Counterintuitive properties of relativistic relative motion for accelerated observers

Relative velocities of inertial frames play an important role in special relativity, and thus feature prominently in all expositions of the theory. Relative motion as measured by non-inertial observers, however, is not a standard topic in introductory accounts of special relativity. With the transition to general relativity, relative motion takes a back seat: in that theory, relativity is not about relative motion, but more generally about covariance and the freedom to choose general coordinates.

That said, accelerated motion per se does have its place in expositions of either theory. In special relativity, the simplest form, hyperbolic motion, was originally introduced by Max Born to explore the concept of rigidity, specifically relativistic rigid motion.¹ Accelerated motion of clocks has been used to resolve the so-called twin paradox—the fact that, in special relativity, there is no symmetry between inertial observers and observers that undergo acceleration, so that a twin departing on a round trip near light speed will have aged less than their stay-at-home sibling.^2–4 Closely related to the topic of the current article is Bell's Spaceship Paradox, arguably the best-known example for a counterintuitive property of relativistic accelerated motion:^5,6 Tie a rope between two spaceships that are, from the perspective of some inertial observer, accelerating in an identical fashion, and the rope will tighten and, eventually, break. Many introductions to general relativity feature the equivalence principle in a version that compares the effects of constant acceleration with those of a homogeneous gravitational field.^7–9 

The aim of this article is to provide an elementary review of the concept of non-inertial relative velocity in special relativity, as well as its counterpart in general relativity. For special relativity, two more examples are provided of how relativity runs counter to our intuition. In general relativity, these examples prove helpful for understanding static gravitational fields, notably the exterior Schwarzschild solution.

In the context of classical mechanics, most of us will have formed intuitive notions about relative motion that one might be tempted to regard as universal:

1. When the physical distance between two objects is constant, their relative speed is zero.

2. Objects in relative motion are moving either away from each other (positive radial velocity) or toward each other (negative radial velocity), but not both at the same time.

Both statements are meant to refer to measurements—this article is not concerned with apparent effects, e.g., how an idealized camera would record what is happening.

As we shall see in Secs. IV and V, once we allow for acceleration, the straightforward generalizations of these statements both turn out to be wrong. In contrast with the standard examples of counterintuitive relativistic effects, which are associated with inertial systems (relativity of simultaneity, time dilation, and length contraction), the two examples presented here, concerning relative velocity, do not seem to be accounted for in the existing literature, although a similar effect was found by Rashidi and Ahmadi¹⁰ for how different observers measure acceleration. Key building blocks for these effects are already present in classical physics, in situations with accelerated observers, as we will see in a toy model in Sec. II.

In Sec. VI, we examine how the two effects can help us to better understand the gravitational redshift in general relativity. The analysis also throws new light on the equivalence principle and horizons.

Any realistic and consistent description of light propagation requires relativistic physics. In classical mechanics, constant acceleration necessarily leads to unrealistic superluminal motion. Nevertheless, we can construct a helpful toy model for accelerated observers in classical physics, as follows: We consider an inertial coordinate system $I$ in which we assume light to propagate isotropically at the usual speed c, and we restrict our description to those times when all our accelerated observers move slower than light in $I$⁠. For simplicity, we restrict ourselves to motion along the x axis.

If we accept radar distances as physical distances, and the Doppler shift as an indicator of relative motion, this yields a classical counterexample to both of the main intuitive statements from Sec. I: Radar distances between our observers are constant, yet the Doppler effect indicates relative motion. Equation (5) shows that one observer in each pair will claim the other is moving away, while the other will conclude the observers are moving toward each other.

Without a framework of absolute space and absolute simultaneity, however, radar distances and Doppler measurements take on a more fundamental meaning. So, motivated by this classical toy example, let us consider the same kind of situation in special relativity.

We begin by reviewing the concepts of relative speed and relative radial velocity in special relativity. Four-vectors provide us with a coordinate-independent language that allows for statements about arbitrary motion. Such statements can typically refer to either observers or objects; for brevity, I will only talk about observers in the following.

It is straightforward to obtain the velocity of one inertial frame $I1$ relative to another such frame $I2$⁠: the ordinary three-space velocity assigned to the origin of $I2$ in the frame $I1$ is the relative velocity. If we align the spatial axes of the two systems, the reciprocity principle holds: The velocity vector of $I2$ as expressed in $I1$ is the opposite of the velocity vector of $I1$ as expressed in $I2$⁠. In consequence, the relative speed, defined as the magnitude of the relative velocity, is the same, regardless of which of the two systems we use as our reference frame.

