arXiv:0808.1081v2[physics.pop-ph]14Apr2009
The kinematic origin of the cosmological redshift
Emory F. Bunn∗
Department of Physics, University of Richmond, Richmond, Virginia 23173
David W. Hogg
Center for Cosmology and Particle Physics and Department of Physics,
New York University, 4 Washington Place, New York, New York 10003
A common belief about big-bang cosmology is that the cosmological redshift cannot be properly
viewed as a Doppler shift (that is, as evidence for a recession velocity), but must be viewed in terms
of the stretching of space. We argue that, contrary to this view, the most natural interpretation of
the redshift is as a Doppler shift, or rather as the accumulation of many infinitesimal Doppler shifts.
The stretching-of-space interpretation obscures a central idea of relativity, namely that it is always
valid to choose a coordinate system that is locally Minkowskian. We show that an observed frequency
shift in any spacetime can be interpreted either as a kinematic (Doppler) shift or a gravitational
shift by imagining a suitable family of observers along the photon’s path. In the context of the
expanding universe the kinematic interpretation corresponds to a family of comoving observers and
hence is more natural.
I. INTRODUCTION
Many descriptions of big-bang cosmology declare that the observed redshift of distant galaxies is not a Doppler
shift but is due to the “stretching of space.” The purpose of this paper is to examine the meaning of such statements
and to assess their validity. We wish to make clear at the outset that we are not suggesting any doubt about
either the observations or the general-relativistic equations that successfully explain them. Rather, our focus is on the
interpretation: given that a photon does not arrive at the observer conveniently labeled “Doppler shift,” “gravitational
shift,” or “stretching of space,” when can or should we apply these labels?
Arguably an enlightened cosmologist never asks this question. In the curved spacetime of general relativity, there is
no unique way to compare vectors at widely separated spacetime points, and hence the notion of the relative velocity
of a distant galaxy is almost meaningless. Indeed, the inability to compare vectors at different points is the definition
of a curved spacetime.1,2,3,4
In practice, however, the enlightened view is far from universal. The view presented
by many cosmologists and astrophysicists, particularly when talking to nonspecialists, is that distant galaxies are
“really” at rest, and that the observed redshift is a consequence of some sort of “stretching of space,” which is distinct
from the usual kinematic Doppler shift. In these descriptions, statements that are artifacts of a particular coordinate
system are presented as if they were statements about the universe, resulting in misunderstandings about the nature
of spacetime in relativity.
In this paper we will show that the redshifts of distant objects in the expanding universe may be viewed as kinematic
shifts due to relative velocities, and we will argue that if we are forced to interpret the redshift, this interpretation is
more natural than any other.
We begin with examples of the description of the cosmological redshift in the first three introductory astronomy
textbooks chosen at random from the bookshelf of one of the authors.
• The cosmological redshift “is not the same as a Doppler shift. Doppler shifts are caused by an object’s motion
through space, whereas a cosmological redshift is caused by the expansion of space.”5
(Emphasis in original.)
• “A more accurate view [than the Doppler effect] of the redshifts of galaxies is that the waves are stretched by
the stretching of space they travel through . . . If space is stretching during all the time the light is traveling,
the light waves will be stretched as well.”6
• “Astronomers often express redshifts as if they were radial velocities, but the redshifts of the galaxies are not
Doppler shifts . . . Einstein’s relativistic Doppler formula applies to motion through space, so it does not apply
to the recession of the galaxies.”7
More advanced textbooks often avoid this language. For instance, the books by Peacock8
and Linder9
give particularly
careful and clear descriptions of the nature of the cosmological redshift. However, statements similar to those
we have cited can be found even in some advanced textbooks. For example, a leading advanced undergraduate level
text states that Doppler shifts “are produced by peculiar and not by recession velocities.”10
In this paper we argue,
as others have before us,11,12,13,14
that statements such as these are misleading and foster misunderstandings about
the nature of space and time.
