Adv. Studies Theor. Phys., Vol. 6, 2012, no. 7, 339 - 354

Theory of Dynamic Gravitational

Electromagnetism

Shubhen Biswas

G.P.S.H.School, P.O.Alaipur, Pin.-741245(W.B), India shubhen3@gmail.com

Abstract The change of gravitational potential due to moving mass particle creates gravitational varying extra potential at the observation point. So for a moving mass particle there will be a dynamic change in extra potential at the observation point. It is postulated that the changing potential will be transformed into electromagnetic field. In a four dimensional space time continuum the electromagnetic field tensors are constructed with the help velocity dependents extra potential. The relations are very much consistent with dimensions and as well as Maxwell equation. With the help of these relations I have explained the planetary magnetic field, and calculation provides somehow a good agreement with the observed value for Earth and then for others. Excluding the ions core dynamo theory calculation for typical Neutron star (Magnetar) shows magnetic field in the order of giga tesla, gives a strong support to the prescribed theory of dynamic gravitational electromagnetism. Keywords: Dynamic gravitations, Extra potential, Electromagnetic field tensor, Maxwell relations, Earth’s magnetic field, Neutron star’s magnetic field, Intoduction From special theory of relativity diminishing of mass is equivalent to the radiation of energy also it is the fact that the energy withdrawn from a body becomes energy of radiation makes no difference[1].Suppose a rest mass ’m’ is moving uniformly with velocity ‘v’ and it is compelled to stop suddenly ,then it will encounter a instant loss of mass ( 1)mγ − ,where [1] .If it doesn’t allow to evolve in to mechanical or chemical energy then an amount of energy will be radiated due to mass energy equivalence. This is practically observed in high energy particle accelerators.

340 S. Biswas

The above fact of radiation can be explained over gravitational field. The general field equation actually constructed for the static mass and hence the corresponding metric [5] also. Now if the moving mass continues to move uniformly then there will be variation of metric which ultimately leads to a dynamic variation of potential at the observation point. This dynamic variation of extra potential generates vector potentials in four dimensional space time continuums. Hence from conservation of energy this variation of gravitational potential need to be identified with electromagnetic vector potential, because at the stopping of mass the electromagnetic radiation must be followed by the sudden unfolding of induced velocity dependent extra potential. Thus there have the scope to construct anti symmetric contra variant second rank electromagnetic field tensor [2] from the rank one velocity dependent extra potential. These facts will be formulated and supported with example in the following sections. The notations used in the following section are i, j, k, or usually 1, 2, 3 for spatial coordinates. , ,α β γ For four spaces time inertial coordinates. , ,μ ν λ Generally for general coordinates. 2. The variation of metric for moving mass From general theory of relativity around a static mass ‘m’ at any point gravitational field is defined by the space time curvatures. And around any point in a small region in the domain of gravitational field defines own space time co-ordinates [5]. Thus domain of entire field constitutes an inhomogeneous space time continuum. Now for weak static gravitational field the space time at any point may be represented by nearly Cartesian Co-ordinate system in which the metric tensor for space time are

(1) The inequality suggests that we ignore powers of hαβ higher than 1 in the

field equation which ultimately leads 12uv uv uvR g R G+ = .

At present it is to be aimed to unfold something new except the gravitational radiation and quantization of field. Let it to be considered what will be the actual happening of uniformly movement of source point with respect to field points i.e. point of observation. If the source moves with respect to observation point it is obvious that field point will face a dynamic change of gravitational field. That is for the uniformly moving source the observation point will encounter a variation of the metric tensor.

Theory of dynamic gravitational electromagnetism 341 From (1)

, [ 0]

g hor g h as

αβ αβ αβ

αβ αβ αβ

δ δη δ

δ δ δη

= +

= = (2)

αβ

η represent the metric of inertial Co-ordinates, but gαβδ can be

expressed as the derivatives of the metric.

.g

g αβ γαβ γδ δχ

χ∂

=∂

(3)

From (2) and (3)

g

h αβ γαβ γδ δχ

χ∂

=∂

(4)

Thus new metric can be represented by the initial metric as the following.

'

( )

g h h

gh

gg

αβ αβ αβ αβ

αβ γαβ αβ ν

αβ γαβ ν

η δ

η δχχ

δχχ

= + +

∂= + +

∂∂

= +∂

(5)

Then from (4) and (5) the variation of metric tensor at the observation point due to relative change of position of the source is just hαβδ .

