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Gravitational Forces with Strongly Localized Retardation

We solve the linearized Einstein equations for a specific oscillating mass distribution and discuss the usual counterarguments against the existence of observable gravitational retardations in the"near zone", where d/r<<1 (d = oscillation amplitude of the source, r = distance from the source). We show that they do not apply in the region d/r \approx 1, and prove that gravitational forces are retarded in the immediate vicinity of the source. An experiment to measure this retardation is proposed, which may provide the first direct experimental observation of propagating gravitational fields.


I INTRODUCTION AND SURVEY OF RESULTS
Between the 1940s and the early 1960s, several researchers proposed the possibility of measuring the retardation of gravitational interaction using laboratory experiments [1,2,3,4]. Because of the poor technology at that time and contradictory theoretical predictions about the retardation, interest in the subject declined. No experiments were performed during this period, and after 1962, the topic was never discussed again in the literature. At the 1996 Virgo conference, a new experimental set-up was described for the purpose of testing the retardation of gravitational interaction in the immediate vicinity of a vibrating mass distribution [5]. Discussions following the talk indicated that there is still ambiguity about the existence of such a retardation. In this paper, we will discuss the common arguments which lead to the conclusion that gravitational forces generated by oscillating mass distributions are instantaneous in the near-field. All of these arguments have the weakness that they are not valid at extremely small distances from the oscillating source. We will analyze the gravitational forces in the immediate vicinity of a specific oscillating mass distribution, for which the dominant part of the corresponding linearized Einstein equations can be solved. It shows that there is a possibly detectable retardation of the gravitational forces in the immediate vicinity of the source, which converges to zero with increasing distance. The dominant length scale for the decay of the retardation is given by the oscillation amplitude of the mass distribution and can be very small.
Experimental verification of this local retardation effect would provide the first direct observation of propagating gravitational fields. We propose a laboratory experiment which requires a highly accurate phase measurement to test our result. Given the state of modern technology, this measurement may now be possible.