2
In general relativity the “stretching of space” explanation of the redshift is quite problematic. Light is governed by
Maxwell’s equations (or their general relativistic generalization), which contain no “stretching of space term” and no
information on the current size of the universe. On the contrary, one of the most important ideas of general relativity
is that spacetime is always locally indistinguishable from the (non-stretching) spacetime of special relativity, which
means that a photon doesn’t know about the changing scale factor of the universe.15
The emphasis in many textbooks on the stretching-of-spacetime interpretation of the cosmological redshift causes
readers to take too seriously the stretching-rubber-sheet analogy for the expanding universe. For example, it is
sometimes stated as if it were obvious that “it follows that all wavelengths of the light ray are doubled” if the scale
factor doubles.10
Although this statement is correct, it is not obvious. After all, solutions to the Schr¨odinger equation,
such as the electron orbitals in the hydrogen atom, don’t stretch as the universe expands, so why do solutions to
Maxwell’s equations?
A student presented with the stretching-of-space description of the redshift cannot be faulted for concluding,
incorrectly, that hydrogen atoms, the Solar System, and the Milky Way Galaxy must all constantly “resist the
temptation” to expand along with the universe. One way to see that this belief is in error is to consider the problem
sometimes known as the “tethered galaxy problem,”16,17
in which a galaxy is tethered to the Milky Way, forcing
the distance between the two to remain constant. When the tether is cut, does the galaxy join up with the Hubble
flow and start to recede due to the expansion of the universe? The intuition that says that objects suffer from a
temptation to be swept up in the expansion of the universe will lead to an affirmative answer, but the truth is the
reverse: unless there is a large cosmological constant and the galaxy’s distance is comparable to the Hubble length,
the galaxy falls toward us.13,14
Similarly, it is commonly believed that the Solar System has a very slight tendency to
expand due to the Hubble expansion (although this tendency is generally thought to be negligible in practice). Again,
explicit calculation shows this belief not to be correct.18,19
The tendency to expand due to the stretching of space is
nonexistent, not merely negligible.
The expanding rubber sheet is quite similar to the ether of pre-relativity physics in that although it is intuitively
appealing, it makes no correct testable predictions, and some incorrect ones such as the examples we have given. It
therefore has no rightful place in the theory. (Some authors20,21
have argued that considerations such as these do not
refute the notion that space is really expanding. We agree with the calculations in these papers but differ regarding
the most useful language to use to describe the relevant phenomena.)
In one set of circumstances the proper interpretation of the redshift seems clear. When the curvature of spacetime
is small over the distance and time scales traveled by a photon, it is natural to interpret the observed frequency shift
as a Doppler shift. This interpretation is the reason that a police officer can give you a speeding ticket based on the
reading on a radar gun. As far as we know, no one has successfully argued in traffic court that there is an ambiguity
in interpreting the observed frequency shift as a Doppler shift.22
In the expanding universe, spacetime curvature is
small over regions encompassing nearby objects, specifically those with z = ∆λ/λ ≪ 1. There should be no hesitation
about calling the observed redshifts Doppler shifts in this case, just as there is none in traffic court. Surprisingly,
however, many people seem to believe that the “stretching of space” interpretation of the redshift is the only valid
one, even in this limit. We will examine the interpretation of redshifts of nearby objects more carefully in Sec. II.
Aside from low-redshift sources, there is another case in which spacetime curvature can be neglected in considering
cosmological redshifts, namely low-density cosmological models. An expanding universe with density Ω = 0 (often
known as the Milne model23
) is merely the flat Minkowski spacetime of special relativity expressed in nonstandard
coordinates.24
In an Ω = 0 universe there are no gravitational effects at all, so any observed redshift, even of a
very distant galaxy, must be a Doppler shift. Furthermore, for low but nonzero density (Ω ≪ 1), the length scale
associated with spacetime curvature is much longer than the horizon distance. Hence, spacetime curvature effects
(that is, gravitational effects) are weak throughout the observable volume, and the special-relativistic Doppler shift
interpretation remains valid even for galaxies with arbitrarily high redshifts.