If, instead of inertial frames, we want to describe the relative motion of two observers 1 and 2, each of which is in arbitrary motion, more information is needed. After all, when the state of motion of either object changes, the relative velocity will, in general, change as well. Even in classical physics, we would need to state at which moments in time we are evaluating a relative velocity. If, for instance, we want to calculate the Doppler effect for a specific signal, we would need to take into account the emitting observer's state of motion at the time of emission, and the receiving observer's state of motion at the time of reception of the signal, and determine the relative speed linking those two states. In this sense, relative velocity is not a property merely of two observers we are talking about. Even in classical physics, there is an additional dependence on two moments in time, one for each observer.

In special relativity, where simultaneity is relative, there isn't even a coordinate-independent way of talking about the relative velocity “at a given moment in time” anymore. For any coordinate-independent description of relative motion, we need to specify two events $E1$ and $E2,$ one on each observer's worldline, in order to define relative speed. Only then do we have sufficient information to be able to talk about “the speed of observer 1 at $E1$ relative to observer 2 at $E2$⁠.”

At this point, there is a natural way of defining relative speed in special relativity: the speed v of observer 1 at $E1$ relative to observer 2 at $E2$ is defined as the speed of the inertial frame $I1$ that is momentarily co-moving with observer 1 at $E1$ relative to the inertial frame $I2$ that is momentarily co-moving with observer 2 at $E2$⁠. Note that this cannot be generalized unambiguously to a fully fledged relative velocity, as the three-vector components of such a velocity will still depend on the (arbitrary) spatial orientation of the two systems.

An interesting special case is purely radial motion. In terms of the momentarily co-moving inertial systems $I1,2$⁠, chosen here for simplicity so that $E1,2$ are at their system's spatial origin, this is equivalent to the two spatial origins moving directly away from or directly toward each other. We define the relative radial velocity $vR$ so that when the origins are moving away from each other, $vR≡+vrel$⁠, whereas for motion toward each other, $vR≡−vrel$⁠.

A number of hyperbolic worldlines from our family are plotted in Fig. 1. Some of them are labeled with their X value. The worldlines are evenly spaced in X, showing clearly how the hyperbolas are crowded against their boundary, namely parts of the boundary of the past and future light cones at the origin of $I$⁠. Dividing Eq. (11) by Eq. (12), it is straightforward to see that curves of constant T correspond to straight lines through the origin in $I$⁠. Some of those straight lines are shown and labeled in Fig. 1 as well.

Using the model described in Sec. IV, we can demonstrate ways in which common intuition goes wrong for accelerated objects in special relativity.

If we were merely given the coordinates X, T and the corresponding metric (15), we might be tempted to argue as follows: Clearly, that metric is static, and the coordinates X, T are adapted to that property—none of the metric coefficients depends on T, and there are no mixed terms in X and T. Thus, two objects that stay fixed at some given X value, say, at $X1$ and $X2$⁠, should be considered as being at rest relative to each other.

If we distrust that line of argument because it relies on specific coordinates, we can go one step further: We let the observer at constant $X1$ send light signals to the observer at constant $X2$⁠, receive reflected light signals back, and document the interval of proper time they measure between the emission event and the reception event. The observer at $X1$ would find that it always takes the same time for their light signals to travel to $X2$ and back. From the external perspective of our inertial system $I$⁠, this constancy is the result of different circumstances, which just happen to yield the same result every time. Figure 2 shows an example. From the perspective of our inertial system, the worldline segment corresponding to light emission at $X=−1$ at time $T=−4$⁠, and reception at $X=−3$ at $T=−2$⁠, looks markedly different from, say, the worldline segment linking $X=−1$ at time $T=0$ and $X=−3$ at time $T=2$⁠. In X, T coordinates, however, light propagation really does follow the simple constant-speed relation (16). The corresponding radar distances determined by either observer are indeed constant.

Given that we are in special-relativistic spacetime, surely a constant radar distance is as close as we can come to making a physical statement about the distance between $X1$ and $X2$⁠: The two observers are at rest relative to each other. Whenever the observer at $X1$ repeats their radar measurement of the observer at $X2$⁠, they will get the same distance. The observer at $X2$ could also perform the experiment, and would find a distance that does not change over time.

By these criteria, namely constant static coordinate values and constant radar distance, the two observers at $X1$ and $X2$ are at rest relative to each other, at all times. However, there is the expression (23), which shows that while the two observers in question indeed have relative speed zero when evaluated at the same time, $T1=T2$⁠, they have non-zero relative speed for comparisons at different times, and in particular, at the pairs of events probed by sending a light signal in either direction. Each observer will see the other's signals to be Doppler-shifted. Being “at rest relative to each other,” as determined by either criterion—static coordinates, or constant radar distances—is not a coordinate-independent property. Even for the best candidate for a “physical distance,” namely constant radar distance, constancy over time does not mean vanishing relative speed (23).