MATHEMATICAL MODEL
We consider an idealized experiment in a laboratory environment, consisting of a mass-balanced, mechanical two-body oscillator, which generates disturbances in the space-time metric. The oscillator consists of a point mass m which is connected by way of a spring with negligible mass to a second point mass M ≫ m. The resulting mass-spring system oscillates with eigenfrequency ω about its center of gravity, and does not interact dynamically with additional, external masses. The distance between the masses m and M is assumed to be much larger than the oscillation amplitude d of the smaller mass m. The oscillation amplitude D = m M d of M, which is calculated with classical mechanics, is much smaller than d and leads to negligible time-dependent gravitational effects in the immediate vicinity of m, as will be shown in section II.1. We will mainly consider the time-dependent gravitational forces generated by m, and will thus refer to m as the "source", or the "source mass". Another mechanical oscillator, tuned to resonance to detect these gravitational forces, is placed on the oscillator axis at a distance of order O(d) from m. The purpose of this paper is to show that the gravitational forces generated in this extreme near-field laboratory experiment are retarded, which may be verifiable with modern technology. Due to many objections against experiments of this kind in the past, we will first discuss the common counterarguments.
The occurrence of a measurable retardation in the proposed system may seem problematic for several reasons. One may suspect that the theorem about the non-existence of gravitational dipole radiation is violated by the mass-balanced, mechanical two-body oscillator described above. It states that isolated mechanical systems cannot generate any dipolar gravitational radiation, due to momentum and angular momentum conservation within the system [6]. However, this argument does not contradict the existence of retarded gravitational forces which are strongly localized around the source. We will refer to the immediate vicinity of m as a region around m with a diameter of order O(d). In section II.1, we will prove that the time-dependent gravitational forces due to the larger mass M can be neglected in the immediate vicinity of the source mass m, although the static field due to M may dominate over the static field generated by m. The mass m, which is by itself not an isolated system, appears locally to be an isolated oscillator. Because the momentum of m is by itself not conserved, m is allowed to radiate gravitationally in its immediate vicinity without contradicting the overall conservation of momentum. At larger distances from the oscillator, the time-dependent gravitational forces due to M and m have equal strength, and we will show in section II.2.1 that their radiative contributions cancel. Hence, there is no contradiction to overall momentum conservation.
Near-field set-ups similar to the proposed one are usually analyzed with the "near zone" limit used in radiation theory, typically defined as a distance range comparable to the wavelength of the radiated signals, which is usually very large. At these large distances from the source, the vibrating mass distribution can be very accurately modelled as a series of multipole coefficients, which neglect all retardation effects within the source. But in the immediate vicinity of the source, i.e. at distances comparable to its vibration amplitude, it is not possible to make this simplification. Instead, one must take local retardation effects into account, even if they disappear at large distances.
Another common objection against the existence of observable retardations in the proposed set-up arises from an electrodynamic analogy. Retarded electromagnetic potentials may result in almost instantaneous electromagnetic fields, due to certain cancellation effects [7]. This phenomenon occurs in the "near zone" introduced above, and is consistent with our results. We will show in section II.2.2 that the retardation of the gravitational forces decreases continuously to zero with increasing distance from the source, and that in the immediate vicinity of m, the retardation has possibly measurable values.
We will first show that our proposed system can be analyzed with the linearized Einstein To keep track of order-of-magnitude estimates, we introduce dimensionless variables. We define the dimensionless four-vector (τ, ξ) := 1 d (ct, x) and use the oscillation amplitude d to represent the length scale. The energy momentum tensor T µν is related to the dimensionless T µν by T µν =: mc 2 d 3 T µν , and the Einstein tensor G µν which has the dimensions [m −2 ] introduces G µν =: 1 d 2 G µν . The Einstein equations, written in terms of dimensionless variables, yield in which G ≈ 6.67 · 10 −11 [ m 3 sec 2 kg ] is the universal gravitational constant. The number 8πGm c 2 d ≈ 10 −25 estimates the strength of gravitational signals due to m. We bound the mass of M by Consequently, one expands the space-time metric g µν = η µν + h µν around the flat Minkowski metric η µν , and introducing γ µν := h µν − 1 2 η µν h λ λ , Eq. ( 2) reduces to by use of the Hilbert gauge condition γ ν µ,ν = 0 [8]. ∆ ξ is the Laplace operator in the dimensionless spatial coordinates ξ. Eq. ( 3) is solved by the superposition of a particular solution with a homogenous solution. The full particular solution includes the influences of both M and m. We impose the boundary condition γ µν → 0 in the limit | ξ| → ∞. Since the full particular solution is asymptotically zero, as will be shown in section II.2, this requires the homogenous solution, given by a superposition of plane waves with constant amplitudes, to be zero. Therefore, γ µν is the particular solution of Eq. ( 3). Due to linearity, the contributions from M and m to γ µν and T µν are additive and can be discussed separately, thus µν . We will first analyze the influences due to m, determined by γ (m) µν and T (m) µν . We obtain in which δ is the Dirac-delta distribution and Θ is the Heaviside function. The perturbation parameter of our analysis is the ratio β = ωd c ≈ O(10 −8 ) between the maximal velocity ωd of the oscillating mass m and the speed of light c. Our analysis will show that the retardation is an effect of the order O(β). Therefore, a relative accuracy of the order O(β 2 ) is sufficient for our calculations. T (m) 00 can be decomposed into is of order O(β i ). Note that this decomposition does not correspond to the full Taylor expansion with respect to β. T (m,0) 00 is the only contribution of order O(1 = β 0 ) to T µν , and is determined by the dimensionless rest energy density ρ(τ ′ , ξ ′ ) of m, which satisfies d ξ ′ ρ(τ ′ , ξ ′ ) = 1. In all physical systems, ρ(τ ′ , ξ ′ ) can be assumed to be a sharply-peaked, Note that the space and time dependent terms on the RHS of Eq. (5) and are of order O(β). The components T (m) ij , i, j = 1, 2, 3, given by will only contribute to error terms of the order O(β 2 ).
We evaluate γ (m) µν (τ, ξ) at the point ξ = r dẑ on the positive z axis, such that The δ-integration in ξ 1 and ξ 2 can be readily carried out, yielding Further reduction is possible using the formula in which we identify the zeros of the argument in the delta distribution as the roots of θ in the equation Note that θ(τ, ξ 3 ) is a function of the combination τ − ξ 3 . The derivative of the argument of the delta in Eq. (6) yields 1 − β cos βθ(τ, ξ 3 ) at θ(τ, ξ 3 ). We obtain for γ (m,0) 00 In Eq. ( 8), β is not only a small perturbation parameter, but also a "dimensionless angular frequency". In contrast to β, the arguments βθ(τ, ξ 3 ) of the trigonometric functions need not be small for arbitrary τ or ξ 3 , and cannot be used as perturbation parameters. Since T Hence it is sufficient to analyze γ The solutions for γ (m) ij (τ, ξ 3ẑ ), i, j = 1, 2, 3, which are given by is the oscillation amplitude of M, and R is its average distance to the point of measurement. This is because both masses oscillate sinusoidally with the same frequency ω and opposite phases along the z axis.
It follows that ∂ τ γ (m,M ) 0i (τ, ξ 3ẑ ) is of order O(β 2 ) for both M and m. We will show below that the only Levi-Civita connection coefficients needed for our analysis are Thus, where U(ξ i ) is the dimensionless potential energy due to the spring. For the moment, we assume that the second mass of the pick-up system is "infinitely" larger thanm, and static.
The space-time trajectory of the test massm is determined by the Euler-Lagrange equation where Γ µ ρσ are linearized Levi-Civita connection coefficients. The dominant terms in the sum Γ µ ρσ ξ ρ ,0 ξ σ ,0 arise from Γ µ 00 ξ 0 ,0 ξ 0 ,0 . All terms Γ i ρσ ξ ρ ,0 ξ σ ,0 which involve ξ j ,0 , j = 1, 2, 3, are at least a factor of order O(β 2 ) smaller, and can be neglected. From the equation of motion for ξ 0 follows that ξ 0 ,00 (ξ 0 ,0 ) 2 + Γ 0 00 = 0 with a relative error of order O(β 2 ). Using the fact that Γ 0 00 is a total derivative with respect to τ , and that γ µν ≪ O(β 2 ), one obtains ξ 0 ,0 = 1 + O(β 2 ) and g µν ξ µ ,0 ξ ν ,0 = 1 + O(β 2 ). Consequently, the motion ofm is determined by with a relative error of the order O(β 2 ). The contributions of M and m to the "gravitational force" 1 4 γ (0) 00,i are additive, and have the same functional form. This weak-curvature limit is usually referred to as the post-Newtonian approximation, or the Newtonian limit. The pick-up oscillator is typically driven at resonance, but has a finite Q factor. In all realistic systems, the condition that the velocity is of order ≈ O(β 2 ) is satisfied, because the gravitational excitation of the pick-up system is very weak, and Eq. ( 10) can be used.
The Levi-Civita connection coefficients Eq. ( 9) depend on our specific choice of coordinates, or gauge. It is not a priori clear that these coefficients are the correct expression for the physical gravitational force, since they could be influenced by meaningless coordinate effects. The physically measurable force is determined by the change of these coefficients in the absence of the source, i.e. for T µν = 0, calculated in the same gauge, and using the same boundary conditions.
In the Newtonian limit, c 2 4 times (8) is interpreted as the classical gravitational potential generated by m, which acts on the external mass pointm by a gravitational force that is proportional to the gradient of V , according to Eq. ( 10). The physical meaning of the solution Eq. ( 11) becomes more apparent if one expands the first fraction in a geometric series, and the second fraction in a Fourier series in which all coefficients c n and d n are of order O(( d r ) 2 ). The leading terms of order O(1 = β 0 ) can be obtained by setting β equal to zero in the first bracket. The time-independent contribution mG r {1 − d r 2 } −1/2 in the Fourier expansion is the static potential of a resting mass distribution centered at the origin, and converges to the classical Newtonian potential mG r for large r. The next, first harmonic term has the asymptotic form mGd . For large r, it describes a propagating, sinusoidal oscillation proportional to 1 r 2 , which decays more rapidly than the usual radiative 1 r solutions. All following terms are higher harmonics that decay even faster with increasing distance. The contributions of O(β) exhibit the same asymptotic behaviour with respect to d r .

