Unified approach to redshift in cosmological /black hole spacetimes and synchronous frame

TL;DR

This paper presents a unified framework for understanding redshift in both static and cosmological spacetimes by transitioning between frames, aiding in the conceptual clarity of gravitational effects in general relativity.

Contribution

It introduces a method to unify redshift interpretation in static and cosmological spacetimes through frame transformations, generalizing known frames like Lemaitre and Painlevé–Gullstand.

Findings

Unified interpretation of redshift in different spacetimes

Construction of generalized spherically symmetric frames

Clarification of gravitation's role in general relativity

Abstract

Usually, interpretation of redshift in static spacetimes (for example, near black holes) is opposed to that in cosmology. In this methodological note we show that both explanations are unified in a natural picture. This is achieved if considering the static spacetime one (i) makes a transition to a synchronous frame, (ii) returns to the original frame by means of local Lorentz boost. To reach our goal, we consider a rather general class of spherically symmetric spacetimes. In doing so, we construct frames that generalize the well-known Lemaitre and Painlev\'{e}--Gullstand ones and elucidate the relation between them. This helps to understanding in an unifying approach, how gravitation reveals itself in different branches of general relativity. This can be useful for general relativity university courses.

Equations

$$d{s}^2=-c^{2}d{t}^2+a^2(t)d{\chi }^2.$$

$$d{s}^2=a^2(d{\chi }^2-d{\eta }^2).$$

$$d{s}^2=0.$$

$$d\chi /dt=c/a.$$

$${\chi }_{em}-{\chi }_{obs}=c{\int }_{t_{em1}}^{t_{obs1}}{a}^{-1}dt,$$

$${\chi }_{em}-{\chi }_{obs}=c{\int }_{t_{em2}}^{t_{obs2}}{a}^{-1}dt.$$

$${\int }_{t_{em1}}^{t_{obs1}}{a}^{-1}dt-{\int }_{t_{em2}}^{t_{obs2}}{a}^{-1}dt=0.$$

$${\int }_{t_{obs1}}^{t_{obs2}}{a}^{-1}dt-{\int }_{t_{em1}}^{t_{em2}}{a}^{-1}dt=0\text{.}$$

$$\frac{t_{obs2}-t_{obs1}}{a(t_{obs})}-\frac{t_{em2}-t_{em1}}{a(t_{em})}=0.$$

$${\lambda }_{em}/a(t_{em})={\lambda }_{obs}/a(t_{obs}).$$

$$\frac{dr}{dt}=\frac{d(\chi a)}{dt}=\frac{d\chi }{dt}a+\frac{da}{dt}\chi =c+\frac{da}{dt}\chi =c+\frac{\dot{a}}{a}a\chi =c+v_{fl},$$

$$\frac{dr}{dt}=\frac{dP(\chi ,t\mid 0,t)}{dt}=v_{loc}+v_{fl},$$

$$v_{loc}=\underset{\Delta t\to 0}{\lim }\frac{P(\chi +\Delta \chi ,t\mid \chi ,t)}{\Delta t}=\frac{\partial P(\chi ,t\mid 0,t)}{\partial \chi }\frac{d\chi }{dt}\text{,}$$

$$v_{fl}=\frac{\partial P(\chi ,t\mid 0,t)}{\partial t}\text{.}$$

$$d\Delta r/dt=H\Delta r.$$

$$v=\frac{d}{dt}{\int }_{{\chi }_1}^{{\chi }_2}\sqrt{g_{\chi \chi }}d\chi =v_{fl}+v_{loc}\text{,}$$

$$v_{fl}={\int }_{{\chi }_1}^{{\chi }_2}\frac{d}{dt}\sqrt{g_{\chi \chi }}d\chi$$

$$v_{loc}=\sqrt{g_{\chi \chi }({\chi }_2)}d{\chi }_2/dt$$

$$d{s}^2=-f(r)d{t}^2+\frac{d{r}^2}{f(r)}+r^2(d{\theta }^2+{\sin }^2\theta d{\varphi }^2)\text{.}$$

$$\rho =t+\int \frac{dr}{f\sqrt{1-f}}\text{,}$$

$$\tau =t+\int \frac{dr}{f}\sqrt{1-f}$$

$$d{s}^2=-d{\tau }^2+(1-f)d{\rho }^2+r^2(\rho ,\tau )(d{\theta }^2+{\sin }^2\theta d{\varphi }^2)\text{.}$$

$$l={\int }_{{\rho }_1}^{{\rho }_2}d\rho \sqrt{1-f}=r_2-r_1\text{.}$$

$$d{s}^2=-d{\tau }^{2}f+d{r}^2+2drd\tau \sqrt{1-f}+r^2(d{\theta }^2+{\sin }^2\theta d{\varphi }^2)\text{.}$$

