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An alternative method for calculating the energy of gravitational waves

In the expansive nondecelerative universe model, creation of matter occurs due to which the Vaidya metrics is applied. This fact allows for localizing gravitational energy and calculating the energy of gravitational waves using an approach alternative to the well established procedure based on quadrupole formula. Rationalization of the gradual increase in entropy of the Universe using relation describing the total curvature of space-time is given too.

(1) where K αβ is the tensor of quadrupole mass distribution in the source of emission. For K αβ it holds: K αβ(t) = ρ (t,x) .(3X α X β −δ αβ X κ X κ )∆V (2) In the expansive nondecelerative universe (ENU) model [1], the creation of matter and of gravitational energy simultaneously occur. The laws of energy conservation still hold since the energy of gravitational field is negative in ENU. The total energy of the Universe is thus exactly zero [2] and the Universe can continuously expand with the velocity of light c. This postulate is expressed in the ENU by equation: (3) where a is the gauge factor, t c is the cosmological time, M u is the mass of ENU.
In such a model, due to the matter creation the Vaidya metrics [3] must be used which enables to localise gravitational energy. Weak fields are in first approximation described by Tolman's relation [4] ǫ g = − R.c 4 8π.G = − 3m.c 2 4πa.r 2 (4) in which ε g is the density of the gravitational energy being emitted by a body with the mass m at the distance r, R denotes the scalar curvature (contrary to the more frequently used Schwarzschild metrics, in the Vaidya metrics, R = 0). Using relations (3) and (4), equation expressing gravitational output P g is obtained: If the total mass of universe, M u is substituted from (3) to (5), relation for the total output of the gravitational energy P tot (its absolute value) by the Universe appears This output is time independent and represents at the same time the upper limit of the gravitational energy output. No machine or physical phenomenon may create a higher output than that of P tot . To illustrate the meaning of the above statement, radiant output of our galaxy is about 10 37 W, number of galaxies in the perceptible part of the Universe is about 10 11 which corresponds to the radiant output amounting 10 48 W.
Sources of the gravitational waves can be both periodical and aperiodical. As an example of aperiodical sources, accelerated direct motion, burst of supernova or nonspheric gravitational breakdown can serve. Further we will focus our attention to periodical sources such as planets rotating around the Sun or a double star rotating around the common centre of inertia. The emitted gravitational energy cannot exceed P tot value from (6). In any case, a condition (7) must be observed P gw ≤ E k .ωP ≤ c 5

2G
(7) where P gw is the energy of gravitational waves emitted within a time unit, E k means the kinetic energy of a body moving on circular or elliptic orbit with the angular velocity ω. In these circumstances (the excentricity of an elliptic orbit is omitted in deriving the following relations), the ratio of the emitted gravitational energy P gw and the kinetic energy E k of such body will be comparable to that of E k and P tot (in the limiting case, both value are identical and equal to unity), i.e. Pgw E k .ω ≈ E k .ω Ptot (8) It follows from (8) that the output of emitted gravitational waves is The validity of equation (9) was tested on a system consisting of two bodies with the nearly identical masses m 1 . = m 2 = m (10) These bodies rotate around the common centre of inertia on circular orbit with diameter r and angular velocity ω. The kinetic energy of either of the bodies is E k = 1 2 m.r 2 .ω 2 (11) Taking the identical masses of the bodies into account, both bodies must emit the same quantity of gravitational energy. Then it follows from (9), (10) and (11) Further domain of application of our approach lies in possibility to rationalize some questions which are still open, the entropy of the Universe in its begining being one of them. A problem arises [5] in interpretation of the individual elements in a simplified equation expressing the total curvature of space-time where R is the Riemann tensor representing the total curvature of spacetime, W is the Weyl's tensor describing deformation and slap forces, R o is the Ricci's tensor, g is the metrics tensor.
Application of the Vaidya metrics manifests (4) that the scalar curvature decreases in time. Since this curvature is a reduction of the Ricci's tensor, the latter must decrease in time too. At identical conditions the Reimann's tensor is time independent which leads to conclusion stating that the Weyl's tensor must be gradually increasing in time and, in turn, the Universe had to start its expansion being in a highly ordered state, i.e. in the state with a minimal entropy [5].
Conclusions The values of the energy of gravitational waves obtained using our alternative simple approach based on first approximation applied to the domain of weak gravitational fields are comparable to those derived by exact quadrupole relation. In addition to our previous results, the prospectiveness of the ENU model and Vaidya metrics is newly underlined.