Physics of the Hubble expansion Sergei M. Kopeikin

Local gravitational physics of the Hubble expansion.
Einstein’s equivalence principle in cosmology.
By Sergei M. Kopeikin,
Of the Department of Physics and Astronomy, University of Missouri, Columbia, Missouri, and
Siberian State Geodetic Academy, Novosibirsk, Russia.Show more

Abstract.
We study physical consequences of the Hubble expansion of Friedmann-Lemaıtre-Robertson- Walker (FLRW) manifold on measurement of space, time and light propagation in the local inertial frame. We use results of this study to analyse the solar system radar ranging and Doppler tracking experiments and time synchronization. The FLRW manifold is covered by the coordinates (t, and y), where t is the cosmic time coinciding with the proper time of the Hubble observers and identified with the barycentric coordinate time (TCB) used in ephemeris astronomy. We introduce local inertial coordinates x alpha equals (x zero, x i) in the vicinity of a world line of a Hubble observer with the help of a special conformal transformation that respects the local equivalence between the tangent and FLRW manifold. The local inertial metric is Minkowski flat and is materialized by the congruence of time-like geodesics of static observers being at rest with respect to the local spatial coordinates x i. The static observers are equipped with the ideal clocks measuring their own proper time which is synchronized with the cosmic time t measured by the Hubble observer.
We consider the geodesic motion of test particles and notice that the local coordinate time x zero, equals x zero (Of t) taken as a parameter along the world line of particle, is a function of the Hubble’s observer time t. This function changes smoothly from x zero equals t for a particle at rest (observer’s clock), to x zero equals t plus a half H t squared for photons, where H is the Hubble constant.
Thus, motion of a test particle is non-uniform when its world line is parametrized by the cosmic time t. NASA JPL Orbit Determination Program operates under assumption that spacetime is asymptotically flat which presumes that motion of light (after the Shapiro delay is excluded) is uniform with respect to the time t but it does not comply with the non-uniform motion of light on cosmological manifold. For this reason, the motion of light in the solar system analysed with the Orbit Determination Program appears as having a systematic blue shift of frequency, of radio waves circulating in the Earth-spacecraft radio link. The magnitude of the anomalous blue shift of frequency is proportional to the Hubble constant H that may open an access to the measurement of this fundamental cosmological parameter in the solar system radiowave experiments.

In other words, the assumption of a local, asymptotically flat Minkowski space-time in which the solar system physics evolves does not take into account the effect of the expansion of the universe, and measureable deviations can be obtained.

Introduction.
Modern physics is intensively looking for the unified field theory that might explain the origin of the universe and the underlying fundamental nature of space-time and elementary particles. This work requires deeper understanding of the theoretical and experimental principles of general relativity. It is challenging to find a new type of experiments that broaden the current knowledge. An appealing problem is to examine a presumable link between the local gravitational phenomena and the global cosmological expansion of the universe that is to test the foundational basis of the Einstein equivalence principle (EEP) in application to a conformal cosmological metric with a time-dependent scale factor.

The EEP in general relativity is universally valid because the gravitational field in general relativity has a purely geometric nature. It is always mathematically possible to find a local diffeomorphism which reduces any global metric to a Minkowski metric in a sufficiently small neighborhood of a time-like world line of an observer if tidal forces are neglected. This mathematical fact was a clue that led Einstein to the formulation of his general principle of relativity, also known as the principle of covariance, and, later on, to the discovery of general relativity as a physical theory of gravitational fields.
The apparent mathematical nature of EEP caused some physicists to deny its physical significance. The present paper neither shares this extremal point of view nor confronts the solid mathematical foundation of EEP. We focus on physical aspects of EEP, namely:

One. The comparison of the inertial motion of test particles on cosmological manifold considered from the local point of view of a Hubble observer,
Two. The derivation of experimental consequences that can be used for testing the Hubble law in local solar system experiments.
So far, all gravitational experiments in the solar system have been interpreted under a rather natural assumption that the background spacetime geometry is asymptotically flat covered by coordinates (t, and y) with the background Minkowski metric:

D s squared equals minus d t squared plus delta I, J d y I, d y j, equation one.
With Delta I, J the unit matrix, with a diagonal of one, one, one, and we have used a convention for the speed of light, c equals one.

The time t entering the metric is identified with the barycentric coordinate time (TCB) of the solar system according to the IAU 2000 resolutions.

On the other hand, theoretical and observational cosmology postulates that the background space-time is described by the Friedmann-Lemaıtre-Robertson-Walker (FLRW) metric:

D s squared equals minus d t squared plus R squared (Of t) over the square of one plus one quarter k r squared, times delta I, J, d y I, d y J, equation two.
Which is to say that the Minkowski metric is effectively multiplied by a function of t, r, and k,
Where: t is the universal cosmic time, y i are the global isotropic coordinates, k is either minus one, zero or one and defines the curvature of space, and the scale factor R( Of t) is a function of time found by solving Einstein’s equations.

The cosmic time t is the proper time measured by observers having fixed spatial coordinates y i. Therefore, it is exactly the same as time t in the flat metric (equation one) and identifies with TCB of fundamental astronomy in the solar system.

In what follows, we admit k equals zero in accordance with observations and limit ourselves with the linearized Hubble approximation. In other words, we consider only terms being linear with respect to the Hubble constant H and neglect all terms that are quadratic with respect to H or proportional to its time derivative H dot, as H dot is around H squared.
We shall also neglect post-Newtonian gravitational effects of the solar system which must be included in the realistic data analysis of observations in the solar system. These effects are described in other references, and can be easily accounted for by superposition, if necessary.