The more interesting cases are when z and Ω are not small; that is, when the source is sufficiently distant that
gravitational effects are important over the photon’s trajectory. The consensus is that the Doppler shift language
must be eschewed in this setting. In Sec. III we review a standard argument8,13
that even in this case the redshift
can be interpreted as the accumulation of infinitesimal Doppler shifts along the line of sight, and we further argue
that there is a natural way to interpret the redshift as a single (non-infinitesimal) Doppler shift.
A common objection to this claim is that the coordinate velocity is not related to the redshift in accordance with
the special-relativistic Doppler formula.25,26
However, the velocity referred to in this claim is a mere artifact of a
particular choice of time coordinate. Specifically, it is the rate of change of the distance to the object with respect to
the cosmic time coordinate, as measured at the present cosmic time. This coordinate velocity is an unnatural quantity
to discuss, because it depends on data outside of the observer’s light cone. The more natural velocity is the velocity of
the object at the time it crossed our past light cone, relative to us today. This velocity is also a coordinate-dependent
concept, but as we will show in Sec. III, the most natural way to specify it operationally27,28
leads to a result that is
consistent with the special-relativistic Doppler formula.
3
In Sec. IV we widen our focus to consider frequency shifts in arbitrary curved spacetimes. In any curved spacetime
the observed frequency shift in a photon can be interpreted as either a kinematic effect (a Doppler shift) or as a
gravitational shift. The two interpretations arise from different choices of coordinates, or equivalently from imagining
different families of observers along the photon’s path. We will describe this construction explicitly, and show that
the comoving observers who are usually used to describe phenomena in the expanding universe are the ones that
correspond to the Doppler shift interpretation.
II. REDSHIFTS OF NEARBY GALAXIES ARE DOPPLER SHIFTS
We begin by returning to the parable of the speeding ticket, mentioned in Sec. I.
“A driver is pulled over for speeding. The police officer says to the driver, ‘According to the Doppler shift
of the radar signal I bounced off your car, you were traveling faster than the speed limit.’
“The driver replies, ‘In certain coordinate systems, the distance between us remained constant during the
time the radar signal was propagating. In such a coordinate system, our relative velocity is zero, and the
observed wavelength shift was not a Doppler shift. So you can’t give me a ticket.’ ”
If you believe that the driver has a legitimate argument, then you have our permission to believe that cosmological
redshifts are not really Doppler shifts. If, on the other hand, you think that the officer is right, and the redshift
can legitimately be interpreted as a Doppler shift, then you should believe the same thing about redshifts of nearby
galaxies in the expanding universe.
Why is the police officer right and the driver wrong? Assuming the officer majored in physics, he might explain the
situation like this: “Spacetime in my neighborhood is very close to flat. That means that I can lay down space and
time coordinates in my neighborhood such that, to an excellent approximation, the rules of special relativity hold.
Using those coordinates, I can interpret the observed redshift as a Doppler shift (because there is no gravitational
redshift in flat spacetime) and calculate your coordinate velocity relative to me. The errors in this method are of
the same order as the departures from flatness in the spacetime in a neighborhood containing both me and you. As
long as I’m willing to put up with that very small level of inaccuracy, I can interpret that coordinate velocity as your
actual velocity relative to me.”
The principle underlying the officer’s reasoning is uncontroversial. It is no different from the principle that lets
football referees ignore the curvature of Earth and use a flat coordinate grid in describing a football field.