$$v=\sqrt{1-f}\text{.}$$

$$v_{12}=v_2-v_1\text{.}$$

$$\frac{dr}{d\tau }=\pm 1-v\text{,}$$

$$d\Delta r=\Delta vd\tau =v^{\prime }(r)\Delta rd\tau ,$$

$$\frac{d\lambda }{dr}(1+v)=\frac{dv}{dr}\lambda ,$$

$$\frac{d\lambda }{\lambda }=\frac{dv}{1+v}.$$

Full text

synchronous frame, Lorentz boost

I Introduction

Though the calculation of redshifts in General Relativity (GR) has no principal difficulties, their interpretation continues to be a source of debates. It is well known that gravitational field can induce a blue/red shift in cases which do not have counterparts in Special Relativity (SR). For example, a photon emitted near a black hole horizon reaches a distant observer severely redshifted, even in cases where emitter and observer do not move with respect to each other. The gravitational redshift of this type can be obtained as follows. In the static field, one can introduce such a time variable $t$ whereby different static observers measure the same time intervals $dt$ between a pair of corresponding events (say, emission and absorption of photons). Then, the intervals of the proper time differ by the ratio of the factors $\sqrt{g_{tt}}$, where $g_{tt}$ is the corresponding component of the metric tensor. One can also describe this redshift in a more formal fashion. In the static gravitation field, there exists a timelike Killing vector $\xi^{\mu}$ (see, for example, wein ). Let a photon have the wave vector $k^{\mu}$. Then, the quantity $\omega_{0}=-k_{\mu}\xi^{\mu}$ is conserved along the path of a photon due to the geodesic equation for $k^{\mu}$ and Killing equations. This has the meaning of a frequency measured at infinity. Let us imagine a set of static observers with the four-velocities $u^{\mu}$; they measure the local frequencies $\omega=-k_{\mu}u^{\mu}$. For a static observer, $u^{\mu}={\xi^{\mu}}/{N}$, where $N=\sqrt{-\xi_{\mu}\xi^{\mu}}$ is a lapse function. Then, we obtain $\omega N=\omega_{0}$ which gives a redshift in a static field. These two methods imply staticity of the field and are completely equivalent.

However, the cosmological redshift does not fall into this scheme. In an expanding universe, one is tempted to turn to explanation in the spirit of the Doppler effect. However, it cannot be simply reduced to this effect (at least globally, see discussion below) since, in particular, coordinate recession velocities can exceed the speed of light. Therefore, in GR, one cannot use straightforward generalization of the Doppler effect in SR which is based on the treatment of physical velocities of the observer and emitter. Meanwhile, we cannot calculate the redshift as it is done in the black hole spacetime because in expanding or contracting Universe there is no timelike Killing vector. Gravitational time delay is also absent in the Friedmann-Robertson-Walker (FRW) coordinate system because this system is synchronous, which means that all particles in the Hubble flow share the same time that coincides with the proper time. For this reason, the cosmological redshift is sometimes treated as a third type of redshift which is distinct from both the Doppler and gravitational ones (see, e.g. Sec. 15 of har ).

Such a classification of redshifts, though being rather useful from a practical point of view (since it prevents the use of incorrect formulae for inappropriate types of redshifts), has its own conceptual problems. The reason for this is, in this picture, cosmology appears to be separated from other GR issues (which seems rather strange), and more importantantly, such a division cannot be done technically in many situations which are slightly more sophisticated than the FRW metrics. For example, should redshifts in the Schwarzschild-de Sitter solution be interpreted in cosmological terms, or it is better to apply terminology from black hole physics?

The goal of the present methodological paper is to show that apparent “cosmological” features of cosmological redshift are not limited only by situations related to cosmology, but, rather, follow from properties of the synchronous coordinate system (for example, the FRW metric), and therefore can be present in any other applications of GR (including black holes) when an appropriate coordinate system is chosen. We think that any graduate student who has chosen General Relativity for a deeper study, as well as young researcher starting to work in this field, should be able to interpet physical phenomena in different frames and understand the relationship between them (our recent experience of supervising graduate students confirms this), and we hope that this paper will help to reach this goal. This ability is especially needed while describing effects connecting with the black hole horizon crossing, because coordinate singularity of commonly used stationary coordinate system has sometimes induced incorrect statements. One of the most famous example is the concept of “ghosts” emitted at a black hole horizon, seen from the zero distance while an observer crosses the horizon ghost1 ; ghost2 . This concept has already been criticized in Kas . We will comment on this point later on.

