Gravitational and Electromagnetic Field of an Isolated Rotating Charged Particle

A charged isolated particle with spherically symmetry is considered at origin in empty space. The particle has both mass and charge; therefore it is under the influence of both gravitational and electro-magnetic field. So to find out a line element especial attention is given in Einstein’s gravitational and Maxwell’s electro-magnetic field equations. Initially Einstein’s field equations are considered individually for gravitational and electro-magnetic fields in empty space. In this work initially starts with Schwarzschild like solution and then a simple elegant, systematic method is used. In these methods the e-m field tensor is also used from Maxwell’s electro-magnetic field equations. Finally thus a new metric is found for both positive and negative charged particles. The new metric for electron is not same as the metric is devised by Reissner and Nordstrom. The new metric for proton is used to find another new metric for rotating charged particle. The new metric is extended for the massive body and gives us some new information about the mass required to stop electro-magnetic interaction. It gives interesting information that planet having mass more than 1.21 times of Jupiter mass, live cannot survive. Also gives information that the mass greater than the aforesaid mass there is no electrically charged body in the universe.


Introduction
Einstein's field equations [1] are a set of nonlinear differential equations, so these equations are difficult to find the exact analytical solution. The exact solution has got physical meanings only in case of some simplified assumptions; among these solutions which are Schwarzschild solution [2], Kerr's solution [3,4] and FRW models [5][6][7][8] for cosmology. T Kaluza [9] in 1921 and later Oskar Klein [10,11] in 1926 try to solve the relativity as a geometrical theory of gravitation and electro-magnetic (e-m) field. The gravitational field due to an isolated electron was given by Reissner [12,13] and Gunner Nordstrom [14] in 1921 and later by G. B. Jeffery [15]. Schwarzschild metric was understood to describe a black hole [16] in the year 1958 and Kerr [3,17] generalised the solution for rotating black hole in the year 1963. Newman [18,19] try to describe the metric for charged, rotating body on the basis of Reissner-Nordstrom solution in the year 1965.

Schwarzschild Metric
The original field equation of Einstein's in empty space is given by The solution of the above Einstein's field equations in empty space of an isolated particle continually at rest at the origin was first given by Schwarzschild This metric is spherically symmetric and may be regarded as the gravitational field of a non-rotating point mass M at rest at origin.

Reissner-Nordstrom Metric
The metric for a non-rotating isolated electron was given by Reissner and Gunner Nordstrom as, Equation (5) gives the Newtonian potential φ given as, In equation (6) if we consider 0 m = then the force is inversely proportional to the cube of the distance, which is impossible. This means that there may have some discrepancy in mathematical derivation process in the Reissner-Nordstrom metric. This compelled me to think this problem seriously. The author [25,26] is trying to solve this problem but still no satisfaction. Again considering this problem a more simple elegant and systematic method is used to modify the new metric which is derived in this research paper. Later taking this new metric of non-rotating charged body a more enlighten way to find a solution for rotating charged body is attempted.

Kerr Metric
The Kerr solution in Boyer-Lindquist coordinates [3,17,19] In the above equations considered,

Newman Metric
The metric for rotating charged particle according to In these above equations 2 The symbol q is taken as the charge of the particle.

Derivation of the New Metric for Non-rotating Charged Particle
A simple elegant and systematic method is used here to determine the metric for non-rotating isolated charged body. Let us consider an isolated proton which is positively charged is placed at origin at rest in empty space. The proton has both gravitational and e-m field since it has both mass and charge. The gravitational and e-m fields of the proton are assumed to be spherically symmetric. The interaction range of the gravitational field is from infinity to 10 -33 cm and for em field it is from infinity to 10 -8 cm; therefore both gravitational and e-m fields have the interaction range from infinity to 10 -8 cm. Since the proton is under the influence of both gravitational and e-m fields in empty space, hence Einstein field equation (1) is applicable.
The fundamental metric or line element is given by, 2 ds g dx dx µ ν µν = (13) The most general form of the metric or line element for gravitational field of an isolated proton at the origin at rest in empty space satisfying the condition of spherically symmetric can be written as, Here and λ ν are the functions of r only.
Similarly the most general form of the metric for the e-m field of an isolated proton at origin at rest in empty space satisfying the condition of spherically symmetric is given by, In this above equation and a b are the functions of r only.
The equations (14) and (15) are individual metric or line element for each field of the isolated proton at origin at rest. Actually the isolated proton at rest at origin is under the influence of both self-gravitational and self-electromagnetic fields together. Therefore the most general form of metric or line element is, The solution of (14) was given by Schwarzschild as, The relation between m and the proton mass p m is, Again the solution of (15) becomes, In above equation the integrating constant is put as B .
Similarly the solution of (17) we can find out the solution as, Here we have taken D as an integrating constant. But still the value of B and D are unknown to us. To find out the value of B let we first find out an equation of motion of a particle [26] with very low velocity in static field, and then we find the e-m field tensor [26] and finally find out the value of B . Then we are able to find out the value of D.

