Gravitational Waves in a Universe with Time-Varying Curvature

In this paper, we present a complete solution of Einstein’s equations for the gravitational wave (GW) problem. The full metric is taken in the usual way to be the sum of a background vacuum metric plus a perturbation metric de-scribing the GW. The background metric used is characterized by time-varying curvature as described in a recent paper. The solution we develop here does exhibit some features found in the standard model but it also contains others that are not found in the standard model. One difference is that the solution with time-varying curvature only allows for outward-directed waves. While this might seem a minor point regarding the GW equations, it is actually a significant verification of the solution presented in our earlier paper. A more obvious difference is that the solution demands that the vacuum along with all matter must experience transverse motion with the passing of the waves. This fact leads to the idea that a new approach to the detection problem based on the Doppler effect could well be practical. Such an approach, if feasible, would be much simpler and less costly to implement than the large-scale interferometer system currently under development.


Introduction
In a recent paper [1], we proposed a new model of cosmology based on the idea that vacuum has content and serves as its own source. One consequence is that the curvature of spacetime must vary with time and these together require that the present-day scaling of the universe must be expanding exponentially. A number of predictions are made in that paper that agree with observation leading support to the idea that the model is correct. In this paper, we will examine the effect that time-varying curvature has on gravitational waves (GW) and for this reason, it would be useful to review at least Section 8 of our original paper for the background needed to understand some of the discussion presented here.
Because the present-day curvature is not large, one would not expect to see dramatic differences between the results found here and those of the standard model (see e.g. [2]) but there are differences that are interesting. For one, with time-varying curvature, the equations only allow outgoing waves from the source. As will be explained in Section 5, this is in direct contrast to the situation with the standard model. In that case, the equations by themselves allow both outgoing and incomings waves to exist and it is a matter of choice to ignore the incoming waves. With time-varying curvature there is no choice; the dynamics must evolve in the direction of increasing entropy. Another consequence of the time variation is that the phase velocity of the waves is not exactly the speed of light. In other words, space exhibits an index of refraction that is not exactly unity. The most obvious difference, however, is that the complete solution demands that the vacuum and all ordinary matter must undergo transverse oscillations with the passing of the GW and this motion suggests the possibility of a new detection scheme based on the Doppler effect.
Here, we will be working with the full Einstein equations so our development is more rigorous than that of the standard model. The equations are fully constrained which makes it possible to determine both the magnitude of the GW metric and its functional dependence on parameters such as the angular velocity and size of the source.
As we just noted, things are moving. Because the wavelength of the GW is on the order of twice the Earth-Sun distance, on a terrestrial distance scale, any device such as an interferometer or, in fact, the entire Earth, is oscillating as a unit of fixed coordinate dimension and what an interferometer detects is the variation in the travel time of photons over a fixed coordinate distance due to the oscillating curvature. For much greater distances on the order of a half-wavelength, on the other hand, one also needs to account for the fact that, for example, the mirrors of a gigantic interferometer would be in relative motion. The travel time of a photon would then vary as a consequence of both the motion of the mirrors and the oscillation of the curvature along its path.

