
"On
the
Influence of Gravitation on the Propagation of Light"
Albert
Einstein (1911)
Translated
from
"Über den Einfluss der Schwercraft auf die Ausbreitung des Lichtes", in
Das Relativitätsprincip, 4th edition
Originally
from Annalen der Physik [35], 1911
IN
a memoir published four
years ago ^{[note
1]}
I
tried to answer the question whether the propagation of light
is influenced by gravitation. I return to this theme, because
my previous presentation of the subject does not satisfy me, and
for a stronger reason, because I now see that one of the most
important consequences of my former treatment is capable of being
tested experimentally. For it follows from the theory here to
be brought forward, that rays of light, passing close to the sun,
are deflected by its gravitational field, so that the angular
distance between the sun and a fixed star appearing near to it
is apparently increased by nearly a second of arc.
In the course of these reflexions further
results
are yielded which relate to gravitation. But as the exposition
of the entire group of considerations would be rather difficult
to follow, only a few quite elementary reflexions will be given
in the following pages, from which the reader will readily be
able to inform himself as to the suppositions of the theory and
its line of thought. The relations here deduced, even if the
theoretical
foundation is sound, are valid only to a first approximation.
1. A
Hypothesis as to the
Physical Nature of the Gravitational Field
IN
a
homogeneous gravitational
field
(acceleration of gravity γ) let there be a
stationary
system of
coordinates K,
orientated so that the lines of force of the gravitational field
run in the negative direction of the axis of z.
In a space free of gravitational fields let there be a second
system of coordinates K',
moving with uniform acceleration ( γ )
in the positive direction of its axis of z.
To avoid unnecessary complications, let us for the present disregard
the theory of relativity, and regard both systems from the customary
point of view of kinematics, and the movements occurring in them
from that of ordinary mechanics.
Relatively to K,
as well as relatively to K',
material points which are not subjected to the action of other
material points, move in keeping with the equations
d²x/dt²
= 0,
d²y/dt² = 0, d²z/dt² =
γ
For the accelerated
system K'
this follows directly from Galileo's
principle, but for the system K,
at rest in a homogeneous gravitational field, from the experience
that all bodies in such a field are equally and uniformly accelerated.
This experience, of the equal falling of all bodies in the
gravitational
field, is one of the most universal which the observation of nature
has yielded, but in spite of that the law has not found any place
in the foundations of our edifice of the physical universe.
But we arrive at a very
satisfactory interpretation
of this law of experience, if we assume that the systems K
and K' are
physically exactly equivalent, that is, if we assume that we may
just as well regard the system K
as being in a space free from gravitational fields, if we then
regard K
as uniformly accelerated. This assumption of exact physical equivalence
makes it impossible for us to speak of the absolute acceleration
of the system of reference, just as the usual theory of relativity
forbids us to talk of the absolute velocity of a system; ^{[note
2]}
and it makes the equal falling of all bodies in a gravitational
field seem a matter of course.
As long as we
restrict
ourselves to purely mechanical
processes in the realm where Newton's
mechanics holds sway, we are certain of the equivalence of the
systems K
and K'. But this view of ours will not
have any deeper significance
unless the systems K
and K' are
equivalent with respect to all physical processes, that is, unless
the laws of nature with respect to K
are in entire agreement with those with respect to K'.
By assuming this to be so, we arrive at a principle which, if
it is really true, has great heuristic importance. For by theoretical
consideration of processes which
take place relatively to a system of reference with uniform
acceleration,
we obtain information as to the career of processes in a homogeneous
gravitational field. We shall now show, first of all, from the
standpoint of the ordinary theory of relativity, what degree of
probability is inherent in our hypothesis.
2. On the
Gravitation of
Energy
ONE
result
yielded by the theory of relativity
is that the inertia
mass of a body increases with the energy it contains; if the increase
of energy amounts to E,
the increase in inertial mass is equal to E/c²,
when c
denotes the velocity of light.
Now is there an
increase of gravitating mass
corresponding
to this increase of inertia mass? If not, then a body would fall
in the same gravitational field with varying acceleration according
to the energy it contained. That highly satisfactory result of
the theory of relativity by which the law of the conservation
of mass is merged in the law of conservation of energy could not
be maintained, because it would compel us to abandon the law of
the conservation of mass in its old form for inertia mass, and
maintain it for gravitating mass.
But this must
be regarded as very improbable.
On
the other hand, the usual theory of relativity does not provide
us with any argument from which to infer that the weight of a
body depends on the energy contained in it. But we shall show
that our hypothesis of the equivalence of the systems K
and K' gives
us gravitation of energy as a necessary consequence.
Let
the two material systems S_{1}
and S_{2},
provided with instruments of measurement, be situated on the zaxis
of K at
the distance h
from each other, ^{[note
3]}
so that the gravitation
potential in S_{2}
is greater than that in S_{1}
by γh. Let a definite quantity
of energy E
be emitted from S_{2}
towards S_{1}.
Let the quantities of energy in S_{1}
and S_{2}
be measured by contrivances which – brought to one place in the
system z
and there compared – shall be perfectly alike. As to the process
of this conveyance of energy by radiation we can make no a
priori assertion because we do not know
the influence of the gravitational field on the radiation and
the measuring instruments in S_{1}
and S_{2}.
But
by our postulate of the equivalence of K
and K' we
are able, in place of the system K
in a homogeneous gravitational field, to set the gravitationfree
system K',
which moves with uniform acceleration in the direction of positive z,
and with the zaxis
of which the material systems S_{1}
and S_{2}
are rigidly connected.
We judge of the process of the transference
of energy
by radiation from S_{2}
to S_{1}
from a system K_{0},
which is to be free from acceleration. At the moment when the
radiation energy E_{2}
is emitted from S_{2}
toward S_{1},
let the velocity of K'
relatively to K_{0}
be zero. The radiation will arrive at S_{1}
when the time h/c
has elapsed (to a first approximation). But at this moment the
velocity of S_{1}
relatively to K_{0}
is γh/c
= v. Therefore by the ordinary theory of relativity
the
radiation arriving at S_{1}
does not possess the energy E_{2},
but a greater energy E_{1},
which is related to E_{2}
to a first approximation by the equation ^{[note 4]}
E_{1}
= E_{2 }(1 +
v/c) = E_{2 }(1 + γh/c²)