The FLRW metric (equation two) is not asymptotically-flat and has a non-vanishing space-time curvature tensor R alpha, beta, gamma, delta where the Greek indices alpha, beta, gamma, delta take values zero, one, two, three. Nonetheless, the Weyl tensor of FLRW metric, C alpha, beta, gamma, delta is by definition zero.
Hence the metric of equation two can be reduced to a conformally-flat metric for any value of space curvature k.

When k equals zero, it is achieved by transforming the cosmological time t to a conformal time, eta equals eta (Of t), defined by an ordinary differential equation:
D t equals “A.” (Of eta) d eta,
Where the scale factor “A.” (Of eta) is R Parameterized by T (Of eta).
This time transformation allows the rewriting of the LMRW metric into the conformally Minkowski form of equation four:

D s squared equals “A.” squared (Of eta) f alpha, beta d y alpha, d y beta.
Here y alpha is y zero, y I, equal to eta, y I, the global conformal coordinates,
F alpha, beta has the diagonal of (minus one, one, one, one), and is the Minkowski metric.

In other words, although a flat space Minkowski metric is used by the IAU, theoretical and observational cosmology postulates the LMRW metric, and by transforming by the use of a scale factor “A.” (of eta), the LMRW can be approximated by a Conformally Minkowskian metric, with the Minkowski metric merely multiplied by a function of eta.

According to Einstein’s general relativity and the definition of FLRW metric, the cosmological time t is a physical proper time of the Hubble observer and can be measured with the help of the observer’s atomic clock while the conformal time eta is a convenient coordinate parameter which is calculated from the clock’s reading but cannot be measured directly.
Typically, the cosmological metric (equation four) is applied to describe the properties of spacetime on the scale of galaxy clusters and larger. On small scales of the size of the Milky Way, the solar system and terrestrial lab, the background spacetime is believed to be flat with any cosmological effect being strongly suppressed. Nevertheless, the question remains open: if we admit FLRW metric to be valid on any scale, can the cosmological expansion be detected in local gravitational experiments?
Following others, we postulate that FLRW metric (equation four) is a physical metric not only in cosmology but for the description of the local physics as well. It describes the background spacetime geometry in the global coordinates y alpha on all scales spreading up from the cosmological horizon to the solar system and down to a local observer. The small parameter in the approximation scheme used in the present paper, is the product of the Hubble constant, H, with the interval of time used for physical measurements.
All non-linear terms of the quadratic order with respect to the small parameter (formally, the terms being quadratic with respect to H) will be systematically neglected because of their smallness.
We introduce the reader to the concepts associated with the Einstein equivalence principle in section 2 and discuss construction of the local inertial coordinates in section 3. The inertial frame is built in sections 4. Light geodesics in local coordinates are derived in section 5. We solve these equations in section 6 and employ them for investigation of observability of cosmological effects in the solar system. Discussion is provided in section 7.

Two: Einstein’s principle of equivalence.

A thorough treatment of the local astronomical measurements on cosmological manifold inquires a scrutiny reexamination of Einstein’s equivalence principle (EEP) which states: “In a given gravitational field, the outcome of any local, non-gravitational experiment is independent of the freely-falling experimental apparatus’ velocity, of where and when in the gravitational field the experiment is performed and of experimental technique applied”. Mathematical interpretation of EEP suggests universality of local geometry in the sense that at each point on a spacetime manifold with an arbitrary gravitational field, it is possible to choose the local inertial coordinates such that, within a sufficiently small region of the point in question, all laws of nature take the same form as in non-accelerated Cartesian coordinates.

EEP is applicable in general relativity to any kind of spacetime manifold, in particular, to the manifold of the FLRW universe, which is described by the metric of (equation four).
We noticed in previous work, that due to the expanding nature of space in an FLRW manifold, the parametric description of the propagation of light given in local inertial coordinates in terms of the proper time of observer, differs from that in the Cartesian coordinates of flat spacetime.

Let us consider a Hubble observer who is at the origin of a local inertial coordinate system (LIC), x alpha equals (x zero, x i). The Physical metric g alpha, beta equals “A.” squared (Of eta) f alpha, beta, given by (equation four) in the global coordinates, is reduced to the Minkowski metric, f alpha, beta, at the origin of LIC with the affine connection being nil, Gamma alpha, mu, nu (Of x) equals zero, on the observer’s world line.
EEP asserts that the worldlines of freely falling (electrically-neutral) test particles and photons are geodesics of the physical metric g alpha, beta with an affine parametrization. Because the affine connection is nil in LIC, it presumes that the geodesic equations of motion of all test particles – massive and massless, can be written down as follows:

D two x alpha, d sigma squared equals zero,

Where sigma is the affine parameter along the geodesic. Equation (five) neglects the tidal (caused by the Riemann curvature of FLRW spacetime) effects, which produce terms of the order of H squared which we discard. Solving (equation five) for the time component shows that the parameter sigma can be chosen equal to the coordinate time x zero of LIC, that is:
Sigma equals x zero.
The local coordinate time, x zero, must be further operationally connected to the proper time t measured in LIC by the central Hubble observer.

The time x zero is often identified with the proper time of observer t but one must keep in mind that this identification is true only for static observers being at rest with respect to the central Hubble observer. In general, the local coordinate time x zero is a nonlinear function of t on world-lines of moving test particles.
Therefore, changing the affine parameter sigma to the non-affine (but directly measurable) parameter t brings (five) to the following form, of equation six:

D two x alpha, d t squared equals d x alpha, d x zero, times d two x zero, d t squared.
The cosmic time t coincides with the proper time of the central Hubble observer in the absence of any gravitational perturbations caused by massive bodies of the solar system like Sun and planets. Real experiments demand to include the effect of gravitational field of the solar system on time transformations but they are well-known and can be easily taken into account. In ephemeris astronomy the time t is identified with the barycentric coordinate time (TCB) which is considered as a uniform global time scale. Equations for transformation of the proper time tau of any observer within the solar system to TCB are given by the corresponding IAU resolutions. This transformation shows that tau differs from t by small relativistic terms which are not essential for further discussion.

