Consider the fall of the apple that, by legend, hit Newton on the head. Instead of using this to introduce Newton’s law of gravity, we will discuss the point of view that the fall of the apple was caused by a minuscule curvature of spacetime in the region surrounding Newton and the apple tree. We will calculate this curvature using concepts and equations entirely appropriate for students in an introductory physics course. The result introduces the idea that if gravity curves spacetime, perhaps curved spacetime can ripple in such a way as to create a gravitational wave that tells us what happened to two black holes 1.3 billion years ago.

1.
For Wheeler’s description of general relativity, see p. G11 of
Edwin F.
Taylor
and
John Archibald
Wheeler
,
Exploring Black Holes: An Introduction to General Relativity
(
Addison-Wesley
,
Longman
,
2000
). This book brings the subject of black holes alive, with only some use of four vectors.
2.
We have included the downloadable Appendix A17 to emphasize that the result follows from the mathematics found in an introductory calculus course. The derivation is from the beginning of Chap. 2 of
E.
Huggins
,
Calculus2000
, freely available at Physics2000.com.
3.
See
Gordon L.
Aubrecht
,
Anthony P.
French
, and
Mario
Iona
, “
About the International System of Units (SI): Introduction and bibliography
,”
Phys. Teach.
49
,
473
(
Dec.
2011
).
4.
Charles E.
Misner
,
Kip G.
Thorne
, and
John A.
Wheeler
,
Gravitation
(
W. H. Freeman and Company
,
New York
1973
). In Gravitation on p. 33, the authors calculate the same nearly 1 l-y radius of spacetime curvature for a ball and a bullet that go up and come back down 10 m away. Their purpose is to introduce a general approach to motion in four or more dimensional curved space, which leads to a discussion of the Riemann curvature tensor on their next page (34). In contrast, our purpose in this paper is to find an example simple enough that we can introduce the idea of curved spacetime in an introductory physics course. Because smooth curvature is easy to describe in two dimensions, but can become messy in more dimensions, we limited our discussion of the motion of stones and photons to just the (y-ct) plane.
5.
A text that begins at the level of our paper (also discussing the Misner, Thorne, Wheeler derivation on its pp. 3-9), but then leads students through the standard general relativity topics, is
Thomas
Moore
,
A General Relativity Workbook
(
University Science Books
,
Mill Valley, CA
2013
). This text has been able to lead junior and senior undergraduates through tensor calculus to the traditional view of general relativity.
6.
For a more complete discussion of curvature in a plane, Google “curvature and acceleration” then select the Wikipedia article. The important result is κ = y″ / (1+y2)3/2, where κ = 1/R, y = y(x), y ′ = dy/dx, and y″ = d2y/dx2. Setting y ′ = 0 gives κ = y″, and we immediately see that y′ must begin to approach 1 before it has an effect on the curvature.
7.
R. V.
Pound
, and
G. A.
Rebka
 Jr.
, “
Apparent weight of photons
,”
Phys. Rev. Lett.
4
(
7
),
337
341
(
April
1,
1960
).
8.
The photographs of Pound and Rebka are stock photographs U1224764A and U1224764 © Bettmann/CORBIS.
9.
For a discussion of the Mössbauer effect, we recommend Googling “Mossbauer effect–Wikipedia.”
10.
For a discussion of the Doppler effect, we recommend Googling “ Doppler effect–Wikipedia.”
11.
Elisha
Huggins
, “
Special relativity in week one: 4) Lack of simultaneity
,”
Phys. Teach.
49
,
340
342
(
Sept.
2011
).
12.
Ira Mark
Egdall
, “
Teaching special relativity to lay students
,”
Phys. Teach.
52
,
406
409
(
Oct.
2014
). We especially appreciated the diagrams at the top of p. 408 on the lack of simultaneity.
13.
Figure 7 was inspired by the 1970 Ealing film loops on the electric field of moving charges. Unfortunately, these film loops appear to exist only in some physics demo centers. These films were inspired by the diagrams of the fields of moving charges on pp. 162–165 of the 1963 text
Edward M.
Purcell
,
Electricity and Magnetism Berkeley Physics Course
, Vol.
II
(
McGraw-Hill
).
14.
The expansion of both the electric and magnetic fields is illustrated on p. 32–29 of E. Huggins, Physics2000 Part 2. (Physics2000.com and ISBN 9780970783622.)
15.
LIGO stands for the Laser Interferometer Gravitational-Wave Observatory, which first observed gravitational waves on Sept. 14, 2015. A second observation was made in June 2016. For details we searched “Event GW 150914 LIGO Open Science Center” and “GW 150914: Factsheet.”
16.
For a clear discussion of the LIGO experiment, see
Don
Lincoln
and
Amber
Stuver
, “
Ripples in reality
,”
Phys. Teach.
54
,
398
403
(
Oct.
2016
).
17.
Readers can download the appendices at TPT Online, https://doi.org/10.1119/1.5080570 under the Supplemental tab.
18.
Brian
Greene
,
The Elegant Universe
(
Vintage Books
,
New York
,
2003
).
19.
Richard
Wolfson
,
Simply Einstein
(
W. W. Norton & Company
,
New York
,
2003
).

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