The Adventures of the Rocketeer:

1
The Adventures of the Rocketeer

• Accelerated Travel in an Expanding Universe
• Juliana Kwan
• Supervisor Geraint Lewis
• Honours Talk, 2007

Image credit STS-41B, NASA
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Introduction

• An accelerated observer feels non-inertial forces
  from a source other than gravity e.g. thrust from
  a rocket.
• We have set up the equations of motion to include
  acceleration in various spacetimes and then
  solved them for accelerated trajectories.
• We have considered two main situations requiring
  a rocket
• Falling into a black hole Can accelerating
  improve longevity?
• Travelling across the universe How far can we go
  in a human lifespan?

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Introduction

• The acceleration four-vector is defined as
• in a general coordinate system, where
  is the four-velocity and are
  Christoffel symbols.
• Four coupled second order differential equations
  to solve for path of rocketeer
• But we can integrate these numerically.
  

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Introduction

• Equivalently (Rindler, 1965)
• where
• La and Ma are orthogonal unit tensors that have
  the same orientation and magnitude along the path
  of the rocketeer.
• Can solve the above as a coupled set of
  differential equations for the path of the
  rocketeer.

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Falling into a Black Hole

• Since proper time is maximised along a geodesic,
  you will live the longest if you dont struggle,
  but just relax as you approach the singularity
  (Carroll, Spacetime and Geometry)
• Simplest metric that describes external space
  time around a spherically symmetric, non-rotating
  massive body is in Schwarzschild coordinates
• A singularity at r 2M means that we need to
  change to Eddington-Finkelstein coordinates

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Killing Vectors

• A Killing vector points in the direction of a
  symmetry in the metric.
• Each Killing vector is also associated with a
  conserved quantity that is calculated by taking
  its dot product with .
• Both metrics are invariant in t, leading to a
  Killing vector
• xa (1,0,0,0) and a conserved quantity, e,
  that is constant along a geodesic
• where r0 is the initial radius from which the
  rocketeer falls and the last equality only holds
  for a zero initial radial velocity.
• Because e arises from time symmetry, it is
  related to conservation of energy

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Maximising Proper Time

• Radial coordinate against proper time (solid
  lines) shows that a geodesic (black) does not
  necessarily experience the longest proper time.
• Maximum proper time occurs for rocketeer with e
  0. This is already satisfied by free falling from
  r 2M.
• However, e is non-zero if rocketeer falls from
  elsewhere. The rocketeer needs to accelerate on
  to a path with e 0 to maximise proper time.

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Maximising Proper Time

• A geodesic only maximises the proper time for a
  set paths between two fixed endpoints.
• Thus, determining which path is best depends on
  the
• initial conditions of the rocketeer.
• For further details, see Lewis and Kwan, 2007

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Travel in an expanding universe

• How far a rocketeer can travel in an expanding
  universe?
• Describe the universe using the
  Friedmann-Robertson-Walker metric
• in comoving coordinates, where k is the
  curvature and a(t) is the scale factor.
• An expression for a(t) comes from solving the
  following equations
• Geometrised units, c G 1. Also, we have set
  H0 1, so unit for length and time is 1/H0. In
  these units, t 1 1/H0 13.6 x 109 years and
  a 1g 1.46 x 1010 H0.

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Conformal Coordinates
Doing a conformal transformation converts a t-r
spacetime diagram to

• Convenient to perform a conformal transformation,
  by using
• where h is the conformal time.
• Conformal time is finite for accelerated
  expansion because
• quickly as .
• Light on radial paths will now
• travel at 45.
• Timelike paths remain timelike
• after a conformal transformation.

... an h-r spacetime diagram.
Davis and Lineweaver (2004)
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(No Transcript)
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Results

• Path of rocketeer has slope 45 in conformal
  diagram
• 4680 Mpc (99 of maximum distance) is travelled
  in 30 years with a 1g.

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• Matter dominated universe ? expansion is
  decelerating ? conformal time is unbounded.
• The rocketeer overshoots because the radial
  component of is greater in a decelerating
  universe so the rocketeer is travelling faster on
  the return journey.

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Squiggly Plots

• Expansion is accelerating conformal time is
  finite.
• When the conformal time saturates, the rocketeer
  stops on a comoving radius.
• Radial component of acceleration becomes small
  compared to time component

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Coaster vs. Rocketeer

• What happens if our rocket cant sustain a
  constant acceleration how far would a coaster
  get compared to a rocketeer?
• Rocketeer in green continually accelerates while
  a coaster in red only has an initial
  acceleration then coasts on a constant velocity.
• Paths are similar in conformal representation,
  both reach a similar radial comoving coordinate
  but the rocketeer does so in a much shorter
  proper time.

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• As in the previous slide, but this universe is
  accelerating.
• The coaster and the rocketeer end up at similar
  comoving distances and both settle on a
  particular coordinate.
• The difference in time dilation between the
  rocketeer and the coaster is less pronounced in
  this universe.

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Conclusion

• Below the Schwarzschild event horizon, the
  maximum proper time is measured on a path with e
  0. For observers not falling from the event
  horizon at rest, they will need to accelerate on
  this path.
• Maximum distance travelled is 4685 Mpc but this
  takes forever. However, we can travel 99 of the
  way in about 30 years.
• Return journeys overshoot in a matter dominated
  universe and undershoot in a vacuum dominated
  universe.
• Constant accelerated motion is always better in
  terms of minimising proper time but this becomes
  less noticeable for universes whose expansion is
  accelerating.