Model
Unlike the case with electromagnetic radiation, GW consists of oscillations of the already existing background curvature of spacetime so, in some ways, GW has more in common with water waves than with radiation. Because of the nonlinearity of the geometry, we cannot simply add a perturbation solution to the background solution but must instead deal with the full metric. We form the full metric in the usual way by adding a small perturbation metric to the background metric. The process of reducing the original equations to a set of linear equations for the perturbation components involves a sequence of steps that must be com-pleted in the correct order. Starting with the full metric, we calculate the Riemann and Ricci tensors in the normal manner and, only afterward, do we simplify by dropping all terms but those with the 1 st order powers of the perturbation. This is not a trivial matter because it is essential that the action of GW is described in terms of the actual background metric with curvature rather than with an idealized flat metric. Unlike the standard model formulation which is based on a flat space metric, we are starting with the non-trivial background metric from [   The next step is to assume a form for the perturbation metric. The background metric is expressed in spherical coordinates and we considered formulating the analysis in those coordinates but it turned not to be convenient so we instead did this analysis in Cartesian coordinates. In the standard model in which the background is assumed to be a flat Minkowski space and with the vacuum described by 0 µν = T , a number of simplifications are possible, such as making judicious use of gauge freedom to be simplified the model down to a simple wave equation. In our case, the non-vanishing energy-momentum tensor fixes the background vacuum which eliminates the possibility of any arbitrary simplifications. As a consequence, for our starting point, we assume a completely general symmetric metric and leave it entirely up to Einstein equations to determine the final form of the solution. Thus, We now wish to calculate the Ricci tensor. We write µν µν µν λδ = + g g g where µν g is the background metric and then compute the connection coefficients, We now need to consider just what is going on here. (Again, a review of Section 8 of [1] will make this more understandable.) First, we must remember that Einstein's equations are local or, in other words, they are evaluated at a single point. From the point of view of the GW, at each point, (x, y, z), the wave experiences the background in the limit that the distance from that point vanishes. This means that if we take the immediate origin of our calculation to be that point, we should evaluate the background at that point, or in other words, we should set the background coordinates to zero. However, if we do so before calculating the Ricci tensor, we will lose all the curvature structure because the spatial derivatives do sample a region away from, albeit infinitesimally close to, the point. At the same time, we know that the background is the same everywhere so the spatial derivatives of the background in (2-5) must vanish. The time derivatives of the background, on the other hand, do not vanish because the background does depend on time. To actually develop and solve these equations, we used Mathematica © Wolfram Research, Inc (and it would be impossible to complete the calculations by hand.) Accordingly, we needed to prevent the evaluation of spatial derivatives of the background. We accomplished this by replacing the spatial coordinates ( ) , , x y z in the background metric with the coordinates ( ) , , x y z . The background spatial derivatives in the connection coefficient cal-culation now vanish but the curvature structure is retained.
With the connection coefficients in hand, we next calculate the Ricci tensor and after doing so, because each point of the perturbation must experience the background evaluated at that point and because we have fixed our origin to be that point, we now set the background coordinates to 0 x y z = = = . Finally, because the equations (but not the metric functions) must be independent of location (one point in the vacuum is exactly the same as any other point), the equations will be dependent on the spatial derivatives with respect to the coordinates ( ) , , x y z but they cannot contain any explicit values of the coordinates (think of the standard wave equation) so we set those to zero which is again just evaluating the equations at the origin of each point.
The Riemann tensor is where µν T is the background tensor. The perturbation tensor is then where we have introduced the dimensionless velocity c = v u . The Einstein equations can be written in two ways and here, in keeping with the convention used in [1], we will work with the EM trace reversed form so we have The final step is to include the source by combining its contribution with the vacuum EM tensor, µν T . Lowering the indices of (2-9), trace reversing the result, and cancelling the background terms give us the final equations, where µν S is the source and S is its trace. The source tensor with be developed in detail in the next section.
In addition to these equations, we have the EM conservation condition, which becomes after expansion Because the source is "over there", it does not contribute to EM conservation at the location of the observer.
Also, because any small volume of the vacuum acts like ordinary matter under the influence of the curvature, we have an additional set of constraints given by the geodesic equations, To first order in small values, the vacuum velocity has the following form Since the velocity is small, we can approximate d dt τ ≈ and after changing the time coordinate to ct , we end up with, In these equations, each chunk of the vacuum is concerned only with the connection coefficients at its location and first, because there are no spatial derivatives in (2-17) and second, because the background curvature is the same everywhere, we can immediately set 0 = x . After doing so, we find that the only nonvanishing coefficients are 0 , 1, 2, 3  With the equations now specified at least symbolically, we will turn to the source.

The Source
Because we are intending to find a complete solution for this problem, we must be explicit about the source and the choice we made was to consider a compact binary star system. The choice is, in fact, fairly general since most sources will involve an orbiting system of one sort or another. To keep things simple, we will consider a system of two stars of equal mass for which the frequency is Journal of High Energy Physics, Gravitation and Cosmology Here, M is the mass of each star and s l is the radius of the system. With stars of unequal mass, the formula of (3-1) would be modified but, since the final equations depend only on the angular frequency and the radius of the system, the choice of equal masses is not restrictive. From [3], , , x y z coordinate system. We also want to allow for an arbitrary orientation of the source so we introduce a second system ( ) , , x y z ′ ′ ′ obtained by a rotation about the shared , y y′ axis. Figure 1 illustrates the two coordinate systems.
The stars orbit in the where the subscripts refer to each of the two stars. For the source EM tensor, however, we need the velocities in the ( ) , , x y z system. A simple rotation gives We will approximate the stars as point objects so the density of the system becomes Integrating this overall space gives Equation (3)(4)(5) gives the density in terms of an origin at the center of the rotating system. The observer, however, sees the system from a distance on the order of Journal of High Energy Physics, Gravitation and Cosmology It is now a straight-forward matter to work out the components of the EM tensor. The complete tensor is the sum of the tensors for two stars with each adding a contribution of

