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Free fall in gravitational theory
W. Engelhardt
a)
Fasaneriestrasse 8, D-80636 Mu¨nchen, Germany
(Received 17 January 2017; accepted 12 July 2017; published online 31 July 2017)
Abstract: Einstein’s explanation of Mercury’s perihelion motion has been verified by
astronomical observations. His formula could also be obtained in Schwarzschild metric and was
published already in 1898. Motion along a straight geodesic, however, namely, free fall into a
gravitational center with vanishing angular momentum, is incorrectly described both by Einstein’s
and by Schwarzschild’s equation of motion. A physical solution for free fall may be obtained by
taking into account the dependence of mass on velocity in Newton’s gravitational law as adopted
in the physics of accelerators. V
C2017 Physics Essays Publication.
[http://dx.doi.org/10.4006/0836-1398-30.3.294]
Re´sume´: Re´sume´: L’explication que donne Einstein sur le mouvement du pe´rihe´lie de Mercure a
e´te´ve´rifie´e graˆce a` des observations astronomiques. L’on pourrait e´galement obtenir en me´trique
de Schwarzschild sa formule qui fut de´ja` publie´e en 1898. Cependant, le mouvement tout au long
d’une ge´ode´sie en ligne droite, a` savoir en chute libre vers un centre gravitationnel sans moment
cine´tique, est de´crit de manie`re incorrecte dans les deux cas: par l’e´quation de mouvement
d’Einstein et par celle de Schwarzschild. Une solution physique pour la chute libre peut s’obtenir
en tenant compte du fait que la masse de´pend de la ve´locite´ dans la loi sur la gravitation de Newton
comme elle est adopte´e dans la physique des acce´le´rateurs.
Key words: General Relativity; Schwarzschild Metric; Perihelion Motion; Space-Time Geodesics.
I. INTRODUCTION
In 1915, Einstein published his famous paper on the
explanation of Mercury’s perihelion motion
1
as a first
application of his new gravitational theory. He arrived at the
identical formula for the advance of the perihelion which
Gerber had derived in 1898 (Ref. 2) on the basis of a velocity
dependent gravitational potential. The velocity dependence
chosen by Einstein was somewhat different, but in the
approximation considered his result agreed with Gerber’s.
Since then the accordance between the Gerber formula and
the astronomical observations on Mercury’s orbit is consid-
ered as a corner stone confirming Einstein’s geometrized the-
ory of gravitation (GR) that was published in 1916 (Ref. 3)
quoting the foregoing paper of 1915.
In this paper, we will show in Sec. II that Einstein
actually did not use his GR equation of motion, but used New-
ton’s equation with a slight modification of the gravitational
potential when he calculated Mercury’s anomalous orbit.
Thus, his theory suffers from the same deficiency as New-
ton’s, namely, free-falling mass points may attain superlumi-
nal velocities depending on their initial velocity at infinity.
This problem does not arise in the Schwarzschild solu-
tion of 1916 (Ref. 4) which we discuss in Sec. III. It results
in an equation of motion which is different from Einstein’s,
but it leads also to the Gerber formula. In two letters
to Einstein
5
and to Sommerfeld,
6
Schwarzschild called it a
“miracle” that his abstract idea leads to the same practical
result published by Gerber in 1898 and by Einstein in 1915.
If one applies, however, Schwarzschild’s equation of motion
to the free fall of a mass point with vanishing angular
momentum, one finds that the kinetic energy of the mass
point decreases and the particle comes to a halt at the
Schwarzschild radius. This solution avoids superluminality,
but it is not physical either.
In Sec. IV, we modify Newton’s equation of motion by
introducing the velocity dependent mass that proved appro-
priate in particle accelerators and prevents superluminality in
free fall. The particle’s kinetic energy increases unlimited
when it falls into the assumed singularity at r¼0. The conse-
quences for closed orbits are discussed. One obtains a perihe-
lion advance which is one third of the Gerber-Einstein
formula.
