1. Simple, Interesting, and Unappreciated Facts about Relativistic Acceleration
John Mallinckrodt
Cal Poly Pomona
2003 AAPT Winter Meeting
Austin Texas

2. 2003 AAPT Winter Meeting, Austin
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.
15 January 2003
2003 AAPT Winter Meeting, Austin

3. 2003 AAPT Winter Meeting, Austin
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?)
15 January 2003
2003 AAPT Winter Meeting, Austin

4. 2003 AAPT Winter Meeting, Austin
Outline of Talk
Analyze a simple worldline that will turn out to be that of a point object undergoing constant proper acceleration.
Summarize its geometric characteristics on a spacetime diagram.
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.
Extend the analysis to continuous bodies.
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.
Consider the behavior of clocks on a rigidly, but otherwise arbitrarily moving object.
Watch a simulation of a rod that accelerates from rest to a maximum speed and decelerates back to rest.
Summarize main points.
15 January 2003
2003 AAPT Winter Meeting, Austin

5. 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 I’m lucky, they won’t 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).
15 January 2003
2003 AAPT Winter Meeting, Austin

6. 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
15 January 2003
2003 AAPT Winter Meeting, Austin

7. 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.
15 January 2003
2003 AAPT Winter Meeting, Austin

8. 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.
15 January 2003
2003 AAPT Winter Meeting, Austin

9. 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.
15 January 2003
2003 AAPT Winter Meeting, Austin

10. Generalization to Arbitrary Worldlines
The instantaneous position, velocity, and acceleration of an arbitrarily moving object associates it with a unique “instantaneous constant (proper) acceleration worldline.”
15 January 2003
2003 AAPT Winter Meeting, Austin

11. 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 A’s trajectory is time-reversed except for the fact that …
… that portion of A’s worldline is hidden behind B’s event horizon
15 January 2003
2003 AAPT Winter Meeting, Austin

12. 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 B’s clock runs faster in direct proportion to their respective vertex distances.
15 January 2003
2003 AAPT Winter Meeting, Austin

13. 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.
15 January 2003
2003 AAPT Winter Meeting, Austin

14. 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 can’t a rigid body straddle the vertex?
15 January 2003
2003 AAPT Winter Meeting, Austin

15. 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.”
15 January 2003
2003 AAPT Winter Meeting, Austin

16. 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 body’s 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.
15 January 2003
2003 AAPT Winter Meeting, Austin

17. 2003 AAPT Winter Meeting, Austin
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.
15 January 2003
2003 AAPT Winter Meeting, Austin

18. 2003 AAPT Winter Meeting, Austin
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.
15 January 2003
2003 AAPT Winter Meeting, Austin

19. Simple, Interesting, and Unappreciated Facts about Relativistic Acceleration
John Mallinckrodt
Cal Poly Pomona