The NASA JPL Orbit Determination Program that is used for spacecraft navigation and calculating planetary and lunar ephemerides, assumes that for any particle including photons, x zero equals t. It means that the right side of (six) is postulated to be nil, or, equation seven:

D two x alpha, d t squared equals zero.
This equation yields the photon’s world line as: X zero equals t, X I equals x zero I plus U I t, where U I is a unit vector in the direction of the photon’s propagation and we assumed that light passes through the point x I zero of the local inertial coordinates at instant t equals zero which fixes the integration constants. It establishes a linear relationship between the spatial coordinates x i of the photon and the proper time t of the observer at the origin of LIC, which is a directly measurable quantity, after accounting for the IAU time transformations.

Equation (seven) does not show the presence of the Hubble constant, H. It led scientists to believe that EEP cancels out all cosmological effects of the linear order of the Hubble constant H, and that prevents astronomers from observing them in solar system experiments. However, equation (seven) and its solution are incomplete as the relation between x zero and t is not a linear function of time so that light does not propagate uniformly with constant velocity.

This non-uniform propagation of light in local coordinates may appear as an anomaly and a violation of EEP for photons but this is just a mathematical consequence of the geometric expansion of space in FLRW universe. This effect makes possible measurement of the Hubble expansion in the solar system in the local experiments like the Doppler tracking of spacecraft in deep space.

The local inertial coordinates.
In order to interpret local astronomical measurements, like radar ranging, spacecraft Doppler tracking, and so on, we have to construct the local inertial coordinates (LIC) in the neighborhood of a time-like world line of an observer. We focus on building LIC in the vicinity of a Hubble observer, which by definition has constant spatial coordinates:
Y I equal a constant value, of the FLRW metric and moves along a time-like geodesic world-line. Real observers move with respect to the Hubble flow and experience gravitational forces from the massive bodies of the solar system.
Therefore, construction of LIC for a real observer requires determining additional coordinate transformations which are known and can be found, so that they are not a matter of concern of the present paper.

Let us put the Hubble observer at the origin of the LIC, with X I equals zero, so that the world-line coincides, then, with the timelike geodesic of the observer. The Hubble observer carries an ideal clock that measures the parameter of the observer’s worldline, which is the observer’s proper time t.
The proper time t of the Hubble observer coincides with the cosmological coordinate time t in the definition of the Lemaıtre-Robertson-Walker (FLRW) metric of equation two.
The Einstein equivalence principle suggests that in a small neighborhood of the worldline of the observer, called a tangent spacetime, there exists a local diffeomorphism from the global, y alpha, to local, x alpha, coordinates such that the physical metric:
G alpha, beta (Of y) equals “A.” squared (Of eta), times f alpha, beta,
Is transformed to the Minkowski metric, f alpha, beta, as follows in equation eight:
“A.” squared (Of eta) f mu, nu d y mu, d x alpha, d y nu, d beta equals f alpha, beta.

This is an essential point, wherein the local flat space the metric must transform to the global FLRW metric, in which the transformation is using terms proportional to the Hubble constant H, and ignoring higher order terms. The suggestion is of a local transformation, which therefore will change on position, from global to local inertial coordinates.

Where all tidal terms of the order of Hubble squared have been omitted as negligibly small. In the tangent space-time where (eight) is valid, the physical space-time interval of the conformally-Minkowskian form of equation four can be written down in local inertial coordinates. It reads:

D S squared equals f alpha, beta d x alpha, d x beta.

Where f alpha, beta is understood as the physical, global metric g alpha, beta (Of x) expressed in the local coordinates.

Equation (eight) looks similar to the special conformal transformation establishing a conformal isometry of the Minkowski metric, which is that of equation ten:

Omega squared (Of x) f mu, nu d y mu, d x alpha, d y nu, d x beta equals f alpha beta.
Since the required diffeomorphism is local, it is a function of the local x co-ordinates, and hence the transformation can be identified with the conformal isometry of the Minkowski metric.
“A.” squared (Of eta) is here replaced with Omega squared (Of x).
The transformation Omega (Of X) can be written as:

Omega (Of x) equals one minus two times the contraction of B alpha with x alpha, plus b squared times x squared.
In this transformation:
B alpha is a constant four-vector yet to be specified,
X squared is defined as f alpha, beta X alpha, X beta, and:
B squared is defined as f alpha, beta B alpha, B beta.
The special conformal transformation includes inversions and translations, and is equivalent to equation thirteen:
Y alpha equals x alpha minus b alpha x squared, all divided by Omega (Of x).

Let us assume for simplicity that the origin of LIC, x I equals zero, coincides with the point having the global spatial coordinates, y I equals zero. As the background manifold is assumed to be analytic, equation (eight) should match (ten) in a small neighbourhood of the origin of the LIC. The matching can be achieved by demanding the scale factor of the FLRW metric, “A.” (Of eta) equals Omega. This equality is valid in any arbitrary cosmological model if we discard the curvature terms being proportional to around H squared and, or the time derivative of H. Indeed, for small values of the conformal time eta we have the expansion:

“A.” (Of eta) equals “A.” (Zero) plus d “A.”, d eta (At zero) times eta, plus higher terms in the derivatives.