II. EINSTEIN’S PERIHELION FORMULA OF 1915
In Sec. II of his perihelion paper,
1,b)
Einstein derived the
equations of motion (7 E) of a mass point in the gravitational
field. In lowest order, the equations are identical with
Newton’s, but in higher order Einstein finds from his theory
the equations of motion
d2x
ds2¼a
2
x
r31þa
rþ2u23dr
ds
2
!
:(7b E)
a)
wolfgangw.engelhardt@t-online.de
b)
The title of this paper reads: Erkla¨ rung der Perihelbewegung des Merkur
aus der allgemeinen Relativita¨ tstheorie (Explanation of the Perihelion
Motion of Mercury from General Relativity Theory). As will be shown in
Sec. II and in the Appendix, this title is inappropriate, since Einstein’s deri-
vation of Gerber’s formula is flawed. His “explanation” is not valid.
ISSN 0836-1398 (Print); 2371-2236 (Online)/2017/30(3)/294/4/$25.00 V
C2017 Physics Essays Publication294
PHYSICS ESSAYS 30, 3 (2017)
Together with the exact conservation of angular momentum
r2d/
ds ¼B(10 E)
and the definition
u2¼dr
ds
2
þr2d/
ds
2
;(8a E)
one obtains from Eq. (7b E)
d2x
ds2¼a
2
x
r31þa
ru2þ3B2
r2
:
Einstein claims that this result could be written in the form
d2x
ds2¼1
2
@
@x
a
rþaB2
r3
¼x
2
a
r3þ3aB2
r5
;(7c E)
but this is not the case except under the condition
u2¼a
r:
It refers to vanishing total energy A¼0 corresponding to a
parabolic orbit. Einstein’s equation
dx
d/
2
¼2A
B2þa
B2xx2þax3;(11 E)
which he derived from his Eq. (7c E) refers, however, to a
closed orbit. It is not a consequence of his geometrized grav-
itational law, but follows from Newton’s equation of motion
with a modified velocity dependent potential
U¼a
2r1þv2
/
c2
!
:
When he calculated the perihelion advance from Eq. (11 E),
he miscalculated the last integral, but he reformulated his
final result
e¼24p3a2
T2c21e2
ðÞ
;(14 E)
such that it was identical with Gerber’s formula. It is difficult
to retrace why he transformed his direct and simpler result
e¼3pa
a1e2
ðÞ (13 E)
by introduction of the orbital period Tinto the form (14 E)
which was not known to him according to his own assertion.
For the case B¼0 which corresponds to free fall of a
mass point with vanishing angular momentum, Einstein’s
equation of motion (7c E) is identical with Newton’s and
suffers from the same deficiency. For an initial radial veloc-
ity at infinity very close to the velocity of light, one obtains
from the Newtonian energy equation
v2
r
c2¼v2
1
c2þa
r;(1)
resulting in superluminal velocities for cosmic radiation par-
ticles, for example. This is a result of canceling the mass in
Newton’s law so that the velocity dependence is ignored. We
come back to this problem in Sec. IV. Before, we discuss the
Schwarzschild solution and its consequence for planetary
motion.
III. PLANETARY MOTION IN SCHWARZSCHILD
METRIC
Einstein’s equation R
ik
¼0 in empty space is satisfied by
Schwarzschild’s line element
7
ds2¼1
Kdr2þr2dh2þr2sin2hd/2Kc2dt2;
K¼12GM
rc2¼1a
r;
(2)
when the gravitational field is produced by a point mass M,
where Gis Newton’s gravitational constant and ais twice
the so-called Schwarzschild radius. For motion in the plane
h¼p=2, one has to solve the following equations:
d2/
ds2þ2
r
dr
ds
d/
ds ¼0;(3)
with the integral
r2d/
ds ¼B
i;(4)
reflecting the conservation of angular momentum, and
d2t
ds2þa
K
1
r2
dr
ds
dt
ds ¼0;(5)
with the integral
dt
ds ¼C
icK :(6)
Conservation of energy follows from Eq. (2) in the form
1
K
dr
ds
2
þr2d/
ds
2
Kc2dt
ds
2
¼1:(7)
Substituting the integrals (4) and (6), one obtains with
dr
ds ¼dr
d/
d/
ds and r¼1=x:
Einstein’s equation (11 E) if one substitutes for the constant
C2¼1þ2A. Integrating this equation correctly one obtains
in lowest order of aGerber’s formula (14 E) as given by
Einstein.