There are other unifying approaches to redshifts, one of which is based on parallel transport of the observer’s velocity along the lightlike geodesic narl , synge , dop . In contrast, we do not introduce additional constructions and operate with equations of motion directly. Throughout the paper, we use many concepts from GR and relativistic cosmology. We suggest the use of standard textbooks wein ; mtw as references books in GR.

II Cosmological redshift

Derivation of the cosmological redshift is a source of controversy — different authors follow different routes to obtain this quantity. We briefly mention the most popular of them, and then present another one, which will be used in the rest of the paper.

We start with the usual expression for the interval 111Only radial motions are discussed. (see eq. 14.2.1 in wein ):

[TABLE]

Here $c$ is the light speed. Hereafter, we assume radial motion only and suppress angular cooridnates.

The simplest way to obtain the redshift in the FRW metric is to use the conformal time, $\eta$. Defining $\eta(t)=\int dt/a$ we reduce the metric to the form

[TABLE]

Assuming $\chi_{obs}=0$ we obtain $\chi_{em}=\eta_{obs}-\eta_{em}$. Here, subscripts “obs” and “em” refer to an observer and emitter, respectively. If a signal was emitted during a period $\Delta\eta$, then it is observed during the same interval of conformal time. However, physical time intervals at two moments are different: $\Delta t_{em}=a(t_{em})\Delta\eta$ and $\Delta t_{obs}=a(t_{obs})\Delta\eta$. Thus, $\Delta t_{em}/\Delta t_{obs}=a(t_{em})/a(t_{obs})$, where we neglected the change of the scale factor during the processes of emission and observation.

Then, one can use symmetry of a metric to derive the redshift using conservation laws (see pp. 457-458 of mtw ). It is possible to show that in the FRW metric the momentum changes as $a(t)^{-1}$ (see, e.g. eq. 114.5 in Ref. ll ). Then, as for photons momentum $\sim\lambda^{-1}$, we obtain that $\lambda\sim a(t)$. Thus, we derive the usual expression for the redshift.

Both of these methods exploit properties of a metric (in particular, FRW), and so they are not general derivations.

The third way we want to mention is popular, however, it is not fundamental, as a priori it is not obvious that it might provide the correct answer. This method uses integration of the local Doppler effect along the trajectory of a photon, or even just the statement that locally cosmological redshift can be given by the Doppler formula. Such a simplified approach, most often with appropriate comments that this is not an accurate but just illustrative way of derivation, can be found in some textbooks (for example, Ch. 5.2 of ldl ).

The most general method one can often find in textbooks uses null geodesics (see pp. 777 - 778 of mtw ).

For photons

[TABLE]

Thus, we obtain from (1) the expression for the coordinate velocity:

[TABLE]

Let us then consider two pulses of an electromagnetic wave. They are emitted in two moments of time $t_{em1}$, $t_{em2}$ and observed, correspondingly, at $t_{obs1}$ and $t_{obs2}$. Both are emitted by a galaxy at $\chi_{em}$. The observer is situated at $\chi_{obs}$.

If an emitter and an observer follow the geodesic lines, their values of $\chi$ remain fixed since (1) represents a synchronous system. World lines of the two photons are:

[TABLE]

[TABLE]

Then:

[TABLE]

Eq. (6) can be rewritten in the equivalent form

[TABLE]

Neglecting changes in $a$ during the interval of emission of the two pulses, and during the interval of their observation (i.e. putting $a(t_{em1})=a(t_{em2})$ and $a(t_{obs1})=a(t_{obs2})$), we obtain

[TABLE]

At emission $\lambda_{em}=c(t_{em2}-t_{em1})$. At observation $\lambda_{obs}=c(t_{obs2}-t_{obs1})$. Therefore,

[TABLE]

This corresponds to equations (14.3.5), (14.3.6) in wein ). Finally, we come to the usual expression for the cosmological redshift.

In our paper, we use a variation of this method. In our opinion, it is interesting from the methodological point of view and also provides a clear visualization of the process. For our goals here, this version of derivation is needed because, unlike the preceding method, it can be easily generalized to inhomogeneous metrics, which will be used below. Our method essentially relies on using different definitions of velocity; this contains some subtleties discussed below.

III Definitions of velocity and “light faster than light”

We note that the fact that the coordinate velocity of light, $d\chi/dt$, can be not equal to $c$ in some coordinate systems is obvious, because coordinates have no direct physical meaning. What is less obvious is that the velocity of light, defined as derivative of the proper distance from an observer to a photon over proper time (which in the FRW metric is equal to the coordinate time), can be different from $c$ as well.