(1) 
By
our assumption exactly the same relation
holds if the same
process takes place in the system K,
which is not accelerated, but is provided with a gravitational
field. In this case we may replace γh
by the potential Φ
of the gravitation vector in S_{2},
if the arbitrary
constant of Φ
in S_{1}
is equated to zero.
We then have the equation
E_{1}
= E_{2}
+ E_{2}Φ/c²

(1a)

This equation
expresses the law of energy for
the
process under
observation. The energy E_{1}
arriving at S_{1}
is greater than the energy E_{2},
measured by the same means, which was emitted in S_{2},
the excess being the potential energy of the mass E_{2}/c²
in the gravitational field. It thus proves that for the fulfilment
of the principle of energy we have to ascribe to the energy E,
before its emission in S_{2}, a
potential energy due to
gravity,
which
corresponds to the gravitational mass E/c².
Our
assumption of the equivalence of K and K'
thus removes the difficulty mentioned at the beginning of this
paragraph which is left unsolved by the ordinary theory of relativity.
The meaning of this result is shown
particularly
clearly if we consider the following cycle of. operations: –
 The energy E,
as measured in S_{2}
, is emitted in the form of radiation in S_{2}
towards S_{1},
where, by the result just obtained, the
energy E( 1 + γh/c²
), as measured in S_{1},
is absorbed.
 A body W
of mass M
is lowered from S_{2}
to S_{1},
work Mγh
being done in the process.
 The energy E
is transferred from S_{1}
to the body W while W is in S_{1}.
Let
the
gravitational mass M
be thereby changed so that it acquires the value M'.
 Let W
be again raised to S_{2},
work M'γh
being done in the process.
 Let E
be
transferred from W
back to S_{2}.
The effect of this cycle is simply that S_{1}
has undergone the increase of energy E(1 + γh/c²
), and that the quantity of energy M'γh

Mγh
has been conveyed to the system in the form of mechanical work.
By the principle of energy, we must therefore have
Eγh/c²
= M'γh  Mγh
or
M'
 M = E_{2}
+ E/c²

(1b)