II.1 GRAVITATIONAL EFFECTS OF THE MASS M
In this section, we will discuss the gravitational effects of the mass M in the immediate vicinity of m. The average distance R between M and the point of measurement, lying in the immediate vicinity of m, is much larger than d, and also much larger than r. Due to M ≫ m, the oscillation amplitude D = m M d of M, which is calculated from classical mechanics, is much smaller than d. Both masses oscillate at frequency ω along the z axis, and the gravitational potential due ( 12). The static term is approximately − M G R and can dominate over the static field stemming from m. However, one obtains approximately M G R D R = mGd r 2 ( r R ) 2 for the amplitude of the first time-dependent term in Eq. ( 12), which is a factor of ( r R ) 2 smaller than the corresponding term due to m. The amplitude of the first harmonic term due to M has the opposite sign as compared to corresponding term due to m, because M and m oscillate with opposite phases.
In addition, the amplitudes of the higher harmonic terms, approximately given by M G R ( D R ) n = mG r ( d r ) n ( r R ) n+1 ( m M ) n−1 , are a factor of ( r R ) n+1 ( m M ) n−1 smaller than the corresponding terms due to m. One can therefore neglect the time-dependent gravitational influences due to M if r R is small, which is the case in the immediate vicinity of m. At larger distances from m, the number r R approaches 1, and all radiative effects cancel, which is required by momentum conservation.