To work with proper distance, we need to integrate $\sqrt{g_{\chi\chi}}$ over $d\chi$ at the same moment of proper time. This means that we have to define hypersurfaces $t=const$. To do this, it is necessary to fix a coordinate system; then the answer depends on the coordinate system, i.e. is not invariant.

In the FRW metric (1) the proper distance between $\chi=0$ and some $\chi$ at a given moment of time $t$ is equal to $r=\chi a$. If $\chi~{}$is allowed to vary with time, for the velocity $dr/dt$ we can write a chain of equations:

[TABLE]

where $v_{fl}=Hr$ is the Hubble flow velocity and $H={\dot{a}}/{a}$ is the Hubble constant; the dot denotes differentiation with respect to $t$. This clearly demonstrates so-called “superluminal motion” in cosmology. Note that eq. (8) can be considered as a particular case of the general fact that, in cosmology, peculiar velocity is summed with the Hubble velocity following the Galilean law. It is worth stressing that a local velocity of light defined as $a{d\chi}/{dt}$ (the first term in eq. 8) is equal to $c$ according to (3), as it should be.

To make the difference between two definitions of velocity more pronounced, let us denote $P(\chi_{2},t\mid\chi_{1},t)$ as the proper distance between points 1 and 2 at the moment $t$. Then,

[TABLE]

where the local velocity

[TABLE]

and the flow velocity

[TABLE]

It is the quantity $v_{loc}$ that can be found by pure local measurements. For light, $v_{loc}=c$. The quantity ${dr}/{dt}$ is more involved and cannot be detected by direct local measurements. Not surprisingly, it can exceed $c$ without any contradiction with GR and SR.

Now let us derive the equation for the cosmological redshift. Here we will use the fact that two light pulses, being at slightly different distances from an observer, have different velocities with respect to this observer. Therefore, let us consider two neighboring maxima of a light wave directed toward an observer, separated by a spatial distance $\Delta r=\lambda$. This radial difference results in the velocity difference with respect to the observer $\Delta v=H\Delta r$. This means that the velocity of the maximum which is closer to the observer is less than the velocity of the maximum which is behind it. Due to this difference, the distance between these maxima changes and we can construct a differential equation

[TABLE]

After obvious calculations using the definition of the Hubble parameter $H=\dot{a}/a$ we obtain that the ratio of wavelengths at $t_{em}$ and the moment of observation, $t_{obs}$, is equal to the inverse ratio of scale factors at these moments: $\lambda_{em}/\lambda_{obs}=a_{em}/a_{obs}$. This is the standard result for the cosmological redshift.

Below, we want to show how features typical for cosmology can be considered in a rather generic black hole background if a synchronous system is used. In doing so, we generalize the well-known Lemaitre metric connected to an observer free-falling into a black hole. Then, it is clear that the concept of an expanding or contracting space (just discussed in the cosmological context) by itself is not necessary to understand the physics of cosmological evolution.

First of all, the definition of the Hubble flow and peculiar velocity has a straightforward generalization. Indeed, in a synchronous system coordinate lines $x^{i}=const$ are geodesics (see, e.g. Sec. 11.97 of ll ). Observers with constant spatial coordinates share the same time coordinate which coincides with their proper time. One can define a proper distance between two points at some particular time $t$ and also velocity of one point with respect to another by differentiating the proper distance with respect to $t$. For simplicity, we will consider spherically symmetric metrics.

Let $\chi$ be the radial coordinate, $\chi_{2}$ be its value of the observed particle which changes in time, while the coordinate $\chi_{1}$ of the observer remains fixed. Then we get the following, using the well-known formula for the integral with varying limits:

[TABLE]

The velocity appears to be a sum of two components. The first one,

[TABLE]

is fully due to the non-static character of the metric (and can be considered as an analogue of velocity of the Hubble flow). The second one,

[TABLE]

is non-zero only when the radial coordinate of a particle changes in time, so we can treat it as an analogue of peculiar velocity in cosmology. Eqs. (13) – (15) represent the counterpart of cosmological eqs. (8) – (11). Of course, for light locally $v_{loc}$ is always equal to $c$.

IV Static spacetime and relationship between frames: Lemaitre vs.

Painlevé–Gullstand metrics

In what follows, we consider a static metric in a “cosmological manner”. This includes the black hole spacetime. We show similarity between phenomena in black hole physics and cosmology, and demonstrate that, looking different at the first glance, these processes have the same interpretation in a synchronous metric. We discuss a black hole metric of a rather general form. Without the loss of generality, we take it to be

[TABLE]

Hereafter, we put the fundamental constant $c=1$. This equation includes the Schwarzschild metric ($f=1-{r_{g}}/{r})$, as well as the Reissner-Nordström one, etc. Extension to the case $g_{00}g_{11}\neq-1$ is straightforward. The event horizon lies at $r=r_{g}$, so $f(r_{g})=0$. Near the horizon, the metric becomes degenerate; this can be remedied by introducing a new set of coordinates.