The Complete GW Equations
In this section, we will flesh out the symbolic equations presented above. It would require far too much space to present the entire development and, in any case, it is unlikely that anyone would be interested. Instead, we will outline the steps and present the full list at the end of the section. From this point on, everything was done with Mathematica so from here on out, we will use Mathematica's notation 1 .
In [1], we found that physical quantities such as the curvature and the motion of particles are dependent only on the sum of the vacuum energy density and pressure rather than on either alone. Expecting a similar result here, the two quantities we worked with were that sum and again the pressure. . As it happens, the equations do not depend on just the sum but the influence of the energy density and pressure turns out to be very small.
The first steps were to convert the background metric, (2-1) to Cartesian coordinates and then to compute the inverse metric and the connection coefficients in the usual manner. After doing so, we made the change of names of the coordinates to the barred versions, x x → , etc. We next expanded (2)(3) in the usual manner to obtain the trace-reversed background EM tensor. Turning now to the perturbation metric, after expanding (2-6) and (2-7), we evaluate at each 1 The primary difference is the notion used for partial derivatives. Mathematica uses superscripts with functions of more than one variable, e.g. (  We now want to apply some physical reasoning to simply this result. First, we know from [1] that signals travel along paths of constant angle and that the effect of the background curvature is entirely radial. We also know that any transverse interaction between elements of the gravitational wave would be 2 nd order in the metric components. Since the background vacuum is isotropic, to this level of approximation the anisotropy of the wave is entirely a result of the asymmetry of the source which we have expressed with the tilt angle α . To get some idea of the magnitude of the velocity variance, we use the chain rule, The derivative of any component with respect to α is of the same order of magnitude as the component but the derivative d dx α introduces an additional factor of 1 es l − so the variance in the transverse directions is very small. We have oriented our coordinate system so the source lies on our z-axis and the smallness of the transverse variance then means that we can eliminate the x and y dependence from the equations.
In the far-field limit, the z coordinate differs from the radial distance to the source by an infinitesimal amount so we can also replace z with the radial distance, l. Finally, because the background metric functions vary slowly with time relative to the travel time from the source, we can evaluate them at the present-day time, 0 t . With these changes, we have reduced the problem to one with a single spatial dimension.
We will now list the final equations. For brevity, we have omitted the arguments of the variables.   The lower case ij s in the first 10 equations is a place holder for the source and its value depends on which of the three sets of equations we are solving, 1 S , 2 S p , or 2 S n . Table 1 summarizes the dependences.