II.2 RETARDATION OF GRAVITATIONAL FORCES
It will first be shown that the forces derived from the gravitational potential Eq. ( 11) are not retarded in the region known as the "near zone" in the literature, which is in accordance to [7]. Then, it will be proved that these gravitational forces are retarded in the immediate vicinity of m. As discussed in the derivation of Eq. ( 10), we place a small test massm which is part of a resonant mechanical oscillator at position ξ = rẑ on the z axis to detect the gravitational forces generated by the source m. From F (τ, r dẑ ) = −m ∂ r V (τ, r dẑ ) it follows that the gravitational force acting onm is which follows from Note that the leading term in F (τ, r d ) is proportional to V (τ, r dẑ ) 2 . The expression for the gravitational force generated by M is similar to Eq. ( 13). It has a static component which is much stronger than the static term due to m. However, the time-dependent gravitational forces generated by M are negligible in the immediate vicinity of m, because d r ≈ 1 and therefore r R ≪ 1. At large distances from m, M and m produce time-dependent gravitational forces of the same strength, since d r → 0 and r R → 1.

II.2.1 GRAVITATIONAL FORCES IN THE "NEAR ZONE"
The spherical region {r = | x|; β ≪ d r ≪ 1} is commonly defined as the "near zone" in the literature. For this region, the approximation θ(τ, r d ) ≈ τ − r d can be used, which is familiar from radiation theory. In addition, we have used the identity 1 1+y = 1−y +O(|y| 2 ), with |y| < 1, to simplify the fractions in V (τ, r dẑ ), and to reduce the potential to The trigonometric functions can be approximated by their Taylor  The time-dependent term of order O(1 = β 0 ) in the force acts instantaneously, thus there is no measurable retardation in the "near zone" where d r ≪ 1. The electromagnetic version of this result can be found in [7]. Historically, this result has convinced the physics community that there exists no region at all where any retardation can be observed.
The additional constraint r R ≪ 1 allows the influence of the second mass M to be neglected. As r increases, this condition can not be satisfied, and r R → 1. The complete gravitational potential in the "near zone" limit, which includes the contributions of both m and M is given which follows from the discussion in section II.1. The time-dependent part of this solution vanishes in the limit r R → 1, and there remains a static solution which is dominated by the larger mass M. Note that (m+M )G r is the rotation symmetric limit solution for r → ∞, which corresponds to the asymptotically vanishing particular solution of Eq. ( 3). We conclude that our result is consistent with [7], and that it is in accordance with the conservation of momentum.