Where we assume that the present epoch corresponds to eta equals zero in the conformal time. We normalize the scale factor at the present epoch to “A.” (At zero) equals one. Then, at the present epoch the Hubble constant H equals d “A.”, d eta (At zero).
The second time derivative of the scale factor two “A.”, d eta squared equals d H, d eta plus two H squared, and we drop it off as being negligibly small.
Assuming that the constant vector, b alpha is of the order of the Hubble constant H, we approximate the conformal factor, Omega (Of x) equals one minus two b alpha, x alpha, by neglecting terms of the order of b squared.
In keeping to first order in terms of the Hubble constant H, we have the equivalence that:
“A.” (Of eta) equals “A.” (Zero) plus d “A.”, d eta (At zero) times eta equals:
Omega (Of x) equals one minus two b alpha, x alpha.
Since “A.” (zero) has been set equal to one, and d “A.”, d eta (at zero) equals H, this yields:

“A.” (Of eta) equals: One plus H eta, equals Omega (Of x), which equals: One minus two b alpha, x alpha.
Or eta equals x zero plus linear terms of Order (B).

Equating terms, b alpha equals H over two times: U alpha, equals a four vector of components:
H, over two, zero, zero, zero. It is directed along the four-velocity u alpha equals (one, zero, zero, zero) of the Hubble observer, and is time-like.

The reader may notice that the special conformal transformation has a singular point, x alpha equals minus b alpha over b squared, that goes over to t equals two over H. It means that the special conformal diffeomorphism (thirteen) is approximately limited in time domain by the Hubble time, T H equals one over H, calculated for the present value of the Hubble parameter:
H is approximately two point three times ten to the minus eighteen per second.
However, because LIC have been derived under assumption that the series in “A.” (Of eta) is convergent, H t is far less than one, the period of time for which the local inertial frame is really valid is much smaller than the Hubble time and is given by, t far less than T H. Because of this limitation imposed on the time of applicability of the local frame, the local coordinates are also bounded in space by the radius, r far less than Hubble radius, where R H equals c times T H is the Hubble radius of the universe.
The conclusion of this paragraph is that the LIC can be employed only for sufficiently close objects in the universe with the redshift factor z far less than one, which excludes the most distant quasars and galaxies. Therefore, the formalism of the present paper is not applicable to the discussion of global cosmological properties and, or effects like the red shift of quasars. More stringent results on the domain of applicability of the local inertial coordinates in cosmology can be found in the literature.
In review:
Because the local coordinates X I, utilizing a Minkowski metric, are embedded in a global Friedmann-Lemaıtre-Robertson-Walker (FLRW) that utilizes a coordinate system Y I, and limiting the transformation between them to first order in the Hubble constant, or dimensions small compared to the Hubble radius, there is a linear transformation between the local time t, and the Cosmological time eta.
The conformal factor thus obtained, Omega (Of x) is one minus H over two U alpha, x alpha, or: one minus H, times U four dot X four.

The association ensures that LIC can be constructed in the linearized Hubble approximation from the global coordinates, y alpha, by means of the special conformal transformation (thirteen) that respects EEP as the procedure demonstrates. In what follows, we accept the equalities, Omega (Of x (Of eta)) equals “A.” (Of eta), that are valid in the linearized Hubble approximation. Moreover, we work in the vicinity to the present epoch where “A.” (Of t) equals one plus H t plus terms of Order (H squared, t squared), and in this approximation we are allowed to use t equals eta in terms which are proportional to H. It means we can equate “A.” (Of eta) equals “A.” (Of t).

Four. The local inertial frame.

The local inertial coordinates, x alpha, are mathematical functions on FLRW manifold which have no immediate physical meaning unlike the Cartesian coordinates in Euclidean space. To make the local coordinates physically meaningful they should be further specified and operationally connected with measuring devices (clocks, rulers) of a set of some reference observers. This materialization yields access to the local inertial frame. The corresponding relations between the measuring tools and the local coordinates are known in differential geometry as inertial (or projective) structure. The Minkowski form of the physical local metric:
D s squared equals f alpha, beta d x alpha, d x beta, suggests that LIC can be associated with the Gaussian normal coordinates based on the congruence of time-like geodesics of (electrically-neutral) test particles being at rest with respect to LIC.
The first step, is to find a relation between the coordinate time x zero and the proper time t of the Hubble observer at the origin of LIC. Because the space-time interval:
D s squared equals minus d x zero squared for x I equals zero, and
D s squared equals minus d t squared,
By the definition of the proper time, we come to the conclusion that x zero equals t on the world-line of the origin of LIC. The grid of the Gaussian coordinates start from the initial hypersurface, t equals zero, that is orthogonal to the world line of the Hubble observer. We identify the spatial Gaussian coordinates with the orthogonal (in the Euclidean sense) spatial coordinates X I of LIC on the initial hypersurface.
Extension of the spatial coordinates from the initial hypersurface to arbitrary values of the time coordinate x Zero equals t is performed by means of time-like geodesics. The Christoffel symbols of the local metric are nil in a neighborhood of the origin of LIC in accordance with diffeomorphism (thirteen) by which LIC were introduced. Because all Christoffel symbols are nil, the time-like worldlines of particles having constant spatial coordinates, X I equals constants, are geodesics given by (five).

Or, if X I equals a constant, d two X I, d sigma squared equals zero.
The proper time of the particle with the constant spatial coordinate X I coincides with the time coordinate X zero which was identified with the proper time of the Hubble observer. Hence, the parameter sigma in (five) can be identified with the proper time t as well. After that equation (five) describing worldlines of the static observers takes on the following simple form or equation nineteen:
D two x alpha, d t squared equals zero.

The meaning of the time-like geodesic equation (nineteen) is as follows. The world lines X alpha equals X zero equals t, x I equals constant. These world lines are identified with the network of static reference observers which play a fundamental role in local physical measurements.
We admit that each static observer is equipped with an ideal (atomic) clock measuring their proper time, which coincides with a time-like parameter, x zero, along the observer’s world line. Solving (nineteen) reveals that X zero equals t is the proper time of the Hubble observer located at the origin of LIC. We assume that the ideal clocks of the static observers are synchronized. It can be done with Einstein’s procedure of exchanging light signals as we will confirm in section six point two.

