4  Derivation of the pulsar resonance frequency

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4.1  The electrostatic-gravitational coupling in dynamic systems

The analysis will utilize the quasi-static form of the Maxwell equations, which is often satisfactory for low frequency problems. For the oldest pulsars the wavelength exceeds the diameter by a factor of 105, justifying the simplification. The coupling of the electrostatic and gravitational fields varies with frequency as ω1, and the time-varying part of the magnetic field also varies as ω1 in these solutions, so the magnetic-gravitational coupling varies as ω2, and it can be neglected at low frequencies. The low frequency approximation breaks down if the equations are applied to millisecond pulsars, or to objects with a mass vastly greater than a solar mass.

Vela X-ray image of the 11.2 Hz Vela pulsar [30]. Its inner jets show no indication of being cone shaped. The image is consistent with the pulsar being an aligned rotor.

Begin with two conductive masses connected by a wire. Cut the wire at the midpoint and insert an oscillator. Mount a clock on each mass. Also place a reference clock near the midpoint. Measure time by reading the clock faces from the midpoint. The magnitude of a potential is not locally detectable, so an observer riding with the mass on its journey in time will insist that the local system behaves in an entirely conventional way. ∂Ψ/∂t behaves like the expansion factor of the space part, and the expansion factor is not locally detectable either. A constant velocity in either space or time is not detectable by virtue of the motion itself.

The mass emits a gravitational flux which moves radially outward at c. As the voltage increases with time (with an undetermined polarity) the clock moves into the future relative to the reference clock. The time interval of the moving clock maps into a shorter time interval at the reference clock. An observer at the reference point therefore perceives a greater time-integrated flux from the source. The changing electrostatic field modulates the intensity of the gravitational field.

The step-by-step derivation of the following equations can be viewed in the computer generated files listed near the bottom of this page.

The voltage on one mass is V0 sin(ω t)/2. The voltage on the other mass is -V0 sin(ω t)/2. The distance between the masses is d, and their static gravitational binding energy is -G m1 m2/d. The magnitude of the one-way modulation is ξ dV/dt, and the modulated field strength enters into the energy equation in the same way as the magnitude of either mass, so from the preceding considerations the energy modulation is

(1 - V0 ξ ω cos (ω t)) (1 + V0 ξ ω cos (ω t)).

In being proportional to dV/dt the coupling seems to be linear, and in some sense it is, but in the overall system the coupling is associated with nonlinearity. That is more clear if the same method of analysis is applied to two charges, where the interactions produce fields quadratic in source strength. The significance is that in a fully developed derivation the electrostatic-gravitational coupling will be lost if the equations are linearized, because the coupling is a space-time cross term that is in the same order as the quadratic terms.

Proceeding similarly, if the gravitational flux from a parent body is enhanced in propagating to a test mass then the flux from the test mass propagates to the parent body along a path where the E field is of the opposite sign. The gravitational potential energy of a thin mass shell is -1/2 G m2/r0. The gravitational energy can also be computed by integrating -a2/(8 π G) over all space, where a is the acceleration. The following calculations proceed by multiplying this energy equation by the modulation factor, then integrating. The pulsar is assumed to be a good electrical conductor, so the E field does not penetrate to the interior region.

Without supposing that the appropriate pulsar spherical harmonics are yet known, begin with the lowest order Legendre polynomial that conserves charge, P1. The spherical harmonics represent the charge density on the surface of the sphere. The radial dependency must be computed. The static solution is multiplied by sin(ω t). The effects of retardation are not considered so the result is not an exact solution to the Maxwell equations, but it is quite close when the frequency is low. Then

Ψ = V0 r02 sin(ω t) cos θ/r2

Neglecting A, the E field is -grad Ψ

Er = 2 V0 r02 sin(ω t) cos θ/r3
Eθ = V0 r02 sin(ω t) sin θ/r3
Eφ = 0

The energy density is 1/2 ε0 E dot E

2 V02 ε0 r04 sin2 (ω t)/r6 -3/2 V02 ε0 r04 sin2 θ sin2 (ω t)/r6

Integrating over the exterior region

Ue = 2/3 π V02 ε0 r0 (1 - cos (2 ω t))

The fringing capacitance between the northern and southern hemispheres forms a capacitor. The equation gives the energy stored in the capacitor. The gravitational inductance of the conductive sphere prevents the capacitor from being short-circuited.

The gravitational energy density is -a2/(8 π G), with a = G m/r2. Multiplying by the modulation factor

Ug= -G m2/(8 π r4)
+G V02 m2 r04 ω2 ξ2 cos2 θ cos2 (ω t)/(8 π r8)

Integrating over the exterior region

Ug = -G m2/(2 r0)
+(1 + cos(2 ω t)) G V02 m2 ω2 ξ2/(60 r0)

At time t=0 the voltage is zero but it is changing at its maximum rate. Viewing the equation for the total energy as representing the sum of a constant negative binding energy and a positive periodic energy flow shows that the positive energy is at a maximum when the electrical potential difference is zero.

The energy stored in an inductor is 1/2 L I2, where I is the current. When resonated with a capacitor, the energy stored in the inductor is also at a maximum when the voltage across it is zero. Thus, at any given frequency, the coupling of the electrical and gravitational fields looks like an inductor. The frequency dependence is that of a capacitor, but it is the phase relationships that matter in computing the resonant frequency. For dense objects the gravitational inductance is far greater than the Maxwellian inductance, so the latter will be neglected.

