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

Gödel on Time and Relativity

Dennis DieksHistory and Foundations of Science

Utrecht University

Gödel and Einsteinin Princeton

• Gödel (b. 1906) arrived at the IAS in 1940. He became a permanent member in 1946, a full professor at the Institute in 1953 and an emeritus professor in 1976.

• Einstein (b. 1879) held a position at the IAS from 1933 until his death in 1955.

Einstein and Relativity

• 1905: Special Relativity (SR)

• 1905-1915: work on a generalization of SR that would include gravitation

• 1916: publication of the General Theory of Relativity

• Einstein at the IAS (1933-1955): work on a further generalization, Unified Field Theory, that should supersede Quantum Theory

..

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

t2

t1 P

Qdr

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

ds2=c2dt2-dx2-dy2-dz2

ds

Geometry of Minkowski spacetime

A B

The Minkowski distance definesgeodesics (paths of particles on which no forces work)and, therefore, determines the background ofthe 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 ds2 = Σ gμν dxμ dxν is thus determined as a solution of these equations;

• There is no a priori geometrical structure

The Einstein Equations

1

2R g R T

“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 557-562

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 beenthe same without

Gödel!