The Gaussian normal coordinates form a local inertial frame that is used for doing local physical measurements of time and space along time-like world lines of static observers and on space-like hypersurfaces of constant time.
The frame is defined operationally in terms of the proper time of the ideal clocks and rigid rulers. The rulers are made of an ordinary matter whose rigidity is determined primarily by the chemical bonds having an electromagnetic origin. We have proved in other references that in the linearized Hubble approximation the electromagnetic (Coulomb) forces in an expanding universe remain the same as in a flat spacetime.
For this reason, the rigid rulers and rods are not subject to the cosmological expansion and can serve for physical materialization of LIC. Another physical realization of the local Gaussian coordinates is achieved by the celestial ephemerides of the solar system bodies since their orbits are not affected by the Hubble expansion either.

Five. The light geodesics.

The most precise measurements of spacetime events are made with electromagnetic waves and light. Therefore, we have to solve the equations of light geodesics of equation six:
D two x alpha, d t squared equals: d x alpha, d x zero, times d two x zero, d t squared.
Where equation six is parameterized with the proper time t of the central observer, which is a directly measurable quantity. First of all, we need to evaluate the right side of (six). The function x zero taken on the light cone, is given by x zero equals “A.” (Of eta) times eta. Since the conformal time eta and the cosmic time t are related by:
T equals the integral of “A.” (of eta) d eta, equals eta plus H, over two eta squared plus Order Hubble squared, and the cosmic time coincides with the proper time t of the central Hubble observer, we get on the light geodesic, equation twenty one:

X zero equals t plus H over two t squared.
Taking the second derivative from x Zero in (twenty one) yields d two x zero, d t squared equals H. Hence, equation of light geodesics (six) takes on in the local coordinates the following form, equation twenty two:
D two x alpha, d t squared equals H d x alpha, d t squared.

Here we have made use of a legitimate approximation, that d x alpha, d x zero equals d x alpha, d t, in the right side of (twenty two).

Equation (twenty two) predicts the existence of a cosmological force in the tangent space of FLRW universe, exerted on a freely-falling photon. It should not be misinterpreted as a violation of general relativity or Newtonian gravity like the “fifth force” or whatever else. Equation (twenty two) is a direct consequence of general relativity applied along with the cosmological principle stating that the global cosmological time t is identical with the proper time measured by the Hubble observer. It explains how and why the Hubble expansion of the universe may appear locally. We discuss the observational aspects of this local cosmological effect in the next section in more detail.

Solution of (twenty two) is given by a quadratic function of time:
X alpha equals X alpha, zero plus k alpha times t plus H over two t squared.
Where x alpha zero is the position of photon at time t equals zero, and k alpha equals (one, K i), is a constant null vector with the unit vector K I pointing out in the direction of propagation of light.
The reader may notice that the coordinate speed of light, v alpha equals k alpha times one plus H t, exceeds the fundamental value of c equals one for t greater than zero. There is no violation of special relativity here because this effect is non-local, the speed is given with respect to the origin of the local coordinates. The local value of the speed of light measured at time t at the current position of the photon, is always equal to c equals one. This is because the group of the conformal isometry includes the Poincare group as a sub-group which allows us to change the initial epoch and the initial position on the background manifold without changing the differential equation (twenty two).

Non-uniform propagation of light in the local frame may look counterintuitive as compared with our experience with special relativity. Nonetheless, this is how light propagates in the expanding universe. Equation (twenty three) is just a direct consequence of a standard light propagation formula in cosmology which reads in the global conformal coordinates, y alpha equals k alpha times eta, where k alpha equals (one, k i) is the null vector.
The non-uniform propagation of light in the local frame can be observed in the solar system, thus, making it possible to measure the Hubble expansion rate locally as contrasted to the cosmological observations of distant quasars.

Six. Cosmological effects in the local frame.

Six point one. Radar and laser ranging.

Precise dynamical modelling of the orbital and rotational motion of astronomical bodies in the solar system, major and minor planets, asteroids, spacecraft, and so on, is inconceivable without radar and laser ranging. The ranging is an integral part of the experimental testing of general relativity and alternative theories of gravity in the solar system. We are to check if the Hubble expansion can be measured in the ranging experiments.
The equation of light propagation in the local Gaussian coordinates x alpha is given by equation twenty two. Let us consider the radial propagation of light. The radial, and always positive, spatial coordinate of a photon is:
R equals square root of delta I, J times x I, X J.
Let a light pulse be emitted at time t zero at point r zero, reach the target at the radial coordinate r greater than r zero at time t, and is immediately retransmitted to the point of observation being at radial distance to which it arrives at time t one. Propagation of the outgoing and incoming light rays are obtained from twenty three, where we demand that at the time of emission, t zero, the coordinate speed of light r dot (Of t zero) equals one for both outgoing and incoming light rays.
The equation of propagation for outgoing light ray is:
R equals r zero plus (t minus t zero) plus H over two times the square of t minus t zero.
And propagation of the incoming light ray is described by
R One equals r minus (t one minus t) minus H over two times the square of t minus t zero.
Let us assume for simplicity that the radar ranging is conducted by the Hubble observer at the origin of the local coordinates so that both the points of emission and observation of the light signal are at the origin and have the radial coordinate, r zero equals r equals zero. We define the radar distance L by a standard equation:
L is one half of (T One minus T Zero).
L is a relativistic invariant due to the covariant nature of the proper time t and the constancy of the fundamental speed c equals one in the geometrized system of units adopted in the present paper. After solving for incoming and outgoing r (Of t) we obtain:

T equals a half T zero plus T One, plus a half H r squared, and:
L equals R minus H R squared.
Where the residual terms of the order of Hubble squared have been neglected, r is the radial distance of the point of reflection of a radar signal at time t.
In other words, the approximation of a locally flat Minkowski space is corrected, to first order by linear terms, in which the propagation of light signals is modified by the cosmological expansion.