The increase in
gravitational mass is thus
equal to E/c²,
and therefore equal to the increase in inertia mass as given by
the theory of relativity.
The result emerges
still more directly from
the
equivalence
of the systems K
and K',
according to which the gravitational mass in respect of K
is exactly equal to the inertia mass in respect of K';
energy must therefore possess a gravitational mass which is equal
to its inertia mass. If a mass M_{0}
be suspended on a spring balance in the system K'
the balance will indicate the apparent weight M_{0}
γ
on account of the inertia of M_{0}.
If the quantity of energy E
be transferred to M_{0}, the
spring balance, by the law
of
the inertia
of energy, will indicate (M_{0} + E/c²)
γ.
By reason of our fundamental assumption exactly the same thing
must occur when the experiment is repeated in the system K,
that is, in the gravitational field.
3. Time and
the Velocity of
Light in the Gravitational Field
IF
the radiation emitted in the
uniformly
accelerated system K'
in S_{2}
toward S_{1}
had the frequency v_{2}
relatively to the clock
in S_{2},
then, relatively to S_{1} ,
at its arrival in S_{1} it no
longer has the frequency v_{2}
relatively to an
identical clock in S_{1},
but a greater frequency v_{1},
such that to a first approximation
ν_{1}
= ν_{2}
(1
+ γ h/c²)

(2) 
For
if we again introduce the unaccelerated system of
reference K_{0},
relatively to which, at the time of the emission of light, K'
has no velocity, then S_{1},
at the time of arrival of the radiation at S_{1},
has, relatively to K_{0}, the
velocity γh/c,
from which, by Doppler's
principle, the relation as given results immediately.
In
agreement with our assumption of the equivalence
of the systems K'
and K, this
equation also holds for the stationary system of coordinates K_{0},
provided with a uniform gravitational field, if in it the transference
by radiation takes place as described. It follows, then, that
a ray of light emitted in S_{2}
with a definite gravitational potential, and possessing at its
emission the frequency ν_{2}
– compared with a clock in S_{2}
– will, at its
arrival in S_{1}, possess a
different frequency ν_{1}
– measured by an identical clock in S_{1}.
For γh
we substitute the gravitational potential Φ
of S_{2} – that of S_{1}
being taken as zero – and assume that the relation which we have
deduced for the homogeneous gravitational field also holds for
other forms of field. Then
ν_{1}
= ν_{2}
(1 + Φ/c²)

(2a)

This
result (which by our deduction is valid to a first
approximation)
permits, in the first place, of the following application. Let v_{0}
be the vibrationnumber of an elementary lightgenerator, measured
by a delicate clock at the same place. Let us imagine them both
at a place on the surface of the Sun (where our S_{2}
is located). Of the light there emitted, a portion reaches the
Earth (S_{1}),
where we measure the frequency of the arriving light with a clock U
in all
respects resembling the one just mentioned. Then by (2a),
ν
= ν_{0}
(1 + Φ/c²)
where Φ
is the (negative) difference of gravitational potential between
the surface of the Sun and the Earth. Thus according to our view
the spectral lines of sunlight, as compared with the corresponding
spectral lines of terrestrial sources of light, must be somewhat
displaced toward the red, in fact by the relative amount
(ν_{0}
 ν)/ν_{0 }=  Φ/c²
= 2.10^{6}
If the conditions under which the solar
bands
arise
were exactly known, this shifting would be susceptible of measurement.
But as other influences (pressure, temperature) affect the position
of the centres of the spectral lines, it is difficult to discover
whether the inferred influence of the gravitational potential
really exists. ^{[note 5]}
On
a superficial consideration equation (2),
or (2a),
respectively, seems to
assert an absurdity. If there is constant transmission of light
from S_{2}
to S_{1},
how can any other number of periods per second arrive in S_{1}
than is emitted in S_{2} ? But
the answer is simple. We
cannot regard v_{2}
or respectively v_{1}
simply as frequencies (as the number of periods per second) since
we have not yet determined the time in system K.
What v_{2}
denotes is the number of periods with reference to the timeunit
of the clock U
in S_{2} , while v_{1}
denotes the number of periods per second with reference to the
identical clock in S_{1}.
Nothing compels us to assume that the clocks U in
different
gravitation potentials must be regarded as going
at the same rate. On the contrary, we must certainly define the
time in K
in such a way that the number of wave crests and troughs between S_{2}
and S_{1}
is independent of the absolute value of time: for the process
under observation is by nature a stationary one. If we did not
satisfy this condition, we should arrive at a definition of time
by the application of which time would merge explicitly into the
laws of nature, and this would certainly be unnatural and unpractical.
Therefore the two clocks in S_{1}
and S_{2}
do not both give the "time" correctly. If we measure
time in S_{1}
with the clock U,
then we must measure time in S_{2}
with a clock which goes 1 + Φ/c²
times more slowly than the clock U
when compared with U
at one and the same place. For when measured by such a clock the
frequency of the ray of light which is considered above is at
its emission in S_{2}
ν_{2}(1
+ Φ/c²)
and
is therefore, by (2a),
equal to the frequency v_{1}
of the same ray of light on its arrival in S_{1}.
This has a consequence which is of fundamental
importance
for our theory. For if we measure the velocity of light at different
places in the accelerated, gravitationfree system K',
employing
clocks U
of identical constitution we obtain the same magnitude at all
these places. The same holds good, by our fundamental assumption,
for the system K
as well. But from what has just been said we must use clocks of
unlike constitution for measuring time at places with differing
gravitation potential. For measuring time at a place which, relatively
to the origin of the coordinates, has the gravitation potential Φ,
we must employ a clock which – when removed to the origin of
coordinates – goes (1 + Φ/c²)
times more slowly
than the clock used for measuring time at
the origin of coordinates. If we call the velocity of light at
the origin of coordinates c_{0},
then the
velocity of
light c
at a place with the gravitation potential Φ
will be given by the relation
The
principle of the constancy of the velocity of light
holds
good according to this theory in a different form from that which
usually underlies the ordinary theory of relativity.
4.
Bending of
LightRays
in
the Gravitational Field
FROM
the proposition which has just
been
proved,
that the velocity
of light in the gravitational field is a function of the place,
we may easily infer, by means of Huyghens's
principle, that lightrays propagated across a gravitational field
undergo deflexion. For let E
be a wave front of a plane lightwave at the time t,
and let P_{1}
and P_{2}
be two points in that plane at
unit
distance from each
other. P_{1}
and P_{2}
lie in the plane of the paper, which is chosen so that the differential
coefficient of Φ,
taken in the direction of the normal to the plane, vanishes, and
therefore also that of c.
We obtain the corresponding wave front at time t
+ dt, or, rather,
its line of section with the plane of the paper, by describing
circles round the points P_{1}
and P_{2}
with radii c_{1} dt
and c_{2} dt
respectively, where c_{1}
and c_{2}
denote the velocity of light at the points P_{1}
and P_{2}
respectively, and by drawing the tangent to these circles. The
angle through which the lightray is deflected in the path cdt
is therefore
(c_{1}
 c_{2})dt
= (δc / δn')dt
,
if
we calculate the angle
positively when the ray
is bent toward the side of increasing n'.
The angle of deflexion per unit of path of the lightray is thus
 (1 / c)(δc
/ δn')
,
or by (3)
 (1 / c²)(δΦ / δn') .
Finally,
we obtain for the
deflexion which a lightray
experiences toward the side n'
on any path (s)
the expression