Making the coordinate transformations along

[TABLE]

[TABLE]

one can reduce the metric to the form

[TABLE]

If $f=1-{r_{g}}/{r}$, we return to the well-known Lemaitre metric ll . Therefore, we will call (19) the generalized Lemaitre (GL) frame.

Since the GL metric is synchronous, all free falling radial observers have the constant comoving radial coordinate $\rho$ and share the same proper time. It appears that the proper distance is related with $r$ in a very simple way. Namely, fixing $\tau$, one finds from (17) – (19) after simple calculations that the proper distance between points 1 and 2 equals

[TABLE]

In what follows, we will use $r$ along with the radial coordinate $\rho$. We can see here some analogue of the cosmological situation with the FRW metric where the proper distance between two points is given by the formula $d=a(\chi_{2}-\chi_{1})$, which is valid for any observer moving in the Hubble flow.

It is worth noting that for massive particles composing the GL system $E=m$, $\rho=const$ and thus $v_{fl}=v$, $v_{loc}=0$. For a photon $\left|v_{loc}\right|=1$.

One can also make one more step and write down the metric in a “mixed” form using a new time variable $\tau$ and old radial variable $r$. Then, eliminating $dt$ we have from (17), (18):

[TABLE]

In the Schwarzschild case, the metric (21) is nothing else but the so-called Painlevé–Gullstand metric (see, e.g. regular and references therein). For a generic $f$, it is natural to call it the generalized Painlevé–Gullstand metric (GPG). It is worth noting that it is metric (21) which was used in river to describe “a river model of black hole” as an interesting counterpart of the cosmological concept of “expanding space”. Strange as it may seem, the relationship between the Painlevé–Gullstand and Lemaitre metrics, to the best of our knowledge, was not noticed before.

In these coordinates, some properties of the metric look rather simple. In particular, the hypersurfaces $\tau=const$ are flat. Eq. (20) becomes a direct consequence of (21). For a massive particle falling along geodesic from the rest at infinity, one has ${dr}/{d\tau}=-\sqrt{1-f}$, whence we have for the velocity $v=\left|{dr}/{d\tau}\right|$:

[TABLE]

If points 1 and 2 move along geodesics, their relative velocity is:

[TABLE]

For light propagation we have:

[TABLE]

where $v$ coincides with (22). Here, sign “+” corresponds to propagation outward and “-” corresponds to propagation inward.

V Behavior of wavelength

Now, by the same method as in the FRW cosmology, we derive the redshift of a photon emitted by a free falling particle at $r_{em}$ and absorbed by another free falling particle at $r_{obs}$ . Remember that the redshift in the FRW frame appears because the proper distance between two light pulses changes with time since the speed of light with respect to a distant observer depends on the distance between the pulse and the observer. The same situation appears in the Lemaitre metric, as we will see soon. Our goal is to start with two maxima of light wave separated by $\Delta r=\lambda_{1}$ and trace the evolution of this separation during light propagation. An observer will see the maxima separated by $\lambda_{2}$ which gives us the redshift via $1+z=\lambda_{2}/\lambda_{1}$. As usual for synchronous systems, the coordinate velocity of light is $v_{fl}+c$. Hereafter, we put c=1 for simplicity. For two points with radial difference $\Delta r$ at $r_{em}$ (here, the dependence on radial coordinate at emission is important, because the metric is not homogeneous) corresponding velocities of light differ by $\Delta v_{fl}=v_{fl}^{\prime}(r)\Delta r.$ In this Section, we consider free-falling particles co-moving with respect to the GL system, so peculiar velocities vanish and $v_{fl}=v$.

It is worth noting that while we need to subtract the velocity of the observer in order to get relative velocity (23), this does not affect the difference between velocities of nearby points $\Delta v$, so we can consider the position $r_{0}$ of the observer as arbitrary. For simplicity, we consider free-fall velocities with respect to an infinitely distant observer.

Thus, if at emission the distance between two wave maxima was $\left|\Delta r\right|\equiv\lambda$, it later increases according to

[TABLE]

if $v$ depends upon $r$ only (this is true for all static spherically symmetric spacetimes). This equation is an analogue of the cosmological equation (12). It is useful to integrate over the radial coordinate $r$ (instead of the Lemaitre coordinates $\rho$ and $\tau$). Let us consider an inwardly moving photon so that, according to (24), $d\tau=-{dr}/{(1+v)}$. Then we get:

[TABLE]

and finally

[TABLE]

Note, that in the FRW cosmology $v$ is not a function of time only in the de Sitter universe where $dv/dr=H=const$.

