to. Unlike what might be portrayed in Hollywood movies, a time traveller (merely by being) will travel in time via a closed time-like curve which forms a loop. By travelling along such a loop, the traveller will eventually end up at the same moment in time. Indeed, this is all physically possible in the Cylinder Universe according to GR. But what does it mean for time travel to be 'physically possible'? The analysis next considered the Grandfather Paradox, which arguably calls the very possibility of time travel into question by deriving a contradiction. The paradox, however, forgets that the time-travel universe is quite different to the cosmos. An adequate understanding of time travel being physically possible, this piece has argued, involves some recognition of the difference between our cosmos universe and one that permits time travel (i.e., the cylinder universe). The latter is entirely different, and this is part of what makes it physically possible. The very physical possibility of time travel acts like a kind of global constraint. Hence, the feature has drawn on the global constraint concept here, where all processes in a time travel universe conform to those constraints. Such a universe is entirely different to our own because of the physical possibility of time travel.


Bibliographical References

Arntzenius, F., & Maudlin, T. (2002). Time travel and modern physics. Royal Institute of Philosophy Supplements, 50, 169-200.


Carmeli, M. (2008). Relativity: Modern large-scale spacetime structure of the cosmos. World Scientific Publishing Company.


Collas, P. (1977). General relativity in two‐and three‐dimensional space–times. American Journal of Physics, 45(9), 833-837.


Effingham, N. (2020). Time Travel: Probability and Impossibility. Oxford University Press. doi:10.1093/oso/9780198842507.001.0001


Hobson, M. P., Efstathiou, G. P., & Lasenby, A. N. (2006). General relativity: an introduction for physicists. Cambridge University Press.


Kennedy, J. B. (2014). Space, time and Einstein: An introduction. Routledge.


Lewis, D. (1976). The Paradoxes of Time Travel. American Philosophical Quarterly, 13(2), 145–152.


Lovelock, D. (1971). The Einstein tensor and its generalizations. Journal of Mathematical Physics, 12(3), 498-501.


Matthews, M. R. (2013). Introduction: the history, purpose and content of the Springer international handbook of research in history, philosophy and science teaching. In International handbook of research in history, philosophy and science teaching (pp. 1-15). Dordrecht: Springer Netherlands.


Maudlin, T. (1990). Time-Travel and Topology. PSA: Proceedings Of The Biennial Meeting Of The Philosophy Of Science Association, 1990(1), 303-315. doi: 10.1086/psaprocbienmeetp.1990.1.192712


Norton, J. (2000). ’Nature is the Realisation of the Simplest Conceivable Mathematical Ideas’: Einstein and the Canon of Mathematical Simplicity. Studies in History and Philosophy of Modern Physics, 31B, 135-170.


Norton, J. (2007). General Relativity. Retrieved 15 March 2023, from https://sites.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/general_relativity/index.html


Norton, J. (2007). Time Travel Universes. Retrieved 14 March 2023, from https://sites.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/time_travel_universes/index.html

Weingard, R. (1979). General relativity and the conceivability of time travel. Philosophy of Science, 46(2), 328-332.

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