This calculation reveals that the difference between the coordinate distance r and the invariant radar distance L is of the order of H r squared. Planetary ranging is done for the inner planets of the solar system so we can approximate r around one astronomical unit, A U and H around two point three times ten to the minus eighteen inverse seconds.
Hence the difference H r squared of around zero point one seven millimeters which is a factor of around ten thousand smaller than the current ranging accuracy two meters to interplanetary spacecraft. In case of lunar laser ranging to the Moon, the coordinate radius of the lunar orbit, r of 384 thousand km, and the estimate of the residual term H R squared is about one point one times ten to the minus six mm which is one million times less than the current accuracy (around one millimeter) of LLR.

We conclude that in radar, laser ranging experiments:
(One) Within the measuring uncertainty the coordinate radial distance r equals L,
(Two) The radial distance r in the local frame of reference has an invariant geometric meaning in agreement with the definition of the proper distance accepted in cosmology,
(Three) the radar, laser ranging metrology is insensitive to the Hubble expansion in the local coordinates.
Hence, the celestial ephemerides of the solar system bodies built on the basis of radar, laser ranging data are not crippled by the Hubble expansion. They represent a dynamical reference frame with a fixed value of the astronomical unit, A U, which is not changing in time and can be treated as a rigid ruler for measuring distances between celestial bodies within the solar system in accordance with a recent resolution of IAU General Assembly (Beijing 2012) on the meaning and value of the astronomical unit.

Six point two. Einstein’s synchronization of clocks.
Let us now consider the Einstein procedure of the synchronization of two clocks based on the exchange of light signals between the clocks. We want to synchronize the clock of the central Hubble observer with the clock of a static observer located at a point with the Gaussian radial coordinate r. We apply exactly the same procedure as in the case of radar ranging described above. By Einstein’s definition, when the photon reaches the reflection point with the radial coordinate r at the instant of time t, the clock of the Hubble observer at the point, r equals zero, reads the time:

T star equals a half T Zero plus t one,
Because the time rate of the ideal clock of the Hubble observer is uniform. The instant of time t∗ is defined as being simultaneous with the time reading, t, of a second clock located at the position with a radial coordinate, r, at the instant when the light signal is reflected. The time t star as a function of t, can be found immediately from equation twenty seven:
t star equals t minus a half H r squared.

This relation reveals that in order to synchronize two clocks separated by a radial distance r, we have to subtract the time difference H r squared from the reading t of the clock of the static observer at the point with radial coordinate r in order to make the time readings of the two clocks identical. Because the radial distance r coincides with the invariant radar distance L, which is a measurable quantity, the Einstein synchronization of clocks in such experiment is operationally possible.
The two clocks will remain synchronized as time goes on, if and only if, the radial distance between the clocks does not change. For example, a clock at a geocenter will remain synchronized with clocks on-board of a geostationary satellite moving around Earth on a circular orbit. On the other hand, an ultra-stable clock on board of spacecraft which moves with respect to the primary time standard on Earth may detect the de-synchronization effect due to the Hubble expansion of the universe if the radial distance between Earth and the spacecraft changes periodically.
If the change in the radial distance amounts to delta r, the overall periodic time difference caused by the clock’s de-synchronization amounts to:
Delta t equals delta of t star minus t, which equals:
Two H times R squared, over c squared times delta r, over r.
Expressing r in astronomical units we can find a numerical estimate of the de-synchronization between the readings of the two clocks, equation thirty one:
Delta t equals one point seven times ten to the minus twelve times R in units of A U squared, delta R, over r seconds.

This local cosmological effect may be detectable by NIST and, or other world-leading timekeepers.

Six point three. The Doppler Effect in the local frame.
Next step is to consider the Doppler Effect that is a change in frequency of propagating electromagnetic wave (light) emitted at one spacetime event and received at another one, as caused by various physical reasons, relative motion of observer and the source of light, gravity field, expansion of the Universe, etc. A monochromatic electromagnetic wave propagates on a light cone hypersurface of a constant phase Phi, that is a function of spacetime coordinates, Phi equals Phi (Of x alpha).
The wave one form is L alpha equals d alpha of Phi, and frequency of the wave measured by an observer moving with four velocity, u alpha is:
Omega equals minus L alpha u alpha.

The frequency of electromagnetic wave can be calculated directly as soon as we know L alpha and u alpha, equals: d x alpha, d tau, where tau is the proper time along the world-line of emitter (or receiver) of light. Indeed,
Omega equals minus the contraction of L alpha, and u alpha, which equals, equation thirty three:
D phi, d x alpha, d x alpha, d tau equals d phi, d tau.

Which is just the rate of change of the phase of the electromagnetic wave along the world line of emitter (or receiver).
Let us denote the point of emission of the wave by P one, and the point of its observation as P two, and the emitted and observed wave frequencies as omega one and omega two, respectively. The proper time of the emitter is denoted as tau one, the proper time of receiver is tau two, and the time measured by the central Hubble observer is the cosmic time t. The ratio of received to emitted frequency, equation thirty four:
Omega one over omega two, equals:
L alpha, u alpha P two, over l alpha, u alpha P one.
Which quantifies the Doppler Effect.