We now return to the consideration of redshifts in the expanding universe. As noted in Sec. I, it is instructive in
this context (as in many others) to start with the zero-density expanding universe (the Milne model).11
The Milne
universe consists of a set of galaxies expanding outward from an initial Big Bang, with the galaxies assumed to have
negligible mass, so that the geometry is that of gravity-free Minkowski spacetime. This spacetime can be described
either in the usual comoving coordinates of cosmology or in Minkowskian coordinates. Because there is no spacetime
curvature and no gravity in this universe, it is clear that the observed redshifts should be interpreted as Doppler
shifts. In comoving coordinates, however, the redshifts are easily seen to be the usual cosmological redshift. In this
case there is no distinction between the cosmological redshift and the Doppler shift.
In our actual universe spacetime is not exactly flat, but we can approximate it as flat in a small neighborhood. It
might be tempting to think that “approximating away” the curvature of spacetime is the same as approximating away
the expansion altogether. (It seems to us that people who believe that cosmological redshifts cannot be viewed as
Doppler shifts, even arbitrarily nearby, often believe something like this.) However, this belief is incorrect. When we
approximate a small neighborhood of an expanding spacetime as flat, we make errors of order (r/Rc)2
in the metric,
where r is the size of the neighborhood and Rc is the curvature scale (generally the Hubble length). The redshifts
of galaxies in that neighborhood are of order (r/Rc), so they are not approximated away in this limit. (The Milne
model corresponds to the limit Rc → ∞, in which case no approximation is made.)
In comoving coordinates the spacetime line element for the Robertson-Walker expanding universe is
ds2
= −c2
dt2
+ [a(t)]2
dr2
+ [S(r)]2
[dθ2
+ sin2
θ dφ2
] . (1)
Here S(r) = r for a flat universe. For a closed universe with positive curvature K, S(r) = K−1
sin(Kr), and for an
open universe with negative curvature −K, S(r) = K−1
sinh(Kr). For realistic models K−1
is at least comparable
to the Hubble length, and thus S(r) ≈ r for nearby points in all cases.
Assume that an observer at the origin at the present time t0 measures the redshift of a galaxy at some comoving
distance r. Assume that the galaxy is nearby so that r/Rc ≪ 1, where Rc is the curvature length scale. Over such a
distance scale spacetime can be well approximated as flat. Let the observer lay down coordinates that approximate
4
0.00 0.05 0.10 0.15 0.20
0.80
0.85
0.90
0.95
1.00
r
t
0.00 0.05 0.10 0.15 0.20
0.20
0.15
0.10
0.05
0.00
R
T
FIG. 1: Spacetime in different coordinate systems. (a) Plot of an expanding Robertson-Walker universe in the usual comoving
coordinates (r, t). Coordinates are expressed in units of the Hubble length and time. The observer is located at r = 0. The
dotted curves show the world lines of a particular galaxy and of a light signal from that galaxy reaching the observer at the
present time (t = 1). The dashed and solid curves are contours of constant Riemann normal coordinates (R, T); that is, the
coordinates that approximate flat spacetime as well as possible near the observer. (b) The roles of the two coordinate systems
are reversed: the coordinate axes are the Riemann normal coordinates (R, T), and the dashed and solid curves are contours of
constant comoving coordinates (r, t). As time passes, the galaxy moves to larger values of R; that is, it is not at rest in this
coordinate system.
spacetime in her neighborhood as flat as well as possible. That is, let the observer choose coordinates T, R, θ, φ such
that
ds2
≈ −c2
dT 2
+ dR2
+ R2
(dθ2
+ sin2
θ dφ2
) (2)
with the smallest possible errors in her neighborhood. There are several procedures for making this choice, such as
choosing Riemann normal coordinates. The resulting errors are of order (r/Rc)2
in the metric. If we regard such
errors as negligible, then we can legitimately approximate spacetime as flat. In such a coordinate system a comoving
galaxy has a velocity v relative to the observer, which is related to the observed redshift in the expected way to leading
order:
z = (v/c) + O((r/Rc)2
). (3)
Figure 1 illustrates the coordinate systems. In Fig. 1(a) the horizontal and vertical axes are standard comoving
coordinates (r, t). The solid and dashed curves show contours of the coordinates (R, T ) that best approximate
spacetime as flat. World lines of a galaxy and of a light signal from the galaxy to the observer are also plotted.
