Simple, Interesting, and Unappreciated Facts about Relativistic Acceleration

1
Simple, Interesting, and Unappreciated Facts
about Relativistic Acceleration

• John Mallinckrodt
• Cal Poly Pomona
• 2003 AAPT Winter Meeting
• Austin Texas

2
Question

• Does relativity allow an object to accelerate as
  a rigid body?
• Answer Yes. Simply apply an appropriate
  amount of force to every piece of the object.

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Followup

• If an object is accelerating as a rigid body, how
  does the acceleration of its front end compare
  to that of its rear end?
• a) Obviously, they would have to be the same so
  that all points move with the same speed at all
  times.
• b) Obviously, points nearer the front end
  would have to have smaller accelerations so that
  the object Lorentz contracts properly.
• c) Both of the above. (If not, why not?)

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Outline of Talk

1. Analyze a simple worldline that will turn out to
   be that of a point object undergoing constant
   proper acceleration.
2. Summarize its geometric characteristics on a
   spacetime diagram.
3. Consider pairs of separated particles that start
   from rest, maintain constant proper
   accelerations, and either a) have identical
   accelerations, or b) maintain their proper
   separation.
4. Extend the analysis to continuous bodies.
5. Find a constraint on the length of a rigidly
   accelerating object and understand its connection
   to the existence of an event horizon for rigidly
   accelerating reference frames.
6. Consider the behavior of clocks on a rigidly, but
   otherwise arbitrarily moving object.
7. Watch a simulation of a rod that accelerates from
   rest to a maximum speed and decelerates back to
   rest.
8. Summarize main points.

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Claims, Disclaims, and Acknowledgements

• I believe this material is accessible,
  surprising, uncontroversial, but nevertheless not
  well known.
• On the other hand, there are closely related
  questions that I feel less competent to discuss.

(If Im lucky, they wont come up.)

• A few (incomplete) references
• Hamilton (AJP, 46, 83)
• Desloge and Philpott (AJP, 55, 252)
• Desloge (AJP, 57, 598)
• Desloge (AJP, 57, 1121)
• Nikolic (AJP, 67, 1007)
• Taylor and Wheeler Spacetime Physics
• Mould, Basic Relativity (Especially Chapter 8).

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Classical Features of the Motion of Interest
Consider the motion described in an inertial
frame by In units where c 1 and with s
vertex distance constant.
Using nothing more than the definitions of v and
a it is easy to show that
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Relativistic Features of the Motion of Interest 1
How much does the moving object feel its speed
change when its speed observed in the inertial
frame changes from v to v dv? (Lorentz velocity
addition)
How much time elapses in the frame of the object
during a time dt in the inertial frame? (Time
dilation)
Thus, the proper (felt) acceleration is given by
That is, the proper acceleration is constant and
inversely proportional to the vertex distance.
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Relativistic Features of the Motion of Interest 2
How does the proper time change along the
trajectory?
With tP(v 0) 0, we can integrate to find
Note that this formula for the elapsed proper
time depends only on the velocity as measured in
the inertial frame (which is monotonically
changing) and the vertex distance and that it is
directly proportional to the vertex distance.
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Geometric Consequences

• The worldline for an object undergoing constant
  proper acceleration is a hyperbola that
  corresponds to the locus of all events having a
  constant spacelike separation from the origin.
  (x2 - t2 ?2 constant)

• The proper acceleration is simply the inverse of
  that separation.(ap 1/??

• As the object accelerates, the line of
  instantaneous simultaneity always passes through
  the origin. (dx/dt t/x)

• An event horizon exists and prevents any causal
  connection to events on the dark side of that
  horizon.

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Generalization toArbitrary Worldlines
The instantaneous position, velocity, and
acceleration of an arbitrarily moving object
associates it with a unique instantaneous
constant (proper) acceleration worldline.
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Two Objects with Identical Constant Acceleration

• Each object moves along a hyperbolic path having
  its own asymptotic light cone
• The separation is constant in the inertial frame
• A and B disagree on matters of simultaneity
• In fact, B might even say that As trajectory is
  time-reversed except for the fact that
• that portion of As worldline is hidden behind
  Bs event horizon

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Two Objects with Identical Asymptotic Light Cones

• Since the vertex distances are different, so are
  the accelerations
• The front object, B, has a smaller proper
  acceleration than the rear object, A
• A and B agree at all times on matters of
  simultaneity
• A and B agree at all times on their common
  velocity
• A and B agree that their proper separation is
  constant
• A and B agree that Bs clock runs faster in
  direct proportion to their respective vertex
  distances.

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A Rigidly Accelerating Rod that Flashes
Synchronously

• Consider the sequence of events in both the
  inertial and accelerating frames.
• Note that, even within the frame of the rod,
  flashing synchronously is not the same as
  flashing at a definite time interval because the
  clocks run at different rates.

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Extension to the Dark Side of the Event Horizon

• Consider a family of hyper-bolic worldlines
  sharing the same focus (vertex).
• Any spacelike line through the focus is a line
  of instan-taneous simultaneity and intersects
  all worldlines at positions of identical slope
  (velocity).
• Positive accelerations on the right, negative on
  the left.
• Why cant a rigid body straddle the vertex?

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Accelerating as a Rigid Body up to a Final Speed

• How do we get a Lorentz contracted rod in the
  inertial frame?
• Must accelerate to a uniform velocity in the
  inertial frame.
• Requires the rear end to stop accelerating before
  the front.
• Clocks are not synchronized in the moving frame.
• Note subsequent penetration of the former event
  horizon.

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Generalized Rigid Body Motion

• Under uniform acceleration the worldlines of the
  front and rear are identical but shifted. The
  body does not Lorentz contract.
• Under rigid body acceleration the bodys
  motion is arbitrary as long as the acceleration
  of the front end never exceeds 1/L.
• The worldline for the front end is always less
  curved (smaller a) than that of the rear end.
• Clocks return to synchronization whenever they
  return to the velocity at which they were
  synchronized.

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Simulation
Here is an Interactive Physics simulation of the
end points of a rigid body (of adjustable length)
whose rear end accelerates with constant proper
acceleration from rest to an (adjustable) maximum
velocity and then back to rest with the opposite
proper acceleration.
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Summary of Main Points

• Uniformly accelerating the various parts of a
  body increases their proper separations and
  develops tensile stresses within the body.
• Bodies can accelerate rigidly, but only by
  accelerating nonuniformly.
• The worldlines of different positions (as
  observed in an inertial frame) follow a family of
  hyperbolic curves sharing the same focus (vertex).

• The proper acceleration of a point a distance x
  behind the front end is given by
  
  thus, maximum length
  1/a(0)

• Rigidly accelerating frames of reference have
  stationary event horizons.

• Observers in a rigidly accelerated frame agree on
  almost everything except what time it is. Clock
  rates are proportional to vertex distances.
• When a rigid body returns to the velocity at
  which all of its clocks were synchronized, the
  clocks regain synchronization.
• I believe this material is accessible,
  surprising, uncontroversial, but nevertheless not
  well known.

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Simple, Interesting, and Unappreciated Facts
about Relativistic Acceleration

• John Mallinckrodt
• Cal Poly Pomona
• ajm_at_csupomona.edu
• http//www.csupomona.edu/ajm