It was this fortunate agreement that caused Schwarzschild
to speak of a miracle. If one calculates, however, the motion
along a straight geodesic in the Schwarzschild metric, namely,
free fall into a gravitational center with vanishing angular
Physics Essays 30, 3 (2017) 295
momentum, one obtains a result very different from Einstein’s.
Equation (7) becomes with dr=dt ¼vr
v2
r
c2¼K21K
C2
:(8)
The integration constant Cmay be expressed by the initial
velocity v2
1at infinity where K¼1 so that the radial veloc-
ity becomes
v2
r
c2¼1a
r
2a
r1v2
1
c2
þv2
1
c2
:(9)
As the mass point approaches the Schwarzschild radius, the
velocity tends to zero and the falling mass comes to a halt
with vanishing kinetic energy. This is not a physical solution,
since experience tells us that falling objects gain kinetic
energy continuously.
We must conclude that neither Einstein’s nor
Schwarzschild’s version of a GR-solution for the motion on
a straight geodesic is in agreement with known facts. It is
apparently necessary to take into account the velocity depen-
dent mass as it is applied in accelerator physics. This will be
considered in Sec. IV.
IV. FREE FALL WITH VELOCITY DEPENDENT MASS
Newton’s law with velocity dependent mass reads
d
dt m~
vðÞ¼mrU;m¼m0
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1v2=c2
p:(10)
Differentiation and scalar multiplication with ~
vyields
1
1v2=c2
dv2
dt ¼~
vr a
r
¼dr
dt
@
@r
a
r
¼d
dt
a
r
:(11)
Integrating with respect to time one obtains in Einstein’s
nomenclature
1v2
c2¼exp 2Aa
r
(12)
or
1v2
c2¼1v2
1
c2
exp a
r
;(13)
where the integration constant Ahas been expressed by the
initial velocity at infinity. Equation (13) ensures that the
radial velocity in free fall can never exceed the velocity of
light. On the other hand, the particle’s energy increases
indefinitely when the center of gravity r¼0 is approached
E2¼E2
1exp a=r
ðÞ
:
Solution (12) disagrees both with the Einstein-Newton solu-
tion (1) and with Schwarzschild’s result (9), but it is in
accordance with the velocity dependent mass increase as
established in accelerator physics.
At last we calculate the influence of the velocity depen-
dent mass on the perihelion motion for closed orbits with
A<0. From the conservation of angular momentum
follows:
v2
/
c2¼B2
r21v2
c2
¼B2
r2exp 2Aa
r
(14)
and from Eq. (12) we have
1
r2
dr
d/
2
þ1
!
v2
/
c2¼1exp 2Aa
r
:(15)
Eliminating the tangential velocity, we obtain for the orbit
with r¼1=x
dx
d/
2
þx2
!
B2¼exp 2AþaxðÞ1:(16)
Substituting the constants by the parameters of a Kepler
ellipse where pis the semilatus rectum and eis the eccentric-
ity, one obtains
dx
d/
2
¼2
apexp a
2
e21
pþ2x
1
x2:(17)
Expanding this for a1 yields
dx
d/¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x21e2
p2þ2x
pþa1e22pxðÞ
2
4p3
s:(18)
Integration over xbetween the zeros of dx=d/yields the
perihelion advance for a full period
e¼ap
p¼ap
a1e2
ðÞ
:(19)
This is one third of the Gerber-Einstein formula (13 E). The
same result was already obtained by Gerold von Gleich in
1925.
8,c)
One may wonder why Einstein ignored the variable
mass in Eq. (10) that was proposed by Planck in 1906.
9
It is
often referred to as “relativistic” mass which he certainly
would have been aware of in 1915.