Figure 1(b) shows the view of spacetime in which the metric appears closest to flat spacetime; that is, it is the one to
use when using special relativistic language to describe our cosmic neighborhood. In this coordinate system comoving
galaxies are moving away from the observer, and the observed redshift is a Doppler shift.
If we assume that the observed galaxy is at a redshift much less than one, the errors in approximating spacetime
to be flat are small. As long as those errors are small enough to be neglected, the observer is in the same situation as
in the speeding ticket parable. We don’t merely say that the police officer is allowed to regard the observed redshift
as a Doppler shift. We say that that is the natural interpretation of the shift. The same statement is true in the
cosmological case. The only natural interpretation of the redshift of a nearby galaxy is as a Doppler shift.
III. REDSHIFTS OF DISTANT GALAXIES CAN BE REGARDED AS DOPPLER SHIFTS
For more distant galaxies, that is, those with redshifts of order one or more, the light-travel distances involved
are comparable to the curvature length scale, and thus the Riemann normal coordinate system cannot be used to
approximate spacetime as flat with good accuracy. In this regime we might imagine being required to drop the Doppler
interpretation in favor of the “stretching of space” point of view. We now argue that the Doppler interpretation is
valid even in this regime.
5
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
t
a
FIG. 2: The scale factor of a hypothetical “loitering” universe as a function of time (measured in units of the present time). At
the time indicated by the dot, a galaxy emits radiation, which is observed at the present time. At the times of both emission
and observation, the expansion is very slow, yet the galaxy’s observed redshift is large.
Let us begin by reviewing a standard derivation8,13
of the cosmological redshift. Consider a photon that travels
from a galaxy to a distant observer, both of whom are at rest in comoving coordinates. Imagine a family of comoving
observers along the photon’s path, each of whom measures the photon’s frequency as it goes by. We assume that
each observer is close enough to his neighbor so that we can accommodate them both in one inertial reference frame
and use special relativity to calculate the change in frequency from one observer to the next. If adjacent observers
are separated by the small distance δr, then their relative speed in this frame is δv = Hδr, where H is the Hubble
parameter. This speed is much less than c, so the frequency shift is given by the nonrelativistic Doppler formula
δν/ν = −δv/c = −Hδr/c = −Hδt. (4)
We know that H = ˙a/a where a is the scale factor. We conclude that the frequency change is given by δν/ν = −δa/a;
that is, the frequency decreases in inverse proportion to the scale factor. The overall redshift is therefore given by
1 + z ≡
ν(te)
ν(to)
=
a(to)
a(te)
, (5)
where te and to refer to the times of emission and observation, respectively.
In this derivation we interpret the redshift as the accumulated effect of many small Doppler shifts along the photon’s
path. We now address the question of whether it makes sense to interpret the redshift as one big Doppler shift, rather
than the sum of many small ones.
Figure 2 shows a common argument against such an interpretation. Imagine a universe whose expansion rate varies
with time as shown in Fig. 2. A galaxy emits radiation at the time te indicated by the dot, and the radiation reaches
an observer at the present time t0. The observed redshift is z = a(t0)/a(te)−1 = 1.5, which by the special-relativistic
Doppler formula would correspond to a speed of 0.72c. At the times of emission and absorption of the radiation, the
expansion rate is very slow, and the speed ˙ar of the galaxy is therefore much less than this value. We can construct
models in which both ˙a(te) and ˙a(t0) are arbitrarily small without changing the ratio a(t0)/a(te) and hence without
changing the redshift.
Upon closer examination this argument is unconvincing, because the calculated velocities are not the correct velocities.