This unusual property complements Bell's Spaceship Paradox:^5,6 In that case, two spaceships are accelerated identically as judged from an inertial system, which corresponds to relative speed zero at all times in that inertial system. Yet, the physical distance between the space ships lengthens as judged by stress in a rope linking the two ships. The example presented above shows the other side of the coin: constant distances as determined by radar, but non-zero relative speed.

From this expression for $vR$⁠, we can derive the second counterintuitive property of relative motion in relativity. If an observer at fixed location $X=X1$ sends light to an observer at fixed location $X=X2$⁠, then $vR$ as specified by Eq. (28) will have the same magnitude, but the opposite sign than for signals traveling from $X2$ to $X1$⁠. The direct consequence in terms of wavelength shifts: Observers will see signals from a source at smaller constant X than their own position as redshifted, and from a source at larger constant X as blueshifted.

Why is that not a logical contradiction? Because in order to determine the relative radial velocity, we always need to specify which two events we are comparing. In our example, each observer only considers events that involve light emitted by the other observer, and received by themselves. Thus, each observer is analyzing a different pair of events. There is no logical contradiction involved if one pair of events yields a result of “motion toward,” while the other pair yields a result of “motion away from.” It is in this sense that our two observers are both moving away from each other and also moving toward each other.

The non-intuitive properties of relative velocities described in Sec. V can help with understanding certain aspects of general relativity, including static gravitational fields.

The notion of parallel transport leads to a natural generalization of relative speed in special relativity: the relative speed of two objects evaluated along a path that intersects each object's worldline is obtained by taking the first object's four-velocity at one end of the path, parallel-transporting it to the other end of the path, and applying formula (29) locally. Scalars such as the scalar product in Eq. (29) are not changed by parallel transport, so we can perform the comparison at either end of the path. Radial relative velocity can be defined, as well: As evaluated by a locally flat (free-fall) system associated with the receiving object, purely radial motion is when the spatial components of the parallel-transported four-velocity are parallel or anti-parallel to the direction defined by the transport path. For light-like transport paths, there is again the direct link with the wavelength shift of the associated light, and since the comparison takes place in locally flat spacetime, $vR$ can be recovered directly from the Doppler formula (10).¹⁶ 

Our departure from defining relative speed as a property of two objects, with no additional information needed, has thus progressed one step further. Classical mechanics required us to specify two times, one per object, for the definition of relative speed. In special relativity, we needed to specify two events, one for each object. In general relativity, we need a path linking the two events.

For events that are within a sufficiently small region of spacetime, the following holds: For each pair of events, there is a unique straightest possible line, a geodesic, linking the two events.¹⁸ The maximal size of the small region can be estimated from its matter content.¹⁹ Within such a region, we can define relative speed or radial velocity by specifying the two worldlines, and the two events, and then using the unique geodesic joining the two events for parallel transport. Beyond this region, there is no unique path, and we can end up with more than one relative speed for each pair of objects and events.

Analogs of the two counterintuitive properties presented in Secs. V A and V B play a role in static gravitational fields. The textbook example is the Schwarzschild spacetime as a model for the simplest kind of black hole and, more generally, for the gravitational influence outside any spherically symmetric mass. The conceptual connection should not come as a surprise: According to one version of the equivalence principle, acceleration can be seen as simulating the simplest possible gravitational field.^7–9 A different set of “accelerated coordinates,” namely Rindler coordinates, have long been used as pedagogical tools for exploring black holes, and in particular, black hole horizons.^20–22 

Consider the exterior Schwarzschild metric. The original Schwarzschild coordinates are static: none of the metric coefficients depend explicitly on time. Consider two observers, each hovering at a constant r value, one directly below the other. Each light signal propagating, say, from the lower to the upper observer takes the same amount of coordinate time. In consequence, two signals sent a coordinate time interval $Δt$ apart will also arrive $Δt$ apart. In this description, wavelength shifts stem from relations between proper time and coordinate time at different r values. The result is a (gravitational) redshift for light traveling outward from the lower to the higher observer, and blueshift for light traveling the other way around.

For Narlikar, the main focus is on how parallel transport allows for a unified description of Doppler shifts and gravitational redshifts. However, there is another aspect: the two apparent contradictions, rooted in classical intuition, that could be used to argue against an interpretation of this situation in terms of relative motion.

