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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 describing 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.

In a recent paper [

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. [

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.

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 completed 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 [

Here,

and

with the constants,

Another difference with respect to the standard model is that the background energy/momentum (EM) tensor is not zero. Again, from [

The background vacuum is at rest with the result that no velocities appear in this equation. The resulting Ricci tensor and the solution for the background metric components are given in the paper. (We note that with a flat metric, the Ricci tensor vanishes so in that case, Einstein’s equations for the background become just 0 = 0.)

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

We now wish to calculate the Ricci tensor. We write

The first term references only the coefficients of the background and the Ricci tensor that results will cancel against the background EM tensor. The perturbation coefficients are what remains,

We now need to consider just what is going on here. (Again, a review of Section 8 of [^{©} 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

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

The Riemann tensor is

and the Ricci tensor is then

Turning now to the EM tensor, we have

where

where

where we have introduced the dimensionless velocity

The final step is to include the source by combining its contribution with the vacuum EM tensor,

where

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

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

Since

At this point, we have 18 equations. It happens, however, that the two sets of equations, (2-14) and (2-18) are not all independent. After removing the redundant equations, we end up with a total of 14. Counting the variables, we have 10 metric components,

With the equations now specified at least symbolically, we will turn to 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

Here, M is the mass of each star and ^{11} m.) Because our time variable is ^{20} m which sets the scale for the Earth-source distance,

We now chose to place the source at the point

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

and taking derivatives with respect to

We will approximate the stars as point objects so the density of the system becomes

Integrating this overall space gives

Equation (3-5) 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

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

It now only remains to lower the indices and trace-reverse the result. The indices are lowered with the background metric so

where the tensors

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 [

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,

We now want to apply some physical reasoning to simply this result. First, we know from [^{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

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,

We next turn to the EM conservation equations, (2-14) and the vacuum geodesic equations, (2-18) and (2-19). We will spend a little more time on these because they are not all independent. After removing the transverse dependencies from (2-14), we obtain the following 4 equations. (From here on out, we will drop the bold font indicating a tensor, i.e. from now on

Doing the same with (2-18) gives us

The first of these states that

The last equation states that the pressure does not vary with distance but this is the only place that the spatial derivative of the pressure appears in the equations so we can’t enforce this constraint by eliminating the derivative from the other equations. We will find from the final solution, however, that while this constrain is not satisfied exactly, the numerical value of the derivative is vanishingly small. The 8 equations we began with have now be reduced to 4; (4-3a) and (4-2b)-(4-2d).

We will now list the final equations. For brevity, we have omitted the arguments of the variables.

The lower case

Because this system of equations is now linear, we can use Fourier transforms (FT) to find the solution. We represent each variable by the general form

Here,

We will deal with the time variable first. We take the FT of both sides of the equations. On the LHS, we integrate by parts to replace the time derivatives with factors of

The total solution is thus the sum of three partial solutions. The first with

eq | δ00 | δ01 | δ02 | δ03 | δ11 | δ12 | δ13 | δ22 | δ23 | δ33 | δp | δρp | v_{x } | v_{y } | v_{z } | S_{1 } | S_{2 } |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | x | x | x | x | x | x | x | x | |||||||||

2 | x | x | x | ||||||||||||||

3 | x | x | x | ||||||||||||||

4 | x | x | x | x | x | ||||||||||||

5 | x | x | x | x | x | x | x | x | |||||||||

6 | x | x | |||||||||||||||

7 | x | x | x | x | |||||||||||||

8 | x | x | x | x | x | x | x | x | |||||||||

9 | x | x | x | ||||||||||||||

10 | x | x | x | x | x | x | x | x | x | ||||||||

11 | x | x | x | x | x | x | |||||||||||

12 | x | x | |||||||||||||||

13 | x | x | |||||||||||||||

14 | x | x | x |

can set

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,

Here, the subscript “j” is a shorthand for the

After solving for the

The final solution is then

when we solve the system of equations, we find that

The

With the integrations now over the entire real axis, we can employ contour integration to perform the final integrations. The

which separates the integration into a sum of two integrals. For the first, we close the contour in the upper half-plane and in the second, in the lower half-plane.

At some point during the development of the solution, it becomes necessary to replace the symbolic parameters with numerical values. This, however, introduces a significant problem because of the vast range of magnitudes involved. For example, the scaling ^{th} power and then multiply by ^{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

We first solve the FT version of (4-4f) with the source set to

In this case, the numerator is a constant so the integrand including the extra factor of k from (5-7) vanishes as

Putting in numbers, these become,

We now see one of the consequences of time-varying curvature. The poles are offset from the real axis by an amount proportional to

Because of the energy density and pressure, the real parts are not exactly ^{7} times larger so the refraction offset will also be somewhat larger.

We now perform the contour integrations. The integral involving

The next step is to repeat the whole process for the

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 [

We will now just outline the process for the remaining components. The FT solution for ^{2} in the numerator and a factor of k^{4} in the denominator so the integrand again vanishes as

The first pair is the same as for

The solution for

The situation for ^{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

After completing the solution, we find that

Next is

Again, the real parts are very small and after the derivative in (5-8) is applied, the value of ^{−20} so the result is that

We now consider the energy density and pressure. Both ^{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

It will not be immediately obvious, but in fact, ^{−17} kg·s^{−2}·m^{−1}. The present-day background energy density, on the other hand, has a value at least as large as ^{−7} smaller.

Referring back to the constraint of (4-3b), taking the derivative introduces an additional factor of

What remains now are the two sets of equations, (4-4b, g, and l) and (4-4c, i, and n).

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

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 ^{2}/k^{2} and so need a cutoff. After working through to the final solution, we find,

Comparing the final two trial values of

We find again that

Using the same procedure on the 2^{nd} set of equations, we find,

We see that these are 90˚ out of phase with each other. We also see that, aside from a factor of

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 being 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 ^{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 complete solution is shown below.

where

Because of (3-1), only two of the parameters are independent. In summary, we have a solution that has some characteristics in common with the standard model, viz.

The final metric including the background is

We can simplify the results a little by recognizing that the argument,

We can now make a comparison with the standard model taking as an example, equation 18.19 of [

we see that the motion is elliptical with axes,

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 ^{−18} - 10^{−20}.

Consider for example, an arm oriented along the x-axis. For photons,

Because we are assuming the coordinate distance is very small, we can just solve this equation algebraically without needing to convert it into a differential equation. The result is,

Along the y-axis, we have

What the interferometer measures is the difference. For equal arm lengths, the result is,

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

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 _{s} or about 16 mins. The corresponding frequency would be around 10^{−3} Hz. For a one-way trip, the frequency shift would be ^{−18} - 10^{−20} depending on the distance to the source.

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 order of ^{−9} - 10^{−11}Hz) that is modulated at the frequency of the source,

This scheme would have several advantages over an interferometric system if it can be made to work. For one, all the signal generation and processing would occur close to home. For another, the distance to the satellite only enters insofar that a distance of 1/2 wavelength would maximize the relative velocity. This means that the position would not need to be controlled or even known with any level of accuracy. The only significant constraint on a particular satellite would be the tilt angle dependencies of

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

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 [

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

Botke, J.C. (2021) Gravitational Waves in a Universe with Time-Varying Curvature. Journal of High Energy Physics, Gravitation and Cosmology, 7, 607-631. https://doi.org/10.4236/jhepgc.2021.72036