We show by direct calculation that the common equivalence principle explanation for why gravity must deflect light is quantitatively incorrect by a factor of three in Schwarzschild geometry. It is, therefore, possible, at least as a matter of principle, to tell the difference between local acceleration and a true gravitational field by measuring the local deflection of light. We calculate as well the deflection of test particles of arbitrary energy and construct a leading-order coordinate transformation from Schwarzschild to local inertial coordinates, which shows explicitly how the effects of spatial curvature manifest locally for relativistic trajectories of both finite and vanishing rest mass particles.
REFERENCES
1.
J. P.
Hartle
, Gravity: An Introduction to Einstein's General Relativity
( Pearson
, London
, 2003
); (Fig. 6.5, p. 114
).2.
L.
Susskind
and
A.
Cabannes
, General Relativity
( Allen Lane
, Dublin
, 2023
), pp. 16
–17
.3.
E.
Di Casola
,
S.
Liberati
, and
S.
Sonego
, “ Nonequivalence of equivalence principles
,” Am. J. Phys.
83
(1
), 39
–46
(2015
).4.
5.
M. P.
Hobson
,
G.
Efstathiou
, and
A. N.
Lasenby
, General Relativity: An Introduction for Physicists
( Cambridge U. P
., Cambridge
, 2006
).6.
C. W.
Misner
,
K. S.
Thorne
, and
J. A.
Wheeler
, Gravitation
, 2nd ed. ( Princeton U. P
., Princeton
, 2017
).7.
W.
Moreau
,
C.
Purdie
, and
J.
Wood
, “ The rate of deflection of light in an accelerated frame and a gravitational field
,” Am. J. Phys.
67
(3
), 241
–251
(1999
).8.
W.
Moreau
,
R.
Neutze
, and
D. K.
Ross
, “ The equivalence principle in the Schwarzschild geometry
,” Am. J. Phys.
62
(11
), 1037
–1040
(1994
).9.
A.
Einstein
, “ Über den Einfluss der Schwerkraft auf die Ausbreitung des Lichtes
,” Ann. Phys.
35
, 898
–908
(1911
); translated “On the influence of gravitation on the propagation of light,” in The Principle of Relativity (Dover, New York, 1952).10.
11.
A.
Romano
and
M.
Mango Furnari
, The Physical and Mathematical Foundations of the Theory of Relativity, A Critical Analysis
( Birkhäuser
, Cham
, 2019
).© 2024 Author(s). Published under an exclusive license by American Association of Physics Teachers.
2024
Author(s)
AAPT members receive access to the American Journal of Physics and The Physics Teacher as a member benefit. To learn more about this member benefit and becoming an AAPT member, visit the Joining AAPT page.
