Two examples of circular motion for introductory courses in relativity

As is usual in discussions of the twin paradox at the introductory level, we pretend the Earth is an inertial frame and ignore its spin and motion around the Sun. Although the paradox should be discussed with one of the twins in an inertial frame rather than on Earth, it loses some of its dramatic appeal when presented in this way. We discuss how the Earth’s motion is taken into account when we consider the experiments of Hafele and Keating. (Ref. 4)

J.

Bailey

 et al., “,”  , – ();

J.

 

Bailey

 et al.,“,”  , – ().

J. J.

 

Hay

,

J. P.

 

Schiffer

,

T. E.

 

Cranshaw

, and

P. A.

 

Egelstaff

, “,”  , – ().

J. C.

Hafele

and

R. E.

Keating

, “,”  , – (); and

J. C.

Hafele

and

R. E.

Keating

, “,” , – ();

J. C.

 

Hafele

, “,”  , – ().

C. O.

Alley

, “,” in , edited by

P.

Meystre

and

M. O.

Scully

(, New York, ), pp. –;

W.

 

Sullivan

, “,”  , ().

A. P.

 

Lightman

,

W. H.

 

Press

,

R. H.

 

Price

, and

S. A.

 

Teukolsky

, (, Princeton, NJ, ) pp. 6,7, –.

M. B.

 

Cranor

,

E. M.

 

Heider

, and

R. H.

 

Price

, “,”  , – ().

See Ref. 7, p. .

L. H.

Thomas

, “,”  , ();

L. H.

 

Thomas

,“,”  , – (). The first article is a letter announcing Thomas’ result and the second presents a complete discussion of it. For a more complete set of references see Ref. 32.

J. R.

 

Taylor

, (, Sausalito, CA, ), p. , Problem 15.28.

J. P.

 

Costella

,

B. H. J.

 

McKellar

, and

A. A.

 

Rawlinson

, “,”  , – ().

G. E.

 

Uhlenbeck

, “,”  (), – ().

See ⟨http://einstein.stanford.edu/⟩and ⟨http://einstein.stanford.edu/content/press_releases/SU/pr-aps-041807.pdf⟩.

N. David

 

Mermin

, (, Princeton, NJ, ).

R.

 

Baierlein

, (, Cambridge, ).

For a quantitative discussion of the standard twin paradox see Appendix  B.

A surface on which the ratio of the circumference of a circle to its radius is less than $2π$ has positive curvature, like the surface of a sphere. The Earth provides a simple example of this situation that is easy to visualize. Imagine we are looking down on the Earth from the North Pole. To us, a circle formed by a latitude line above the equator has a radius equal to the length of the longitude line from the North Pole to the circle, which is $Rθ$⁠, where $R$ is the radius of the Earth and $θ$ is the angle the latitude line makes with the $z$ axis. In contrast, the actual radius of the circle is $Rsinθ$⁠. Consequently, to an observer looking down from the North Pole, the ratio of the circumference of a latitude line to its radius is $2πRsinθ∕(Rθ)<2π$⁠.

A.

Einstein

, (, Princeton, NJ, ), pp. –;

A.

 

Einstein

, (, New York, ), pp. –.

This is the experimental precision given in Y. Z. Zhang, (, Singapore, ), p. .

This experimental precision is given in W. Rindler, (, Oxford, ), 2nd ed., p. .

Reference 19, pp. –, and

Ref. 22, pp. – also discuss this point.

H. C.

 

Ohanian

, (, Lakeville, MN, ).

This graph is in Ref. 5, Fig. 47.

Just as the famous experiment with muons created in the upper atmosphere is shown in a film (Ref. 25), so too are film clips of the experiment of Alley et al. (Ref. 5) included in the BBC documentary “Einstein’s universe,” which can be obtained from Corinth Films (Ref. 26). Showing both films in class not only provides a welcome change of pace but also reinforces the reality of time dilation.

D. H.

Frisch

and

J. H.

Smith

, “,” Education Development Center, Newton, MA, .

See also  , – ().

“,” produced by the BBC and WGBH, 1979. Available from Corinth Films, 3117 Bursonville Rd., Riegelsville, PA 19077, ⟨www.store.corinthfilms.com⟩.

This precision is quoted in Ref. 28, p. .

D.

 

Newman

,

G. W.

 

Ford

,

A.

 

Rich

, and

E.

 

Sweetman

,  , – ().

In class we define a gyroscope as simply a mass spinning around an axis. We say that the key to the gyroscope is that its mounting can move in any direction without exerting a torque back on the axis about which the mass is spinning. To the degree that no torques act, angular momentum is conserved and the gyroscope points in a fixed direction no matter how its mounting moves.

This derivation is based on the discussion presented in the appendix of Ref. 31, which the author says is based on an idea of Purcell. The derivation is especially attractive because it only requires knowledge of length contraction, whereas most derivations of Thomas precession are much more complicated (see, for example, the references in Rhodes and Semon—Ref. 32).

Richard A.

 

Muller

, “,”  , – ().

John A.

 

Rhodes

and

Mark. D.

 

Semon

, “,”  , – ().

R. L.

 

Reese

, (, New York, ), pp. 138–140.

M. L.

 

Boas

, (, New York, ), pp. –.

G.

 

Pugh

, “,” 1959 Pentagon Weapons Evaluation Group (WSEG), memo one. This paper is reprinted in Ref. 37, pp. –, and is available at ⟨http://einstein.stanford.edu/content/sci_papers/papers/Pugh_G_1959_109.pdf⟩.

L. I.

Schiff

, “,”  , – ();

L. I.

Schiff

, “,”  , – ().

This paper is reprinted in Ref. 37, pp. – and at ⟨http://einstein.stanford.edu/content/sci_papers/papers/Schiff_LI_1960_110.pdf⟩.

, edited by

R.

 

Ruffini

and

C.

 

Sigismondi

(, Singapore, ).

J.

Lense

and

H.

Thirring

, “,”  , – ().

See also Ref. 37, pp. –, for an English translation of this paper.

B. R.

 

Holstein

, “,”  , – ().

Excellent discussions of the history, technology, and physics of Gravity Probe B can be found at ⟨http://einstein.stanford.edu⟩, ⟨www.nasa.gov/pdf/168808main_gp-b_pfar_cvr-pref-execsum.pdf⟩, ⟨http://books.nap.edu:80/html/gpb/summary.html⟩, and ⟨http://einstein.stanford.edu/highlights/GP-B_Launch_Companion.pdf⟩.

The actual orbit of the satellite has a perigee altitude of $639.5km$ above the Earth and an apogee altitude of $659.1km$⁠. We are using the average value $649.3km$⁠. Note that Francis Everitt, in his talk “Testing Einstein in Space: The Gravity Probe B Mission,” Stanford University, May 18, 2006 (⟨http://einstein.stanford.edu/ ⟩), used the value $642km$⁠.

S.

 

Tomonaga

, (, Chicago, ), Chaps. 2 and 11. The quote at the end of this paragraph is on pp. –.

Not all texts calculate the Lorentz transformation equations for acceleration. One that does is Rindler (Ref. 20), pp. –.

Richard A.

 

Muller

, “,”  , – ().

Reference 14, pp. –,

177, and 178.