Equation of Motion of a Particle
Let we consider a particle is in motion with very low velocity in a static field. Therefore the geodesic equation for a non-relativistic particle is: The metric in Riemannian space is given by the (13) and let us now assume that in (13) In above equation µν η are very small quantities and functions of , , x y z ; but independent of time t. This means that, 4 4 0 g Furthermore, the relation between Christofell's symbols of second kinds with first kinds, , g α ρα µν ρ µν Γ = Γ Then we can write from (23) in covariance form ( ) Neglected ρα η , because it is very small quantity and putting ρ α = we obtain, But the velocity v is non-relativistic and hence v c << then it gives, So we can write, Therefore by virtue of (22) and using (25) the above equation yields   2  44  44  2   1 for Again using the (26), This is the equation for motion of a particle. Now Newtonian equations of motion are Where φ is gravitational potential function and 1, 2, 3 α = . Therefore from (29)

The E-m Field Tensor
We have found the equation of motion for a particle in static field. Now we are going to find out the e-m field tensor for a charged particle which is also required to find out the value of constant B . Since the field in (15) is symmetrical and our assumption implies that the field is purely electrostatic. So the magnetic field intensities are, , , 0 The general potential K µ is defined in terms of electromagnetic potential A and scalar potential ψ as, Now the e-m field tensor F µν is defined as, , , This implies that the only non-vanishing components of 14 14 41 is and The current density J µ can be written as, The value of 0 F αν µ αν Γ = and taking J q µ = , where q is considered as charge, this yield But there is no charge and no current in the space surrounding the charged particle which is at origin. Therefore, The symbol ε is considered as an absolute constant. The constant ε is related with the charge of the particle which is situated at origin. This means 4 ( ) q πε = is charge.
Integrating above equation,

The New Metric
We know from (2) and (3), Considering 4 q πε = , the charge of the particle and Putting (44) From (47) For negatively charged particle or electron, In above equation e m is the mass of electron.

Derivation of the New Metric for Rotating Charged Particle
We consider the (47) and taking p m M = and q Q = for a massive body, Now we can transform (53) in to a null co-ordinate system where the time co-ordinate t is replaced by the null time coordinate u as, The equation (55) Therefore, The contra-variant form of g µν is The metric can now be expressed in terms of null tetrad { } We can easily found the followings: The used the method used by Newman and Janis [4], to make r complex (with r its complex conjugate) and replace the tetrad above by, Here r′ and u′ real. The ' a ' is a parameter and later we shall justify its interpretation as the angular momentum of the body.
For determination the values of and m m µ µ ′ ′ we used the usual formula given below: Now the contravariant components of the metric using (60) and (67) The covariant component is This is the required metric for an isolated rotating positively charged particle. The metric tensor is Putting the above values in below: The e-m potential is stronger than 31 1.29 10 × times of gravitational potential.
Let us consider another proton comes near to the origin particle up to distance 'r' and interacts both electrically and gravitationally. In the above equations we have used gravitational potential energy given by Newton's law since the forces are static and weak. Therefore we have extended the Newton's law in above equations for two protons also. Consider the (46) In equations (81) the gravitational potential energy is very weak then e-m potential energy. Let we consider isolated particle at rest in origin is a massive body , p M Nm = ( 1, 2,3, ..... ) N = ∞ which is nothing but the combination of protons. As number of proton increases the mass of the body increases and gravitational force increases, since all massive particles gravitationally interacts with all massive particles but one charged particle electromagnetically interacts with only one charged particle. Hence the (81) becomes,

Conclusions
In equation (47) to stop electromagnetic interaction the required mass is 1.898 10 gms × and the mass required to stop electromagnetic interaction is just 1.21 times greater than Jupiter's mass. So in any planet more than this mass life cannot survive, because in that planet electromagnetic interaction will be stopped by the gravity. Since the life is nothing but the low energy level electromagnetic interaction. Also this gives another interesting conclusion that more than the mass 0 0.00116 M (=384.63 times of earth's mass) there is no charged planets or stars. On the other hand when electromagnetic interaction stops by gravitational field then starts nuclear interaction and a star is born. This is the reason that stars are electrically neutral. Hence the metric given by the (75) is valid less than the aforesaid mass 30 2.29701 10 gms ×