II.2.2 GRAVITATIONAL FORCES IN THE IMMEDIATE VICINITY OF THE SOURCE
The immediate vicinity of the source is characterized by d r ≈ 1, in contrast to d r ≪ 1 which is a necessary condition for radiation theory. Consequently, radiation theory breaks down. At this distance range, r R ≪ 1 holds, and time-dependent influences of M are negligible, as shown in section II.1. As opposed to the situation in the "near zone", it is not possible to approximate functions of d r by the leading terms in their Taylor series. For this reason, we will have to explicitly calculate the dominant Fourier components of the lowest harmonics in Eq. ( 13). To order O(β), we find The condition β ≪ d r which has defined the "near zone" is also valid for the immediate neighborhood of the oscillating mass point. Therefore, it is still possible to expand the trigonometric functions with respect to β r d ≈ O(β). From the defining equation Eq. ( 7) for θ(τ, r d ), it can be seen that β θ(τ, r d ) is a function of the variable β r d . From Eq. ( 14), one finds for the leading terms of the Taylor series in β r d . We expand the trigonometric functions in Eq. ( 18) with respect to β θ(τ, r d ) ≈ β θ(τ, 0) − β r d , and perform the Taylor series to first order and is of order O(β). At very small distances r from the source, where d r → 1 and Φ( d r ) → 1, the value of the phase shift approximates the result which would occur for a wave that travels at speed of light. At larger distances in the "near zone", we have d r → 0 and r d Φ( d r ) → 0. Hence there is no measurable phase shift, and the gravitational forces seem to act instantaneously, as was shown above. In the intermediate region where d r ≈ 1, the phase shift can be calculated from Eq. ( 24). Note that in the limit d r → 0, ( 23) converges to ( 17), as it should be. The absolute phase shift φ(r) := ω c Φ( d r )r converges rapidly to zero for r → ∞, see Fig. 2. We conclude that there is a possibly measurable retardation of the gravitational forces in the immediate vicinity of the oscillating source in our proposed system. A discussion of extended gravitating sources with the linearized Einstein equations would be a straightforward generalization of the presented analysis. Due to the superposition principle, expressions similar to Eq.
( 23) would then have to be integrated over the spatial spread of the oscillating continuum.
However, we have assumed cylindrical symmetry in our mathematical model, and thus could find a solution for the dominating gravitational force term. For systems without cylindrical symmetry, the analysis will be much more difficult. From the results in this section, we will now discuss how they could be experimentally verified.

III EXPERIMENTAL POSSIBILITIES
Eq. ( 23) suggests that it may be possible to experimentally verify the retardation of gravitational forces in the immediate vicinity of an oscillating mass distribution. This would require both the amplitude and phase detection of gravitationally transmitted vibrations. With the previous mathematical model in mind, we propose the following experiment: A mass m within a mass-balanced apparatus is oscillated with a very stable phase and frequency ω, as described in the mathematical model. If a mechanical resonator with eigenfrequency ω, massm, and quality factor Q is placed at position rẑ, it will be driven into resonance by the first harmonic term in the gravitational force. Given suitably chosen parameters, the amplitude of the sinusoidally vibrating resonator, typically of the order O(10 −9 [m]), can be detected with modern technology [5].
We propose to measure the retardation of the gravitational force in the following way. Place the mechanical resonator at the location r 1ẑ on the oscillator axis, such that d r 1 ≈ 1. From the observed gravitationally induced sinusoidal vibration, the phase difference ∆φ 1 between the phase of the resonator and the phase of the vibrating source mass can be measured. If the mechanical resonator is placed at another location r 2ẑ on the oscillator axis, a different result ∆φ 2 will be observed. By subtracting ∆φ 1 from ∆φ 2 , all internal influences from the experimental measuring instruments can be eliminated, resulting in the relative phase shift Since the 1960s, several researchers have detected the amplitudes of gravitationally induced vibrations in laboratory experiments [5,9,10,11]. It may be possible that these experiments can be analyzed in a similar way as the model in this paper, which would provide information about the magnitude of the retardation, if present. The detection of the retardation of gravitational interaction in laboratory experiments has not been discussed much recently because the usual "near zone" calculations predict instantaneous interactions, as was shown in section II. However, this result does not apply in the immediate vicinity of the oscillating source, as was also shown in section II. Note that our experiment is not related to the gravitational waves of helicity two, which are far more difficult to detect, since they are of order O(β 2 ). Unfortunately, several technical problems must be solved before measurements to test these retardations can be realized. In 1958, Q. Kerns demonstrated an analog electronic circuit capable of measuring 10 −6 degree phase shifts [4]. We have made several modifications to Kerns' original design, and we are currently able to measure 10 −7 degree relative phase shifts over a 30 second measurement period [13]. Currently, the technology is thermally limited, but thermal control of the system and application of noise reduction techniques may soon enable us tho increase the phase sensitivity to the 10 −9 degree phase accuracy required, to measure the retardation to 1% [5]. It is not yet known if the phase stability of the mechanical systems used for the experiments can be maintained to this accuracy over the measurement time.

IV CONCLUSIONS
Using the linearized Einstein equations, we have proved that the gravitational force generated by a specific oscillating mass distribution is retarded in its immediate vicinity. We have proposed an experiment to measure this retardation which may enable the first direct experimental observation of propagating gravitational fields. The amplitudes of gravitationally induced vibrations in systems similar to ours have been successfully detected since the 1960s. The detection of the retardation requires a very sensitive phase measurement, which may now be possible given the state of modern technology.