V. CONCLUSIONS
Our analysis revealed that the problem of free fall into
a gravitational center is not adequately described in the
framework of General Relativity, be it Einstein’s or
Schwarzschild’s version of it. We also found that Einstein
actually did not use the equation of motion he had obtained
from his geometrized gravitational theory when he derived
the formula of Mercury’s perihelion advance that was pub-
lished by Gerber seventeen years before.
c)
A profound and learned review of Einstein’s relativity theories may be
found in: Gerold von Gleich, Einsteins Relativita¨ tstheorien und physikali-
sche Wirklichkeit, Johann Ambrosius Barth, Leipzig 1930. A reprint is in
preparation.
296 Physics Essays 30, 3 (2017)
In view of these findings, it is doubtful whether a geome-
trized gravitational theory—which is reminiscent of Kepler’s
laws—is capable of describing the dynamic phenomena due
to gravitational forces. In practice, only the center of gravity
of a planet could move on a geodesic line in space-time that
is practically coincident with a Kepler orbit, but all other
mass points are barred from this line by internal tensions.
Tidal forces can hardly be described by force-free motion in
space-time.
In equilibrium, it is obviously necessary to balance grav-
itational forces with elastic forces that keep objects on the
earth’s surface, for example. GR does not tell us how the
effects of space-time flexion connect to the everyday experi-
ence of countless forces. It took centuries to develop the
extremely useful and successful physical concept of forces.
One should not abandon it light-heartedly.
APPENDIX: ATTEMPT TO SOLVE EINSTEIN’S
EQUATION OF MOTION IN ORDER TO OBTAIN
GERBER’S FORMULA
Einstein’s use of the variable “s” is not unique in Ref. 1.
This may lead to confusion which, however, can be avoided
by elimination of this variable and attempting a direct deriva-
tion of his Eq. (11) from his equation of motion (7b).
Einstein finds up to second order
d2x
ds2¼a
2
x
r31þa
rþ2u23dr
ds
2
!
;(7b)
which yields with the definitions
u2¼dr
ds
2
þr2d/
ds
2
;(8a)
r2d/
ds ¼B;(10)
the equivalent equation
d2x
ds2¼a
2
x
r31þa
ru2þ3B2
r2
:(7b)
Scalar multiplication with the components dx=ds and sum-
mation yields
1
2
du2
ds ¼a
2
d
ds
1
r
1þa
ru2þ3B2
r2
:
With x¼1=rand elimination of s, one obtains a differential
equation of first order
du2
dx ¼a1þaxu2þ3B2x2
:(*)
From Eqs. (8a) and (10), we have u2=B2¼dx=d/ðÞ
2þx2so
that Einstein’s “solution” (11)
B2dx
d/
2
þx2
"#
¼2AþaxþaB2x3(11)
becomes
u2¼2AþaxþaB2x3:(**)
This expression does not solve the differential equation (*).
While Gerber’s formula may be calculated from
Eq. (11), this equation itself is not derivable from Einstein’s
equation of motion contrary to the claim in the title of Ein-
stein’s famous paper.
1
1
A. Einstein, Ko¨ niglich-Preussische Akademie der Wissenschaften, Sit-
zungsberichte 1915 (part 2), p. 831.
2
P. Gerber, Z. Math. Phys. 43, 93 (1898).
3
A. Einstein, Ann. Phys. (Ser. 4) 49, 769 (1916).
4
K. Schwarzschild, Ko¨ niglich-Preussische Akademie der Wissenschaften,
Sitzung vom 3. Februar 1916, p. 189.
5
K. Schwarzschild, “Letter to Einstein,” The collected papers of
Albert Einstein, Vol. 8, Part A: Correspondence 1914–1917, p. 224,
Document 169.
6
K. Schwarzschild, “Letter to Sommerfeld,” Selbstzeugnisse großer Wissen-
schaftler, Kultur & Technik, Heft 4, 1987, Verlag Deutsches Museum,
Mu¨ nchen.
7
H. Bucerius and M. Schneider, Himmelsmechanik II (Bibliographisches
Institut, Mannheim, 1967).
8
G. von Gleich, Ann. Phys. Bd. 383, 498 (1925).
9
M. Planck, Verhandl. Deutsch. Phys. Gesellschaft 8, 136 (1906).
Physics Essays 30, 3 (2017) 297
