We demonstrate that the de Broglie wave of a particle in a gravitational field turns toward the region of lower gravitational potential, causing the particle to fall. This turning is caused by clocks running slower in the smaller potential. We use the analogy of ocean waves that are slower in shallower water and turn toward beaches. This approach implies that the motion is along a geodesic and explains the free fall qualitatively and quantitatively with only elementary algebra.
REFERENCES
1.
R. R.
Gould
, “
Why does a ball fall?: A new visualization for Einstein's model of gravity
,” Am. J. Phys.
84
, 396
(2016
).2.
3.
K.
Rȩbilas
, “
Comment on ‘Why does a ball fall?: A new visualization for Einstein's model of gravity’ [Am. J. Phys. 84, 396–402 (2016)]
,” Am. J. Phys.
85
, 66
–67
(2017
).4.
D.
Dieks
and
G.
Nienhuis
, “
Relativistic aspects of nonrelativistic quantum mechanics
,” Am. J. Phys.
58
, 650
–655
(1990
).5.
B.
Houchmandzadeh
, “
The Hamilton–Jacobi equation: An alternative approach
,” Am. J. Phys.
88
, 353
–359
(2020
).6.
W.
Stannard
,
D.
Blair
,
M.
Zadnik
, and
T.
Kaur
, “
Why did the apple fall? A new model to explain Einstein's gravity
,” Eur. J. Phys.
38
, 015603
(2016
).7.
8.
M.
Takamoto
,
I.
Ushijima
,
N.
Ohmae
,
T.
Yahagi
,
K.
Kokado
,
H.
Shinkai
, and
H.
Katori
, “
Test of general relativity by a pair of transportable optical lattice clocks
,” Nat. Photonics
14
, 411
–415
(2020
).9.
J. C.
Hafele
and
R. E.
Keating
, “
Around-the-world atomic clocks: Predicted relativistic time gains
,” Science
177
, 166
–168
(1972
).10.
J. C.
Hafele
and
R. E.
Keating
, “
Around-the-world atomic clocks: Observed relativistic time gains
,” Science
177
, 168
–170
(1972
).11.
R. F. C.
Vessot
,
M. W.
Levine
,
E. M.
Mattison
,
E. L.
Blomberg
,
T. E.
Hoffman
,
G. U.
Nystrom
,
B. F.
Farrel
,
R.
Decher
,
P. B.
Eby
,
C. R.
Baugher
,
J. W.
Watts
,
D. L.
Teuber
, and
F. D.
Wills
, “
Test of relativistic gravitation with a space-borne hydrogen maser
,” Phys. Rev. Lett.
45
, 2081
–2084
(1980
).12.
P.
Delva
,
N.
Puchades
,
E.
Schönemann
,
F.
Dilssner
,
C.
Courde
,
S.
Bertone
,
F.
Gonzalez
,
A.
Hees
,
C.
Le Poncin-Lafitte
,
F.
Meynadier
et al, “
Gravitational redshift test using eccentric Galileo satellites
,” Phys. Rev. Lett.
121
, 231101
(2018
).13.
S.
Herrmann
,
F.
Finke
,
M.
Lülf
,
O.
Kichakova
,
D.
Puetzfeld
,
D.
Knickmann
,
M.
List
,
B.
Rievers
,
G.
Giorgi
,
C.
Günther
et al, “
Test of the gravitational redshift with Galileo satellites in an eccentric orbit
,” Phys. Rev. Lett.
121
, 231102
(2018
).14.
T.
Van Baak
, “
An adventure in relative time-keeping
,” Phys. Today
60
(3
), 16
(2007
); see also <http://leapsecond.com/great2005>.15.
J. M.
Espinosa
, “
Physical properties of de Broglie's phase waves
,” Am. J. Phys.
50
, 357
–362
(1982
).16.
D.
Morin
, Introduction to Classical Mechanics: With Problems and Solutions
(
Cambridge U. P
.,
Cambridge
, 2008
).17.
C. E.
Mungan
, “
Relativistic effects on clocks aboard GPS satellites
,” Phys. Teach.
44
, 424
–425
(2006
).18.
N.
Ashby
, “
Relativity in the global positioning system
,” Living Rev. Relativity
6
, 1
(2003
).19.
We thank Arkady Vainshtein for this interpretation.
20.
The Collected Papers of Albert Einstein. Vol. 2: The Swiss Years: Writings, 1900–1909 (English Translation)
, edited by
J.
Stachel
,
D. C.
Cassidy
,
J.
Renn
, and
R.
Schulmann
(
Princeton University Press
,
New Jersey
, 1989
).21.
C. M.
Will
, “
Was Einstein right?: Testing relativity at the centenary
,” Ann. Phys.
15
, 19
–33
(2005
).22.
A.
Sen
,
S.
Dhasmana
, and
Z. K.
Silagadze
, “
Free fall in KvN mechanics and Einstein's principle of equivalence
,” Ann. Phys.
422
, 168302
(2020
).23.
24.
J.
Evans
,
K. K.
Nandi
, and
A.
Islam
, “
The optical–mechanical analogy in general relativity: New methods for the paths of light and of the planets
,” Am. J. Phys.
64
, 1404
–1415
(1996
).25.
J.
Evans
,
P. M.
Alsing
,
S.
Giorgetti
, and
K. K.
Nandi
, “
Matter waves in a gravitational field: An index of refraction for massive particles in general relativity
,” Am. J. Phys.
69
, 1103
–1110
(2001
).26.
V. V.
Nesvizhevsky
et al, “
Quantum states of neutrons in the Earth's gravitational field
,” Nature
415
, 297
–299
(2002
).27.
V.
Nesvizhevsky
and
A.
Voronin
, Surprising Quantum Bounces
(
Imperial College Press
,
London
, 2015
).© 2021 Author(s). Published under an exclusive license by American Association of Physics Teachers.
2021
Author(s)
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