We should not calculate velocities at a fixed instant of cosmic time (either t = te or t = t0). Instead we should
calculate the velocity of the galaxy at the time of light emission relative to the observer at the present time. After all,
if a distant galaxy’s redshift is measured today, we wouldn’t expect the result to depend on what the galaxy is doing
today, nor on what the observer was doing long before the age of the dinosaurs.
In fact, when astrophysicists talk about what a distant object is doing “now,” they often do not mean at the present
value of the cosmic time, but rather at the time the object crossed our past light cone. For instance, when astronomers
measure the orbital speeds of planets orbiting other stars, the measured velocities are always of this sort. There is an
excellent reason for this convention: we never have information about what a distant object is doing (or if the object
even exists) at the present cosmic time. Any statement in which “now” is used to refer to the present cosmic time at
the location of a distant object is not about anything observable, because it refers to events far outside our light cone.
In summary, if we wish to discuss the redshift of a distant galaxy as a Doppler shift, we need to be willing to talk
about vrel, the velocity of the galaxy then relative to us now. Talking about vrel is precisely the sort of thing that
6
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
0.4
0.6
0.8
1.0
rt
FIG. 3: Parallel transport of the velocity four-vector. The solid and dashed lines show the world line of a galaxy and the path
of radiation from the galaxy to the observer in comoving coordinates. The short solid lines show the galaxy’s four-velocity
being parallel transported to the observer along the path of the light.
the enlightened cosmologist described in Sec. I refuses to do because, in a curved spacetime, there is not a unique
way to define the relative velocity of objects at widely separated spacetime events. Determining the velocity of one
object relative to another involves comparing the two objects’ velocity four-vectors. [To be specific, the dot product
of these vectors is γrel ≡ (1 − v2
rel/c2
)−1/2
.] To do that we have to transport one of the vectors to the location of the
other. In a curved spacetime, the result of such a “parallel transport” depends on the path along which the vector is
transported.
Suppose that we wish to defy the purist and talk about vrel. We have to parallel transport the galaxy’s velocity
four-vector to the observer’s location. As others have noted before,27,28
the only natural path to choose for this
parallel transport is the path followed by the light (that is, the null geodesic joining the emission and observation
events). If we follow this procedure to determine the relative velocity, we find that the velocity obeys the rule
c + vrel
c − vrel
=
a(to)
a(te)
. (6)
As noted in Eq. (5), the ratio of scale factors is equal to 1 + z, so Eq. (6) is the special-relativistic Doppler formula.
In other words, the relative speed vrel as defined by parallel transport is related to the observed redshift as it should
be if the redshift is a Doppler shift.
Figure 3 illustrates this process. The solid line is the world line of a galaxy, and the dashed line is the path of
light traveling from the galaxy to the observer. The scale factor is as shown in Fig. 2. The short solid lines indicate
the galaxy’s velocity four-vector as it is parallel transported along the light path to the observer. During the times
when the universe is expanding slowly the direction of this vector doesn’t change significantly, but it does when the
universe expands more rapidly.
Equation (6) can be derived by a straightforward calculation using the rules for parallel transport, but the derivation
is easier if we recast the statement in more physical terms. As we did at the beginning of this section, imagine many
comoving observers stationed along the line from the observed galaxy to the observer. Each observer has a local
reference frame in which special relativity can be taken to apply, and the observers are close enough together that
each one lies within the local frame of his neighbor. Observer number 1, who is located near the original galaxy,
measures its speed v1 relative to him and gives this information to observer 2. Observer 2 measures the speed
u of observer 1 relative to her, adds this speed to the speed of the galaxy relative to observer 1 using the usual
special-relativistic formula,
v2 =
v1 + u
1 + v1u/c2
, (7)
and interprets the result as vrel, the speed of the galaxy relative to her. She passes this information on to the next
observer, who follows the same procedure, as does each subsequent observer. At each stage, the imputed velocity of
the original galaxy relative to the observer will match the redshift of the galaxy in accordance with Eq. (6). The
procedure we have described operationally is equivalent to performing parallel transport of the galaxy’s four-velocity
and using that vector at each stage to calculate vrel.