The first apparent contradiction is that the situation involves a static spacetime, and static observers. As judged by their constant radial coordinate values and by radar measurements, the two observers are at rest relative to each other. So how can those observers possibly be said to be in relative motion, as per (30)? Sec. V A shows that, even in special relativity, this is no contradiction when accelerated observers are involved.

That we have a blueshift for light traveling in one direction, but a redshift for light traveling in the opposite direction could be a second argument against the consistency of the Doppler interpretation: If those observers are in relative motion, surely they must either be moving away from or toward each other—so doesn't the asymmetry preclude an interpretation as relative motion? Sec. V B shows that, on the contrary, this too is a generic property of accelerated observers. The key to avoiding a logical contradiction is the same as in our simple model: relative radial velocity is not a property that can be assigned directly to two observers. It depends on the two events used for the comparison. Blueshift and redshift are associated with two different classes of events, one characterized by ingoing, the other by outgoing light-like geodesics.

In these two ways, understanding the properties of relative motion involving accelerations in special relativity can help us understand what is happening in a static gravitational field. Objections based on our classical intuitions are not applicable even in special relativity.

These examples also provide a slight change in perspective on the equivalence principle. That principle is commonly couched in terms of the following alternatives: We can interpret a situation as taking place either with two observers at rest in a gravitational field, or both observers accelerating in the absence of a gravitational field. The redshift of light traveling from the “lower” to the “higher” observer is interpreted as a gravitational redshift or as a Doppler shift, respectively.

Relative motion adds a twist: Once the path for parallel transport is fixed, relative radial velocity is an invariant. Whether the two observers undergo acceleration in a field-free environment, or else are at rest in a gravitational field, will depend on the perspective, but their relative velocity does not. Even in the latter scenario, where the two observers are at rest in a gravitational field, there is a non-vanishing relative radial velocity. Relative motion is involved in both cases, and in this way, even a pure gravitational redshift is necessarily connected with the Doppler effect of the associated non-zero relative radial velocity.

Last but not least, the relative motion description provides a building block for understanding horizons: boundaries separating spacetime into regions whose light can reach us (if we wait long enough) and regions whose light will never reach us. For an external, stationary observer at $r2$⁠, by Eq. (30), the Schwarzschild horizon at $r1=R$ is always at a relative speed $vR=c$⁠, implying an infinite redshift. Our example situation from Sec. V B provides a simple analog for a horizon: For exchanges of light signals between members of our preferred family of observers, see Fig. 1, there is a horizon for light signals sent from $X1→−∞$ to $X2>X1$⁠, which by Eq. (28) corresponds to $vR=c$⁠.

The same intuitive understanding can be applied to cosmological horizons. Consider the recession velocity $vrec$ taken from the Hubble–Lemaître law for an expanding universe. The simple notion that “behind the cosmological horizons, objects are moving away so fast their light can never reach us because $vrec>c$” is incorrect and misleading.^23,24 The speed one should consider is the relative radial velocity $vR$ of Hubble-flow galaxies. This leads to a consistent interpretation of the cosmological redshift as a Doppler shift.^15–17,25 This $vR$ is not the same as the usual $vrec$⁠. A key difference is that $vR<c$ even for the most distant Hubble-flow galaxies we can observe. However, the existence of a cosmological horizon as a boundary between those regions of spacetime in an expanding universe whose light signals can reach us in the future and those regions whose light signals can never reach us is indeed marked by the limit $vR=c$⁠.²⁶ Defining relative motion in a suitable way turns a misleading intuition about motion and the light-speed barrier into a helpful one.

The author would like to thank Thomas Müller and the anonymous referees for helpful comments on an earlier version of this text.

The author has no conflicts to disclose.

In this section, we outline the calculations leading to key results in Sec. II.

Re-writing the two round-trip times in terms of $a0$ instead of $a1$ using (1) yields the unified form given in the main text as Eq. (3). Equation (A8) is valid only for $d2>d1$⁠, so applying this result to the case $d1>d2$ that is part of the general formula (3) calls for some judicious switching of indices, $1↔2$⁠.

Doppler shift—To obtain Eq. (5), for $d2>d1$ take the derivative the arrival time formula (A4) and use (1) to replace $a1≡a(d1)$ by $a0$⁠. For $d2<d1$⁠, do the same with the arrival time formula (A6), but be careful: since in the Doppler formula (5), the receiver is always the observer that is initially at $d2$⁠, you need to switch indices, $1↔2$⁠, when you apply (A6).

Relative velocity—The relative velocity in Eq. (6) follows directly from taking the t-derivative of the modulus $|x2(t)−x1(t)|$ given by Eq. (A2), and using (1) to express the two accelerations in terms of $a0$⁠.