Because the phase of the electromagnetic wave remains constant along the light rays we can use equation (thirty three) to reformulate (thirty four) in terms of the time derivatives. This is another way of saying, along a geodesic the phase in the proper time of a light pulse is constant.
Omega two over omega two equals d tau one, d t one, times: d t one, d t two, times: d two, d tau two.
We can introduce the unit vector N two one, which is X two minus X one, over the magnitude of X two minus X one, which points out from the point of emission, P one, to the point of reception, P two, of the light signal.
After some algebra, expanded upon in the paper, we obtain the Doppler shift of frequency of electromagnetic wave in the expanding universe for the emitter and receiver being moving with respect to the LIC, equation forty two:

Omega two, over omega one equals:
One minus N two one dot V two,
Over one minus N two one dot V one times:
The square root of one minus v one squared, over one minus v two squared, times:
One plus H t two minus t one.
Where we have dropped off all residual terms of the order of H V One and H V Two as negligibly small. Notice that (forty two) does not depend on the choice of the initial epoch t zero.
Equation (forty two) consists of two groups of terms. The first group depends on velocities of emitter and receiver, and represents a special relativistic Doppler effect. The second group (in square brackets) depends on the Hubble constant H and represents an additional shift of frequency caused by the cosmological expansion of space. The gravitational field of the solar system bodies should be also taken into account in realistic experiments. We have excluded the gravitational shift of frequency as it brings about many more terms to (forty two) and makes it unnecessarily complicated. These terms are well-known and can be found, for example, in the literature.
For a static emitter and receiver we have V One equals V two equals zero, and the Doppler shift equation (forty two) drastically simplifies to:
Omega two, over omega one equals one plus H times T two minus t one.

It tells us that the cosmological Doppler shift measured by the local static observers is blue because T two is greater than t one and, consequently, Omega two is greater than Omega one. It works opposite to the cosmological red shift for distant quasars, but there is no contradiction here. Cosmological red shift is measured with respect to the reference objects (quasars) which have fixed values of the global coordinates, y I, while the local Doppler shift is measured with respect to static observers having fixed Gaussian coordinates x i. Thus, the Doppler shift measurements in the global cosmological spacetime and in the local tangent spacetime refer to two different sets of reference observers moving one with respect to another with the velocity of the Hubble flow. Therefore, it is natural to expect a different signature of the Doppler effect, red shift for light coming from distant quasars and blue shift for light emitted by the astronomical objects, for example spacecraft, within the solar system. Our theory provides an exact answer for the signature and magnitude of the cosmological blue shift effect measured in the local inertial frame.

The Doppler effect in the tangent spacetime of FLRW universe has been considered by a number of other authors, most notably by Carerra and Giulini. They claimed that the cosmological expansion does not produce any Doppler effect in the local radio-wave frequency measurements.

Their conclusion is invalid as they implicitly identified the local Minkowskian time coordinates x zero with the proper time t of the Hubble observer on a worldline of any freely-moving particle including photons. However, this identification is not applied to photons (or any other moving particle) but solely to the static clocks of the Hubble observer.

Six point four. Measuring the Hubble constant with spacecraft Doppler-tracking.

Results of previous section suggest that precise and longterm Doppler tracking of space probes in the solar system may offer a new, fascinating opportunity to measure the local value of the Hubble constant H in the solar system. It is highly plausible that the “Pioneer anomaly” detected by John Anderson with the JPL deep-space Doppler tracking technique in the hyperbolic orbital motion of Pioneer spacecraft has a natural explanation given in terms of the Hubble expansion which changes the frequency of radio waves in spacecraft radio communications in an amazing agreement (both in sign and in magnitude) with our equation (forty three).

We have analyzed the cosmological origin of the “Pioneer anomaly” effect in another paper, making use of the local equations of motion for charged and neutral test particles as well as for photons in the FLRW universe. We have proved that in the local frame of reference the equations of motion for interacting massive neutral and, or charged particles do not include the linear terms of the first order in the Hubble constant, only tidal terms of the order of H squared remain. On the other hand, equations of motion of photons parameterized with the TCB time t do contain such linear terms of the order of H which have dimension of acceleration.
The present paper confirms results of the paper from the point of view of a set of local observers doing measurements in tangent space of the FLRW manifold.
Transformation to the local coordinates x alpha, equals (x zero, x I) allows us to transform the FLRW metric to the Minkowski metric:
D s squared equals minus d x zero squared plus Delta I, J, d x I, d x J, but the coordinate time x zero can be identified with the proper time t of the central Hubble observer only for static observers while for moving particles X zero equals x zero (Of tau) is a non-linear function of time t which is given for photons by (equation twenty one).
Equation (forty two) explains the “Pioneer anomaly” effect as a consequence of the expansion of space bringing about the blue frequency shift of radio waves on their round trip from Earth to spacecraft and back. Indeed, let us denote omega zero the reference frequency emitted from Earth to spacecraft, omega two the frequency received at spacecraft and transmitted back to Earth, and omega two the frequency received on Earth.
Then, according to (forty two), the shift between omega zero and omega two can be determined.
Let us simplify further consideration by assuming that the measurement is done by the central Hubble observer located at the origin of LIC. Then, V zero equals V two equals zero, and the unit vector is N one zero, equals minus N two one points out in the positive radial direction toward spacecraft moving with velocity v equals V One. After noticing that T Two minus T zero is approximately two times (T one minus T zero), and neglecting quadratic with respect to velocity terms, formula for omega two over omega one takes on the following form of equation forty five:

Omega two, over omega zero equals one minus two V over C, minus H times (T One, minus T zero).
Where V equals V dot N is the radial velocity of spacecraft, and we prefer to retain the speed of light c explicitly. The Doppler shift is defined as:
Z equals a half omega two over omega one minus one.
We get from equation forty five:
Z equals minus v over C plus H Two minus t zero.

That shows that the cosmological shift of frequency appears as a tiny blue shift on top of a much larger red shift of frequency caused by the outward motion of the spacecraft.
It was observed by J Anderson, and confirmed in a number of papers.
The time rate of change of the Doppler shift is Z dot equals d X, d t one, which yields:
Z dot equals minus one over c times “A.” minus H C.

