Gödel on Time and Relativity Dennis Dieks History and Foundations of Science Utrecht University

Gödel and Einstein in Princeton Gödel (b. 1906) arrived at the IAS in He became a permanent member in 1946, a full professor at the Institute in 1953 and an emeritus professor in Einstein (b. 1879) held a position at the IAS from 1933 until his death in 1955.

Einstein and Relativity 1905: Special Relativity (SR) : work on a generalization of SR that would include gravitation 1916: publication of the General Theory of Relativity Einstein at the IAS ( ): work on a further generalization, Unified Field Theory, that should supersede Quantum Theory

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Consequence of these postulates: Simultaneity can no longer be an absolute concept! v v v For a moving observer the light does not arrive simultaneously at the two clocks

Time. “Absolute, true, and mathematical time, of itself and from its own nature, flows equably without relation to anything external..” Space. “Absolute space, in its own nature, without relation to anything external, remains always similar and immovable.” Newton’s absolute space and time

In Newtonian space-time there is a spatial distance between any pair of space-time points, plus a temporal distance t2t2 t1t1 P Q dr dt

Einstein started to use four- dimensional spacetime geometry after Minkowski’s pioneering research (1908)

The spacetime structure of Special Relativity is different from that of Newtonian spacetime The relativity of simultaneity demonstrates that there is not one temporal distance between two events. For some observers there is no time difference: for some, P is earlier than Q; for others P is later than Q. Similarly, there is not one objective spatial difference: for some observers P and Q happen at the same spot, for others at different positions. Nevertheless, there is one spacetime distance that is the same for all observers.

In Minkowski spacetime there is one spacetime distance between any pair of points (events) P Q ds 2 =c 2 dt 2 -dx 2 -dy 2 -dz 2 ds

Geometry of Minkowski spacetime A B

The Minkowski distance defines geodesics (paths of particles on which no forces work) and, therefore, determines the background of the dynamics

In Minkowski spacetime all distances are fixed a priori. This “last remnant of absolute space and time” disappears in General Relativity. The geometrical relations become subject to dynamical equations, the “Einstein equations”; The quadratic form ds 2 = Σ g μν dx μ dx ν is thus determined as a solution of these equations; There is no a priori geometrical structure

The Einstein Equations

“Gödel was especially preoccupied by the nature of time, which, he told a friend, was the philosophical question. How could such a ‘mysterious and seemingly self-contradictory’ thing, he wondered, ‘form the basis of the world’s and our own existence’?”

Kurt Gödel (1949) An Example of a New Type of Cosmological Solution of Einstein's Field Equations of Gravitation. Rev. Mod. Phys. 21: 447

Normally, one would expect worldlines to exhibit a linear temporal ordering

x t In the Gödel universe there are closed worldlines!

Peculiarities of the Gödel universe Closed worldlines (closed time-like curves, CTCs) occur There is no global time function: it is not possible to slice up the Gödel spacetime into a sequence of spaces In other words, there is no succession of global “Nows” Matter is in rotational motion

“A Remark About the Relationship Between Relativity Theory and Idealistic Philosophy” In: Albert Einstein: Philosopher-Scientist, 1949, edited by P. Schilpp, pp

Gödel’s argument against the reality of time In the Gödel universe there is no global time function, and no globally consistent time order Therefore, in the Gödel universe there can be no objective “lapse of time” If time is conceptually different from space, in that it lapses, this should be an essential difference, present in all possible universes Therefore, the lapse of time is not real, not even in our universe Any impression of a flow must come from within us; it is “ideal”

Gödel’s argument is controversial.. Why should it be problematic that a global temporal ordering is something contingent? There are lots of things that are contingent but nevertheless real and not “ideal”! Perhaps our universe is objectively temporally ordered in a global way, and perhaps in our universe there is a real difference between space and time If there is no global ordering in some possible universes, a local notion of lapse of time may still represent an intrinsic, essential difference between space and time The notion of a flow of time has its problems anyway (what is the velocity of flow??), but these are independent of Gödel’s argument!

Gödel’s philosophy of time may be unconvincing, his universe is important! Directs attention to unexpected features of the Einstein equations Possibility of strange causal properties of spacetimes Possible non-existence of Cauchy hyperplanes Possibility of going back into the past; time travel!

x t Causal paradoxes... young grandfather my birth

Physics, and the world, wouldn’t have been the same without Gödel!