We can express this derivation in yet another way. Consider the “world tube” obtained by drawing a sphere with
a small radius ǫ around each point on the path of the light ray from the galaxy to the observer. This region of
7
spacetime can be considered as flat Minkowski spacetime up to errors of order ǫ2
. (This statement is true for any
small neighborhood around a geodesic.) Within this tube the observed redshift can be explained only as a Doppler
shift, because the spacetime is flat to arbitrary precision. This argument supplies the simplest proof we can think of
that vrel is related to the redshift by Eq. (6).
We do not expect to have convinced purists who refuse even to talk about vrel to change their minds. They would
dismiss vrel as a mere “coordinate velocity,” not an “actual velocity,” and take no interest in it. If purists have the
courage of their convictions, they would say that it makes no sense to try to attach labels such as “Doppler” or
“gravitational” to the observed redshift. This position is unassailable, and we have no wish to argue against it. We
do claim, however, that if you wish to try to talk about vrel, then the definition proposed in this section is the most
natural way to do so. Because this definition of vrel results in the Doppler formula entirely explaining the redshift,
we claim that you should either be a purist and refuse to try to label the cosmological redshift, or you should label it
a Doppler shift.
It is interesting to contrast the position we advocate with another approach13,29
that describes the cosmological
redshift as a particular combination of Doppler and gravitational terms. In that approach comoving coordinates are
used to specify the velocity of the distant galaxy relative to the observer, and the observed redshift is decomposed
into a Doppler term based on this velocity and a gravitational blueshift, which can be calculated by considering
the gravitational potential due to the matter in a sphere centered on the observer. The difference between the two
approaches is in the way the velocity is defined. Because there is no unique specification of relative velocity for distant
objects in curved spacetime, both approaches are correct. We believe that the approach based on defining vrel through
parallel transport is more natural than the one based on a particular choice of coordinates.
IV. KINEMATIC AND GRAVITATIONAL REDSHIFTS IN GENERAL SPACETIMES
Let us step back from the specific case of the expanding universe and consider the propagation of a photon in an
arbitrary spacetime. A photon is emitted at some spacetime point Pe with a frequency νe and is absorbed at Pa with
frequency νa. The two frequencies are measured by observers in local inertial reference frames at the two spacetime
points. If there is a frequency shift (νe = νa), under what circumstances should we describe it as a Doppler shift or
as a gravitational shift?
A natural way to answer this question is to populate the space between the photon’s emission and observation with
a dense family of observers. Let the first observer be at rest relative to the emitter of the photon at the emission event,
and let the last be at rest relative to the absorber at the absorption event. Finally, let the velocities of observers vary
smoothly along the photon’s path. As in Sec. III, we can explain the observed frequency shift by adding up the shifts
as the photon passes from one observer to the next. We can construct two such families of observers: one in which
each of the shifts is a Doppler shift, and one in which each is a gravitational shift. By reference to these families, we
can interpret the observed shift as the accumulation of either many small Doppler shifts or many small gravitational
shifts.
The Doppler and gravitational families are not the only possibilities. We can construct other families with respect
to whom both Doppler and gravitational shifts would be seen to contribute. As noted in Sec. III, for example, one
way of regarding the cosmological redshift13,29
splits it into a specific combination of Doppler and gravitational terms.
The Doppler and gravitational families provide two of many different ways of interpreting the observed shift. In any
given situation, we can argue about which (if either) of the two families is natural to consider.
To construct the Doppler family of observers, we require all observers to be in free fall. In this case, within each
local inertial frame, there are no gravitational effects, and hence the infinitesimal frequency shift from each observer
to the next is a Doppler shift.