Solution of the GW Equations
Here, ω has the units of inverse length. It is now necessary to pin down the origin and it will be convenient to fix it at the source. In this case, the observer is now located at ( ) The total solution is thus the sum of three partial solutions. The first with 0 ω = is static and hence does not contribute to the GW. The remaining two both contribute and because we are only interested in the inhomogeneous solution, we Table 1. Summary of equation dependencies. eq δ00 δ01 δ02 δ03 δ11 δ12 δ13 δ22 δ23 δ33 δp δρp vx vy vz S1 S2 1 x Turning now to the spatial coordinates, we first replace the delta function in the source by its FT, After moving the source to the LHS, we end up with a list of equations with the symbolic form, (5-4) Here, the subscript "j" is a shorthand for the µν indices taken in sequence. Since this must be true for all k , the integrand in the parentheses must vanish which results in a system of equations for the ˆj g δ . The is dependent only on the magnitude of the wave vector, k , so the ˆj g δ in turn are also only dependent on that magnitude.
After solving for the ˆj g δ , the would appear that we can't extend the lower limit. We can easily get around the problem, however, by defining an auxiliary function, 1.2 10 ct = × raised to the 7 th power and then multiply by Ω to the 7 th power, we overflow the limits of the ability of the computer to represent numbers. The solution in this particular case is to multiply the two factors before raising to the power.
From Table 1, we see that 12 g δ appears alone in the single Equation (4-4f) and because this is the simplest case, we will walk through the steps to the solution. The others follow a similar pattern with variations. The final results are listed in Sec 7.
We first solve the FT version of (4-4f) with the source set to 2 p S and find, . While this result is far too small to have any observational consequences, it does show that the vacuum exhibits an index of refraction that is not exactly unity and because the correction is negative, the phase velocity of the GW is slightly greater than c.
This value was computed using the background vacuum properties far from any matter which of course is not the situation we are in. In [1], we showed that the corresponding parameters in the interior of a galaxy can be expected to be perhaps 10 7 times larger so the refraction offset will also be somewhat larger.
We now perform the contour integrations. The integral involving e ikl closes in the upper half-plane and captures the first pole thus becoming 2 e i l Ω multiplied by a number very close to unity from the imaginary part. The integral involving e ikl − closes in the lower half-plane. The real part of the pole is, in this case, negative which cancels the minus sign in the exponent. Also, its coefficient in (5-9) is −1 but because we are closing in the lower half-plane, there is another factor of −1 coming from the clockwise traverse of the contour. The net result is that the two contributions are the same. Because we are considering the 2 p S contribution, this gets multiplied by The next step is to repeat the whole process for the 2n S source. It is apparent from (5-11) that reversing the sign of Ω will change the signs of the imaginary parts of the two poles. This means that the e ikl integral now picks up a negative real part so the contribution is proportional to − Ω + Ω = which is again an outgoing wave. As before, the e ikl − contribution is the same as the e ikl contribution.
The fact that the solution only allows for an outgoing wave is actually a significant result on more than one account. The standard model is based on a linearization of Einstein's equations [2] that reduces to a simple wave equation, Green's function approach now requires that a choice be made as to how to offset the contours (see e.g. [4]) to avoid the poles. One makes a choice based on one's expectations regarding causality or some other criterion but that is still a choice. The equations don't make the decision for you. There is nothing in the standard model, for instance, that would disallow a scenario in which inward bound waves could add energy to a binary system. With time-varying curvature, there is no choice; the equations make the decision. What this means is that the time-variance of the background curvature fixes the dynamics so that the entropy is always increasing. This result is a striking verification of the model presented in [1]. The first pair is the same as for 12 g δ . The second set is a pair with a very small real value. This corresponds to a solution the varies with time but not with position (aside from the overall 1 es l − that arises from the spherical geometry.) A very small real value corresponds to an apparent index of refraction close to zero which in turn implies a phase velocity vastly larger than c. What we are seeing is a consequence of setting the transverse derivatives to zero. By doing so, we are in essence saying that the medium is infinitely stiff and such a medium would have an infinite phase velocity. This would also imply that the vacuum is acting on itself which we have disallowed from the very beginning. Of course, there is no such signal. The vacuum curvature is oscillating in synchrony without any transverse interaction at all at our level of approximation but from the point of view of the equations, the synchrony is a consequence of a very large phase velocity. It turns out in the end that the final contribution from these poles is extremely small and has no physical significance.
The solution for is different again because it has a factor of k 2 in both its numerator and denominator and because of the extra factor of k in Equation (5-7), the integrand does not vanish as k → ±∞ . In order to make the integral finite, we must introduce a cutoff. Large k corresponds to small distances so introducing a cutoff is equivalent to placing a limit on the minimum meaningful distance. A simple method that preserves the other desirable characteristics of the solution is to introduce another set of poles at ( After completing the solution, we find that We now consider the energy density and pressure. Both p δ and  p δ ρ have the form k 2 /k 2 so again we need to add a cutoff. The poles for the pair are the same and are the same as those of What remains now are the two sets of equations, (4-4b, g, and l) and (4-4c, i, and n).

Velocities
From (4-4l, m, n), we see that the velocities are given by This has the expected pole structure which, in fact, it is the same as that of 12 g δ .
When we solve for 01 g δ and 13 g δ , we find that 13 01 g g δ δ = − but we also find that their magnitudes are wildly too large. The problem is the extremely small denominator in the second line of the equation so clearly, we need to make an adjustment. To get some idea of the magnitude, we set 13 g δ to zero in the FT of (4-4g) This, like (6-1), is an odd function of k so an auxiliary function will be required.
After solving the equations, the derivative of (5-8) will cancel the factor of Ω in the denominator. What we do next is to substitute this guess back into the equations and calculate the We see that these are 90˚ out of phase with each other. We also see that, aside from a factor of c α , these have the same magnitude and that this magnitude is consistent with the magnitudes of the other components.
It is important to appreciate that these results are solutions to the equations.
While there is some uncertainty about the magnitude of the velocities, there is no uncertainty about the fact of the velocities because there is no solution of the equations with zero velocities. Another important point is that these results reflect the perturbation geodetic and so apply to all matter, not just the vacuum. The means, for example, that the entire Earth is undergoing transverse oscillations during the passing of a GW. Clearly, something is missing from these equations which results in their be-ing singular. The likely answer is that higher-order terms must be retained. In developing the equations, we made two assumptions. First, we assumed that we could linearize the equations with respect to the perturbation metric and second, we assumed that we could drop the transverse derivatives. The metric components are of ( ) O l − . We might consider retaining just the transverse derivatives but the 2 nd order metric contributions would be of the same magnitude so if we retain one, it would be necessary to also include the other. These higher-order terms are probably not too important individually but including these would couple these equations to all the other equations. The first equation, for example, contains a term proportional to . The problem we then have is that including the higher-order terms would land us in an entirely new realm of difficulty. Instead of dealing with a 1-dimensional, linear problem, it would become necessary to solve a 3-dimensional, non-linear problem. At some point, it would be useful to explore the 3-dimensional problem but that would require far more computer capacity than we have available.