Where “A.” equals d v, d t one is the magnitude of the radial acceleration of the Pioneer spacecraft due to the attraction of the solar gravity field. The Hubble frequency-shift term, H c, is subtracted from the spacecraft acceleration and can be interpreted as a constant, inwardly directed acceleration, “A.” P equals H c, in the motion of the spacecraft.
In fact, the true cause of the “anomalous” acceleration is associated with the motion of photons but not the spacecraft. This is the reason why the vigorous attempts to find out the explanation for the “anomalous gravity force” exerted on the Pioneer spacecraft were unsuccessful. The observed value of acceleration observed equals eight point five times ten to the minus ten meters per second per second, and is in a good agreement, both in sign and in magnitude, with the theoretical value of:
H C is around seven times ten to the minus ten meters per second per second.
Therefore, we believe that our result provides a strong evidence in favour of general-relativistic explanation of the “Pioneer anomaly” as opposed to numerous attempts to explain it by thermal recoil force.
The thermal recoil definitely makes contribution to the acceleration of the Pioneer spacecraft because the observed value of observed acceleration P exceeds theoretical value by 20 percent. Recent studies indicate that the numerical value of the Pioneer anomalous acceleration may be slightly decreasing over time which may be associated with the radioactive decay of the power generators of the Pioneer spacecraft.
The question about how much the thermal recoil force contributes to the overall effect remains open. The literature states that the Pioneer effect is 100 percent thermal but they have not taken into account the geometric effect of the expanding space on the propagation of light in the local frames in cosmology which suggests that the numerical value of the Pioneer effect cannot be smaller than the theoretical value of seven times ten to the minus ten. The thermal emission always adds to the general-relativistic prediction. Observations indeed show an observed acceleration larger than theoretical values by 20 percent. The theory of the present paper explains 80 percent of the overall effect by the effect of the expanding geometry leaving for the thermal recoil contribution no more than 20 percent.

Seven. Discussion.

One. We have built the LIC by applying the special conformal transformation (thirteen). Comparison with other approaches to build the LIC in cosmology reveals that all of them bring about the same coordinate transformation in the linearized Hubble approximation.
Therefore, there is no difference between various approaches to build the local inertial coordinates in cosmology so far as the quadratic terms in the expansion with respect to the Hubble parameter are not considered. Our approach to build LIC helps to realize that the transformation to the local coordinates on the expanding cosmological manifold is, in fact, an infinitesimal special conformal transformation which establishes one to one local mapping between the local and conformal coordinates.

Two. Introducing a local physical distance X I equals R (Of t) Y I allows recasting equation two into the following form:

D s squared equals minus one minus H squared X squared d t squared minus:
Two H X I, X I d t plus delta I, J, d X I, d X J.
This can be rearranged as:

D S squared equals minus d t squared plus Delta I, J times (d X I minus Chi I d T), times (d X j minus Chi J d t).
Where vector field Chi I equals the Hubble constant times X I. The former metric is exactly the warp-drive metric that was suggested by Alcubierre to circumvent the light-speed limit in general relativity.
All mathematical properties of the warp-drive metric that have been analyzed, and they are valid in the local coordinates (t, x i) where t is the proper time of the local static observers (X I equals constant) coinciding with the cosmic time. The metric is non-inertial but it can be converted to the flat Minkowski metric in a neighborhood of the coordinate origin with the help of an additional transformation of the proper time t to a local time coordinate x zero as shown in (equation thirteen). The local time coordinate x zero coincides with the proper time t of the static observers but deviates quadratically from t on the light cone as demonstrated in equation twenty one.

Three. The analysis of EEP given in the present paper, was focused on the solar system experiments as contrasted with pure cosmological tests. There are other possible tests which can be potentially conducted for testing the formalism worked out in the present paper, for example, with binary pulsars. Timing measurements establish a very precise local frame for the binary pulsar system which is not affected by the Hubble expansion as explained in the literature. On the other hand, we expect that the cosmological expansion influences the time of propagation of radio pulses from the pulsar to observer on Earth, and this effect should be seen in the secular change of the orbital period P b of binary pulsars of the order of:
The time derivative of P B, over P B equals H around two point three times ten to the minus eighteen seconds.
This effect is superimposed on the effect of the orbital decay due to the emission of gravitational waves by the binary system and introduces a bias to the observed value of P B dot in addition to the Shklovskii effect.
However, the orbital decay of binary pulsars with wide orbits is negligibly small, hence, we may expect to observe the Hubble expansion effect in the secular change of the orbital period.

Four. It is worth mentioning that the Cassini spacecraft was also equipped with a coherent Doppler tracking system and it might be tempting to use the Cassini telemetry to measure the universal “anomalous Cassini acceleration” of around seven times ten to the minus ten meters per second per second.
Unfortunately, there are large thermal and outgassing effects on Cassini that would make it difficult or impossible to say anything about the “Cassini anomaly” from Cassini data, during its cruise phase between Earth and Saturn.
Due to the presence of the Cassini-on-board-generated systematics, the recent study of radio science simulations in general relativity and in alternative theories of gravity is consistent with a non-detection of the “Cassini anomalous acceleration” effect.

Local gravitational physics of the Hubble expansion
Einstein’s equivalence principle in cosmology
Sergei M. Kopeikin1,2,a
1Department of Physics & Astronomy, University of Missouri, 322 Physics Bldg., Columbia, MO 65211, USA
2 Siberian State Geodetic Academy, 10 Plakhotny Street, Novosibirsk 630108, Russia
a E-mail: kopeikins@missouri.edu

arXiv:1407.6667v2 [astro-ph.CO] 21 Jan 2015

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