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Gödel on Time and RelativityDennis
DieksHistory and Foundations of ScienceUtrecht
University
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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.

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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

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Consequence of these postulates

• Simultaneity can no longer be an absolute concept!

For a moving observer the light does not arrive
simultaneously at the two clocks
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Newtons absolute space and time
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.
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• In Newtonian space-time there is a
• spatial distance between any pair
• of space-time points, plus a
• temporal distance

t2
Q
dr
dt
t1
P
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• Einstein started to use four-dimensional
  spacetime geometry after Minkowskis pioneering
  research (1908)

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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.

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In Minkowski spacetime there is one spacetime
distance between any pair of points (events)
Q
ds
ds2c2dt2-dx2-dy2-dz2
P
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Geometry of Minkowski spacetime
B
A
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The Minkowski distance defines geodesics (paths
of particles on which no forces work) and,
therefore, determines the background of the
dynamics
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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 S gµ? dxµ dx? is thus
  determined as a solution of these equations
• There is no a priori geometrical structure

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The Einstein Equations
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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
worlds and our own existence?
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Kurt Gödel (1949) An Example of a New Type of
Cosmological Solution of Einstein's Field
Equations of Gravitation. Rev. Mod. Phys. 21
447
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Normally, one would expect worldlines to exhibit
a linear temporal ordering
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t
x
In the Gödel universe there are closed worldlines!
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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

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A Remark About the Relationship Between
Relativity Theory and Idealistic Philosophy
In Albert Einstein Philosopher-Scientist,
1949, edited by P. Schilpp, pp 557-562
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Gödels 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

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Gödels 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 Gödels argument!

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Gödels 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!

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t
my birth
young grandfather
x
Causal paradoxes...
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Physics, and the world, wouldnt have been the
same without Gödel!