Since for the photon moving inward, the sum $v+1$ never crosses zero, the differential equation is regular at the event horizon, so the solution should give us the redshift of such a photon for both possible locations of an emitter and an observer (inside or outside the event horizon).

Then, we have the ratio of the initial and final wavelengths for an inward motion of a photon:

[TABLE]

For the outward motion of a photon, in a similar way one would obtain

[TABLE]

These results are in agreement with those of ras2013 and Kas (in the latter case they were derived with the help of the Kruskal coordinates). As noted above, in FRW cosmology, these formulas are valid only for the de Sitter solution (see more comments on this in the following). For other regimes, the dependence between the redshift and the velocity at emission (in FRW metrics, it is reasonable to set the velocity of observer to zero) is more complicated; see, for example, our1 .

Going back to static spacetimes, we can use (22) to get the general expression for the redshift through radial coordinate of emission and observation:

[TABLE]

where the upper sign is used for ingoing photon, and the lower sign for the outgoing photon.

The latter case can be applied if observer and emitter are either both outside the horizon (in this situation evidently $r_{em}(\tau_{em})<r_{obs}(\tau_{obs})$), or both inside the horizon (and $r_{em}(\tau_{em})>r_{obs}(\tau_{obs})$). On the contrary, there are no restrictions for the locations of emitter and observer with respect to the horizon for the ingoing photon case.

If, for example, an observer crosses the horizon, $f_{obs}=0$, then according to (22), $v_{obs}=1$. Thus, he/she receives a signal from a remote source at infinity ($f_{em}=1$) with the redshift $z=1$.

If the observer approaches the singularity where $r\rightarrow 0$ and $f\rightarrow-\infty$ (like in the Schwarzschild metric where $f=1-{r_{g}}/{r}$), $v_{em}\rightarrow\infty$, so the redshift diverges.

It is clear from the very method of derivation that these results are quite general and are valid far beyond GR. It is necessary, of course, that free massive particles follow geodesics; this holds in metric theories of gravity. Also, it is necessary that photons move along null geodesics. This is achieved if the coupling between electromagnetism and gravity is minimal. This second condition is fulfilled in most popular generalizations of GR, though may be violated for some class of theories studied by Horndeski horn .

It is instructive to illustrate some formulas just derived using the Schwarzschild metric as an example. This is the most simple and, at the same time, physically relevant metric describing a black hole. In this case, $f=1-\frac{r_{g}}{r}$, where $r_{g}=2M$ is the horizon radius in geometric units (the fundamental constants $G=c=1$), $M$ being a black hole mass.

Eq. (30), describing the frequency shift for freely moving particles, gives us

[TABLE]

If a photon is emitted at infinity and registered when the observer crosses the horizon (so $r_{em}\rightarrow\infty$ and $r_{obs}=r_{g}$), (31) gives us $z=1$. Thus the frequency is as two times as little as compared to the original one, in agreement with a general property noted above.

One can ask whether our approach can describe stationary metrics which are not asymptotically flat? The main example of such a case is the Schwarzschild-de Sitter metric $f=1-\frac{2M}{r}-\frac{\Lambda r^{2}}{3}$, where the cosmological constant $\Lambda>0$. In principle, our approach needs some modifications in this case because we cannot use the condition that the velocity of free fall is equal to zero in spatial infinity (we remind a reader that the peculiar velocity is set to zero in the present paper). We leave a careful treatment of this case to a future work.

However, we can rather easily incorporate into our scheme the pure de Sitter metric which, in stationary coordinates, has $f=1-\frac{\Lambda r^{2}}{3}$. In this case, we can demand zero velocity at the coordinate origin (instead of infinity). This corresponds to the usual definition of zero peculiar velocity in cosmology. It is worth noting that since the direction of a “free fall” is changed – it is now directed to bigger values of radial coordinate – the signs in formulae for inward and outward photons interchange.

If, in addition to this, we set an observer at the origin of coordinate system (as usually done in FRW cosmology), we can see that Eq. (29) for the redshift is still valid and coincides with the known cosmological result (see, for example our1 ). As for Eq.(22), it gives us the famous Hubble law $v=\sqrt{\frac{\Lambda r^{2}}{3}}=Hr$. Obviously, this happens because the synchronous frame for the de Sitter metric is just the FRW frame (1).

V.1 Special case: propagation along the horizon

One particular case of an outward propagation of a photon is still beyond our analysis. Namely, if a photon is emitted exactly at the horizon and further propagates along it, we cannot integrate over $r$ because it is constant for such photons (but the Lemaitre coordinate $\rho$ obviously changes).