Summary and Detection
The complete solution is shown below.
Magnitude is uncertain. It could possibly be larger but probably not smaller. , g ct l δ are not the same so the correspondence would be to elliptically polarization with again the same sense of rotation as the source. The solution here, however, is more complex because the metric contains additional components that couple the , x y coordinates to both the time coordinate and the z coordinate. Everything still oscillates at the same frequency but not in a form that can be described as simple circular polarization.
We also have the velocities. If we write but with a sense of rotation opposite that of the source. We will now consider some implications for a detection system. First, we note that because the wavelengths of the GW are very large, everything on earth and, in fact, the earth itself undergoes the oscillation as a single unit. This follows from the fact that the geodetic is the same everywhere on Earth so that all terrestrial matter moves with the same velocity. We argued earlier that the transverse variance of the components is on the order of ( ) As noted earlier, if we consider some detection system on a much grander scale, we must account for the variation in the distance from some chosen origin in the arguments of the metric functions as well as the motions of both our chosen origin and the reflectors. Also, the metric must be treated as the definition of a set of differential equations, i.e., instead of , ct x ∆ ∆ we need to solve for ( ) d d x ct , etc. and integrate over the appropriate paths.

Doppler Detection
The fact that distant sources would be in relative motion suggests that a detection system based on the Doppler effect might work. The idea would be to place a few satellites at distances on the order of 1/2 of the expected wavelength from the Earth and then to detect the Doppler frequency shift of signals returned from the satellites. The reason for that choice of distance is that it would maximize the relative GW velocity between the Earth and the satellite. The shift would be exceedingly small but the fact that its frequency would be known with considerable accuracy for any identified source should allow for the signal processing necessary to detect the GW.
We will set the stage with a few baseline numbers. The wavelength is given by What seems to be the most sensible arrangement would be to generate a signal at a local base station that would be directed outwards toward the satellites that would then act as passive mirrors to send it back. To generate the signal, the outputs of one or more optical atomic clocks operating at a single frequency would be combined and then run through a frequency comb to generate a signal at an intermediate frequency suitable for transmitting. The Mars mission uses an X band (8 GHz) signal for communications so that might be a suitable choice because the large antennas needed already exist. The received signal would nominally be an image of the transmitted signal twice shifted in frequency by the Doppler effect or Because of the extremely small frequency shift, it will be necessary to used interferometry in some form to detect the GW signal. If we heterodyned with a signal at the transmitted frequency, the output would be, The first three terms have frequencies either at the transmitted frequency or at double that frequency and consequently are of no interest. The last term is the one that concerns us. Given that the transmitted frequency would likely be in the GHz range, the last term is a signal with an extremely low frequency (on the or-Journal of High Energy Physics, Gravitation and Cosmology der of ( ) T es O l ω or 10 −9 -10 −11 Hz) that is modulated at the frequency of the source, Ω . Passing the output, (8-2), through a bandpass filter centered at the source frequency would leave us with a signal that would constitute detection of the GW. The additional phase term would also contribute at that frequency but its effect would be much smaller; on the order of ( ) reason for positioning multiple satellites is that at least one or two should be in a favorable position for any particular source. A third advantage is that detection would only require the use of a single satellite so no coordination between multiple satellites would be necessary.
The concept seems to be feasible but, as is the case for any such scheme, noise will be the controlling factor. Optical atomic clocks at present have frequency uncertainties of 17 10 f f − ∆ ≈ at optical frequencies. Our primary concern, however, is with the noise levels at the GW frequency. Each step, all the way from the clocks to the satellite and back, is a potential noise source and all these would need to be analyzed by experts in the various disciplines to determine if this approach would work. As a final thought, we have no idea if it is even possible but if a continuous sampling of the transmitted signal could be stored for the duration of the transit time of the signal, it could be used as the reference signal in the heterodyning process in which case the noise would become part of the signal instead of being a problem.

Conclusion
In this paper, we have presented an analysis of GW in which the background metric is one with time-varying curvature. Some basic features of the standard model are found in the solution but the solution also contains other features that are not found in that model. Notably, the model predicts that both the vacuum and all matter must undergo oscillations with the passing of the GW, a result that leads to the idea of a detection system based on the Doppler effect. The model also predicts that only outbound GW is possible which stands as a significant verification of the time-varying curvature background model of [1].

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.