(4) 
We
might have obtained the
same result by directly
considering
the propagation of a ray of light in the uniformly accelerated
system K', and
transferring the result to the system K,
and thence to the case of a gravitational field of any form.
By
equation (4)
a
ray of light passing along by a heavenly body suffers a deflexion
to the side of the diminishing gravitational potential, that is,
on the side directed toward the heavenly body, of the magnitude
where
k
denotes the constant of gravitation, M
the mass of the heavenly body, Δ
the distance of the ray from the centre of the body. A ray of
light going past the Sun would accordingly undergo deflexion to
the amount of 4 * 10^{^6} =
0.83 seconds of arc. The angular distance
of the star from the centre of the Sun appears to be increased
by this amount. As the fixed stars in the parts of the sky near
the Sun are visible during total eclipses of the Sun, this consequence
of the theory may be compared with experience. With the planet Jupiter
the displacement to
be expected reaches to
about 1/100
of the amount given. It would be a most desirable thing if astronomers
would take up the question here raised. For apart from any theory
there is the question whether it is possible with the equipment
at present available to detect an influence of gravitational fields
on the propagation of light.
original
footnotes:
footnote 1 
A.
Einstein, Jahrbuch
für Radioakt. und Elektronik, 4, 1907. 
footnote 2 
Of
course
we cannot
replace any arbitrary gravitational field by a state of motion
of the system without a gravitational field, any more than, by
a transformation of relativity, we can transform all points of
a medium in any kind of motion to rest. 
footnote 3 
The
dimensions of S_{1}
and S_{2}
are regarded as infinitely small in comparison with h.

footnote
4 
See
A.
Einstein,
"Does the Inertia of a
Body depend on its EnergyContent?"
1905. 
footnote 5 
L.
F. Jewell (Journ.
de Phys., 6, 1897, p. 84) and
particularly Ch. Fabry and H.
Boisson (Comptes rendus, 148, 1909, pp. 688690)
have actually found such displacements of fine spectral lines
toward the red end of the spectrum, of the order of magnitude
here calculated, but have ascribed them to an effect of pressure
in the absorbing layer. 