Instead, we use directly (25) with $\Delta r$ replaced with $\lambda$ and $r=r_{g}$, whence

[TABLE]

and $1+z=\exp[v^{\prime}(r_{g}){(\tau_{obs}-\tau_{em})}].$ Using (22) and the fact that $f(r_{g})=0$, $f^{\prime}(r_{g})=2\kappa$, where $\kappa$ is the surface gravity, we obtain:

[TABLE]

Thus, in this case, the redshift can be expressed through the difference between emitting and observing time, in agreement with the result obtained in along by another method. In the particular case of the Schwarzschild metric, the same problem has been considered earlier using Kruskal coordinates Kas .

Thus, we demonstrated that the Lemaitre coordinates enable us to obtain the final value of the redshift easily. It is also easy to see that the distance from the emitter to the observer is finite and well-defined (either at the time of emission or observation). On the contrary, the radial coordinate $r$ of emission and observation is the same, which sometimes leads to a wrong interpretation mentioned in the Introduction. The Lemaitre coordinates prevent our intuition from making such wrong statements.

VI Frequencies in different frames

In this section, we obtain the same results for redshifts in a more formal way, as it is usually done if problems of interpretations are not concerned.

The geodesic equations for radial light propagation in the static frame are the following (see Eq. 16):

[TABLE]

[TABLE]

where, for definitness, we consider motion in the inward direction only. Here, $k^{\mu}$ is the wave vector, $\omega_{\infty}$ has the meaning of the frequency measured by a remote observer at infinity.

Using (17) and (18), one can find

[TABLE]

[TABLE]

The frequency measured by a local observer with the four-velocity $u^{\mu}$ is:

[TABLE]

where $u^{\mu}={dx^{\mu}}/{d\tau}$ is the four-velocity and $\tau$ is the proper time.

For the GPG observers $\rho=const$, so $u^{\tau}=1$, $u^{\rho}=0$. Then, we have from (37) that

[TABLE]

In terms of the velocity (22), it can be rewritten as

[TABLE]

If light is emitted at $\rho_{em}$ and is absorbed at $\rho_{obs}<\rho_{em}$,

[TABLE]

which coincides with (28), as it should be. If $\rho_{obs}>\rho_{em}$, we again obtain (29).

VII From the GL frame to the static one

In the above analysis, we have followed the usual cosmological pattern to derive the redshift. We can also use these results to get the standard redshift in the Schwarzschild geometry, which is usually obtained using the notation of gravitational time delay, and is described as a typical example of a gravitational redshift. This concerns an observer and emitter with a fixed Schwarzschild radial coordinate $r$. We want to reproduce it using the above formulae without any references to the gravitational “time delay”.

The preceding formulas describe redshift (or blueshift) in the GL frame, when frequencies or wavelengths are measured by the corresponding free falling observers who have $\rho=const$. Now we are interested in a redshift in the static frame. To obtain it, it is sufficient to make a boost from the GL frame. It follows from (17), (18) that, for an observer or emitter to be fixed at a constant $r$, it must have $\rho=\tau+const.$ Then, it moves with the velocity $v={dl}/{d\tau}$ (22) with respect to the GL frame. Therefore, we need to combine the redshift (28) with the Lorentzian boosts corresponding to the peculiar motion of the emitter and the observer (i.e. the Doppler effect).

Let us assume that light propagates in the outward direction. Using the standard formula for the Doppler redshift, we get in the static frame for the emitter

[TABLE]

where subscripts “st” and “GL” indicate the static and GL frames, respectively. The same formula holds for the observer.

Then,

[TABLE]

Taking into account (29), we obtain

[TABLE]

Here, we reproduced the well-known result (see, e.g. Sec. 2.3.1 in fn ) which is ascribed usually to the gravitational time delay. Note that this concept does not show up explicitly in the presented derivation, and we have only a combination of a “cosmological” redshift with the usual SR Doppler formula. If $r_{obs}>r_{em}$, light propagates outward, $f_{obs}>f_{em}$, $z>0$, and we have redshift. If $r_{obs}<r_{em}$, light propagates inward, $z<0$, and the resulting effect is blueshift.