Now connect an oscillator between the polar regions of the conductive mass. The energy equation shows that, when averaged over many cycles, the magnitude of the gravitational potential energy is less than before. The modulation weakens the gravitational field. The object is not in equilibrium, so its resilience will increase the radius. When the oscillator is turned off the object will return to its original radius, and the electrical energy transferred to the system could be recovered during the return.

Next, allow the radius to reach equilibrium with the oscillator on, then slowly compress the mass. The oscillator did work on the system in increasing the radius, so the system must do work on the oscillator when the mass is compressed. Gravitational contraction can power the oscillator.

At time t=0 all of the resonator energy is stored in the inductor. At the 90 degree point on the cycle all of the energy has been transferred to the capacitor, so the time-dependent portion of Ug at the first time must be equal to Ue at the second time. The two equations are solved for ω, then divided by 2 π to obtain the frequency in Hz. The solution is then reparameterized by the substitution r0 = 2 k G m/c2.

F = (40 ε0 G/π)1/2 k/(c2 ξ)

Substituting for ξ from the equation derived in Section 2

F = k b (40 ε0 G/π)1/2 c2 me2/(hbar qe)

hbar=h/(2 π), h=Planck's constant, G = gravitational constant, ε0= vacuum permittivity, me = mass of electron, qe = charge of electron, c = speed of light.

This frequency is for the P1 spherical oscillation mode. The numerical frequencies for if a few other spherical solutions are given below. The first column is the Legendre polynomial or the spherical harmonic that represents the charge distribution on the surface of a sphere. The second column is the resonance frequency. b is an undetermined coefficient but it probably has a magnitude near unity. k is the ratio of the actual to the Schwarzschild radius.
P1 0.3828 k b Hz   Frequency calculation
P2 0.5548 k b Hz   Frequency calculation
P3 0.7264 k b Hz   Frequency calculation
sin(2 φ) sin(θ)    0.5568 k b Hz   Frequency calculation

The power output is calculated in Section 5.10

4.2  Limitations

The gravitational redshift will lower the computed frequency of the oldest pulsars. The calculation is Newtonian, so there are several other inaccuracies near the Schwarzschild radius.

Self-initiated oscillation modes probably cannot evolve on the surface of a sphere without the preferred direction specified by the pre-existing magnetic field. The nonlinear coupling to the static field was not included in these calculations, but its stabilizing influence is probably essential.

The oscillation amplitude will build to the point where some nonlinear limiting process occurs, and that point is probably much less than Em. For example, a field of only 105 V/m would be sufficient to levitate a proton. This field is weak even by laboratory standards, and at some point it may become possible for surface charges to be swept away by the E field. The mass ejection would become so large at that point that further increases in oscillation amplitude would not be possible.

The millisecond pulsars are outside the range of validity of the low frequency approximations utilized if they are of normal mass, and observational data indicate that they are [24]. There may be a second solution representing the coupling of the gravitational and magnetic fields, with the E field playing only an auxiliary role. This kind of duality exists in the Maxwell equations, and it could carry over into the gravitational solutions. The magnetic flux lines within the interior of a highly conductive sphere are frozen, so the magnetic field cannot change without an accompanying mechanical deformation. A periodic mechanical deformation would change the surface elevation, providing a mechanism for energy transfer between the magnetic and gravitational fields. The millisecond pulsars are not just ordinary pulsars that operate at a higher frequency. There are essential differences between the two pulsar classes. For example, pulsars J2019+2425 and J2322+2057 have theoretical ages which exceed that of the universe itself.

These equations predict that young pulsars are much larger than is currently thought, and observational or theoretical data may exist that will not accommodate the larger size. The pulsar radii are difficult to infer, and there is very little observational data on the subject, but one estimate of the 4.2 Hz Geminga pulsar's radius concludes that it is less than 9.5 km [33], which is not consistent with the oscillatory model. The estimate is based on an upper limit for the unpulsed blackbody optical brightness. An essential consideration is that if the surface is highly conductive then it will be specular, which translates into a low thermal emissivity.

At still higher frequencies there is the constraint that the oscillatory calculations become so inaccurate as to be nearly unusable when the diameter is comparable to a quarter wavelength, and, depending on the value of b, that problem can arise with the youngest conventional pulsars. The neglect of the magnetic field in the calculations can also cause the size to be overestimated when the frequency is relatively high. The magnetic field of a highly conductive sphere cannot change quickly, but it may nevertheless be possible for mechanical deformations to generate internal E fields, which cannot exist either. However, the if the E field due to the rate of change of the magnetic field is equal and opposite to the E field due to mechanical deformation then the interior magnetic field can oscillate at the pulsar frequency. Such an effect, if it exists, would significantly affect the higher frequency solutions.

While the gravitational inductance looks like an inductor at any given frequency, the frequency dependence is quite different. In particular, the equations illustrate that if the vacuum surrounding the pulsar is replaced by a material with a higher dielectric constant then the frequency increases. Thus if there are additional energy terms that are in phase with the energy of the E field then the frequency will increase, which will result in a smaller pulsar for a given frequency and mass. There is then the possibility that a more refined derivation (along with some new theory) will predict smaller young pulsars.

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