To construct the gravitational family of observers, we require that each member be at rest relative to her neighbor
at the moment the photon passes by, so that there are no Doppler shifts. Initially, it might seem impossible in general
to satisfy this condition simultaneously with the condition that the first and last observers be at rest relative to the
emitter and absorber, but it is always possible to do so. One way to see that it is possible is to draw a small world
tube around the photon path as in Sec. III. Within this tube, spacetime is arbitrarily close to flat. We can construct
“Rindler elevator coordinates,” the special-relativistic generalization of a frame moving with uniform acceleration,
within this tube, such that the velocities at the two ends match up correctly.30
The members of the gravitational
family are at rest in these coordinates. Because they are not in free fall, the members of the gravitational family all
feel like they are in local gravitational fields. Because each has zero velocity relative to her neighbor when the photon
goes by, each observer interprets the shift in the photon’s frequency relative to her neighbor as a gravitational shift.
Because the two families exist for any photon path, we can always describe any frequency shift as either Doppler or
gravitational. In some situations, one seems clearly more natural than the other. When we discuss the Pound-Rebka
experiment,31
which measured the redshift of photons moving upward in Earth’s gravitational field, we generally choose
8
to regard observers fixed relative to the Earth (the gravitational family) as more natural than free-fall observers, and
hence we interpret the measurement as a gravitational redshift. In contrast, if you were falling past Pound and
Rebka in a freely falling elevator as they performed the experiment, you might choose a Minkowskian inertial frame
encompassing you and the entire experiment. Within this frame, you would, by the equivalence principle, interpret
their results as a Doppler shift. In so doing, you would be choosing to regard the Doppler family as the natural one,
because this family is the one whose behavior is simplest to describe in your chosen frame. Finally, if you tried to
beat a speeding ticket by claiming that the radar results were due to a gravitational redshift, you would in effect be
considering a “gravitational family” of prodigiously accelerating observers, with one at rest relative to the radar gun
and one at rest relative to the driver. Needless to say, the police officer who gives you a ticket regards this family as
extremely unnatural.
In the cosmological context, we almost always work with freely falling comoving observers, a choice which corresponds
precisely to the Doppler family. This choice is natural because it respects the symmetries of the spacetime
and makes the mathematical description as simple as possible. On the other hand, the gravitational family is so
unnatural in cosmology that, as far as we know, it has never been used for anything. Because the Doppler family is
by far the most natural family to work with, it is natural to interpret the cosmological redshift as a Doppler shift,
and it is curious that this interpretation is frequently frowned upon.
V. WHY THE INTERPRETATION MATTERS
There is no “fact of the matter” about the interpretation of the cosmological redshift: what one concludes depends on
one’s coordinate system or method of calculation. Nonetheless, it is instructive to analyze the differing interpretations
of the redshift, partly to improve the understanding of cosmology, but more importantly to improve the understanding
of general relativity. That analysis leads us to conclude that the most natural interpretation of the redshift is kinematic.
One of the key ideas of general relativity is the importance of distinguishing between coordinate-independent
and coordinate-dependent statements. Another is the idea that spacetime is always locally indistinguishable from
Minkowski spacetime. Cosmology instructors, books (especially at the introductory level), and students often fall into
the fallacy of reifying the rubber sheet; that is, treating the expanding-rubber-sheet model of space as if it were a
real substance. This error leads people away from both of these key ideas and causes mistaken intuitions such as that
the Milky Way Galaxy must constantly “resist the temptation” to expand with the expanding universe or that the
“tethered galaxy” described in Sec. I moves away after the tether is cut.
The common belief that the cosmological redshift can only be explained in terms of the stretching of space is based
on conflating the properties of a specific coordinate system with properties of space itself. This confusion is precisely
the opposite of the correct frame of mind in which to understand relativity.
Acknowledgments
We thank Sean Carroll, John Peacock, Jim Peebles, Dave Spiegel, and Ned Wright for helpful comments. This
work was supported in part by NSF grant AST-0507395 (EFB) and a research fellowship from the Alexander von
Humboldt Foundation of Germany (DWH).
∗
Electronic address: ebunn@richmond.edu
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