In the static coordinate system, the interval of time $t$ between two pulses remains the same during light propagation. As a result, the redshift is the ratio of values of the function relating proper time with the coordinate time $t$ at the moments of emission and detection. In this situation, it is possible to think that the time goes slowly in regions of large $f$. Moreover, if we imagine a space traveller who visits a vicinity of the event horizon and then returns, he/she will be younger than his twin staying on Earth. This is true even if the traveller has enough rocket fuel to move towards the black hole and back very slowly, so that it is hard to attribute the different time passed to usual Lorentz time dilation. However, it is necessary to keep in mind that such an interpretation is connected with a particular coordinate system rather than with spacetime itself. In the Lemaitre coordinates, there is no gravitational time delay, and the black hole twin paradox can be explained solely by SR Lorentz time dilation – to return back, the traveller must apply a huge Lorentz boost with respect to his “natural” free-falling frame.

Formula (43) refers to a standard situation when both an observer and an emitter are at rest in a static gravitational field. It is instructive to remind the reader that there is an analogue of this situation in cosmology. This corresponds to the case of so-called tethered galaxies tethered . In this case, the cosmological expansion is exactly compensated due to a peculiar velocity towards an observer, so the proper distance remains constant. However, the detected emission can be redshifted or blueshifted depending on a cosmological equation of state. This fact has been remarked on only recently in tethered and shows clearly that the cosmological redshift cannot be considered as a SR Doppler shift in an expanding Universe. In the case of black holes for free-falling observer and emitter, as we want to use the Lemaitre metric or its generalization we can mimic this situation in the limit $r_{obs}\rightarrow\infty$. Then we can approximately consider that the observer is at rest, as it is in the cosmological picture, where an observer has zero peculiar velocity. To have $r=const$, it is necessary for the emitter to have a peculiar velocity which exactly compensates the free-fall velocity. The emission will be always detected with a redshift. Usually this situation is considered in static coordinates, and it is interpreted in terms of gravitational time delay. However, we see that we can use a different interpretation, saying that in Lemaitre metric the non-zero redshift of an object at a constant proper distance has the same nature as non-zero redshifts of tethered galaxies. In the syncronous system one can also define the light travel distance as $D_{l}=c(\tau_{obs}-\tau_{em})$. It follows from the staticity of the metric and the constancy of $r_{em}$ and $r_{obs}$ that $D_{l}=const$. Therefore, an object with constant light travel distance is always redshifted — the property which is still valid in the FRW cosmology we2 .

VIII Conclusion

In this methodological note, we proposed another interpretation of a redshift in spherically symmetric static spacetimes, including black holes. We split the calculation of the effect into two parts. The first part includes transition to the synchronous system that is formed by free-falling particles. Here, the situation is very similar to that in cosmological FRW spacetime. In the second step, we return to the static system by means of the local Lorentz boost. This enables us to consider it as a kind of the Doppler shift and apply well-known corresponding formulas. It is worth noting that calculations on step 1 are realized by us by two methods. We found (i) the frequency from equations of motion, (ii) the wavelength taking into account that different points move with different velocities (in analogy with cosmology). Thus, we see that there is no crucial difference between interpretations of the redshift in cosmological and black hole spacetimes. Rather, the situation in the latter case can be thought of as a combination of the former one and the Doppler effect.

The obtained results have a rather curious consequence. As is noticed above, an observer crossing the horizon registers a signal from infinity with $z=1$. This allows the observer to identify the moment of crossing the event horizon. In doing so, an observer should use information about the photon emitted at infinity whereas he is unable to do this using only local measurements. (It is instructive to remind the reader that another kind of pure local measurements can in principle locate the moment of the horizon crossing formally, but that method is rather sophisticated and does not operate with entities having direct meaning in physics petrov .)

When speaking about the cosmological spacetimes throughout the paper, we implied the FRW spacetimes as opposed to the static ones such as the Schwarzschild metric. For completeness, we have to mention that there exist also special cases in which cosmological spacetimes themselves admit presentation in a static form. This includes the de Sitter and Milne solutions when both forms (explicitly static and pure cosmological) are possible. In melia , six cosmological metrics with stationary frames have been listed and the corresponding redshifts studied. It was shown that in all these cases the cosmological redshift can be represented as a combination of SR Doppler shift and gravitational time delay, and so it cannot be consider as a “fundamental” shift. In our paper, we did just the opposite, showing that gravitational redshift in a stationary frame can be represented as a combination of the Doppler shift and a ”cosmological” redshift. This means that the question of what type of redshift should be considered the more fundamental one is a matter of convenience. On the other hand, we can argue that since a synchronous system can be constructed for any spacetime and static systems exist only for certain particular metrics, the “cosmological” description of a redshift presented in our paper has more general significance.

Acknowledgements.

We thank anonymous referees for their useful comments and suggestions. This work was funded by the subsidy allocated to Kazan Federal University for the state assignment in the sphere of scientific activities. The work of OBZ was also supported by SFFR (Ukraine), project 32367. SBP acknowledges support from the Russian Science Foundation (grant 14-12-00146).