As, ( )'

gg g

h

αβ γαβ αβ γ

αβ

δχχ

δ

∂− =

∂=

Hence when source is moving at any instant the change of the metric with respect to proper time δτ at the observation point will be

gh

γαβ

αβ γ

χδ δτχ τ

∂ ∂=

∂ ∂ (6)

3. Explanation of relativistic extra potential The line element in presence of gravitational field is given by K. Schwarzschild [4] as

1

2 2 2 22 2

2 21 1 .Gm Gmds c dt drc r c r

−⎛ ⎞ ⎛ ⎞= − − −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

342 S. Biswas 2 2 2 2 2sinr d r dθ θ φ− − (7) Representing (7) in quasi Minkowskian co-ordinates[5] as defined by the transformation   , and 

 

 (8)       When the field is very weak, such that

Then line element of the above will be just like a Minkowskian line element except the temporal part. Thus for weak field approximation we can ignore all hαβ part in the metric except . Then equating metric component with (8)

00 2

21 Gmgc r

⎛ ⎞= −⎜ ⎟⎝ ⎠

Or 00 00 2

21 Gmhc r

η ⎛ ⎞+ = −⎜ ⎟⎝ ⎠

° 

00 2

2Gmhc r

= −

g2

2, =g Gm

c rφ

φ= − (9)

As the only interested terns is using (6) the variation of the metric for

proper time δτ in which information’s of moving source will reach to the observation point leads

oooo

ghγ

γ

χδ δτχ τ

∂ ∂=

∂ ∂

ooh γ

γ

χ δτχ τ

∂ ∂=

∂ ∂

Theory of dynamic gravitational electromagnetism 343

As for =1,2,3o rγχ

τ∂

=∂

Here the observation point supposed to be rest with respect to moving source in its own co-ordinate system.

2

2 cos g

rc rφ γυ

ψ δτ= −

Putting rc

δτγ

=

Here ‘r ‘is the distance between retarded position of moving source and the point of observation.

2

2 h . cosg

oo rorc cφ υδ ψ= − , in terms of present position and

time from Fig.-1 cos at fixed 2r oc

υ πψ ψ= =

Or 2

2 2

2h . g

oo c cφ υδ = −

2 2

2 1h (1 )goo c

φδ

γ= − − (10)

Where 2 2

11 / c

γν

=−

.

Then (10) is the variation of metric due to moving source. And at the observation point associated varying extra potential

2

11g gδφ φγ

⎛ ⎞= −⎜ ⎟

⎝ ⎠ (11)

4. Formation of long rage rank one tensor potential Now when the source moves uniformly with respect to point of observation it is obvious at the observation oohδ term will be a function of proper time. Taking the derivative of oohδ with respect to the proper time τ

( )( ) oo

ood h dhd d

γ

γ

δ χδτ χ τ

∂=

∂

2

1(1 )ooh U γγχ γ

⎡ ⎤∂= −⎢ ⎥∂ ⎣ ⎦

(12)

344 S. Biswas U γ

the four velocity as source is moving uniformly respect to the observation point, then U γ can be taken inside the bracket.

2

1(1 )ooh U γγχ γ

⎡ ⎤∂= −⎢ ⎥∂ ⎣ ⎦

(13)

If the observation point has same velocity as of the source, i.e. both of them are co-moving, and then there will be no variation of as well as of field. In this

case 0oohδ = and hence ( ) 0,ood hd

δτ

= and principle of general equivalence

leads 00( ) 0d hd

δτ

= for any general co-ordinate system.

Then (13) can be written as 00 2

1(1 ) 0h U μμχ γ

⎡ ⎤∂− =⎢ ⎥∂ ⎣ ⎦

Or 2 2

2 1(1 ) 0g Uc

μμ

φχ γ

⎡ ⎤∂− − =⎢ ⎥∂ ⎣ ⎦

From the energy conservation point of view, the four divergence of flow of extra potential gives

Or 2

1(1 ) 0g U μμ φ

χ γ⎡ ⎤∂

− =⎢ ⎥∂ ⎣ ⎦ (14)

Here 2

1(1 )γ

− is not a pure constant is a function of relative velocity so cannot

be dropped out and the term in the square bracket shows the flow of extra potential at the observation point at present time.

Putting 2

1(1 )

0

g U A

A

μ μ

μμ

φγ

χ

− =

∂ ⎡ ⎤ =⎣ ⎦∂

(15)

Equation (15) shows divergence of four vectors ( )Aμ is zero. Thus first order approximation leads

2 2

2 2 2

1 1 1( ) , (1- )= 4 4

A gU higher orderc c

μ μυ φ υγ γ γ

= + (16)

Theory of dynamic gravitational electromagnetism 345

The authenticity of equation (15) can be checked by four divergences taking 2

2

1( ) ,4

A gUc

μ μυ φγ

= as the following.

2

2

1 , as .4

g Uc t

μμ

φ υ υχ γ

⎡ ⎤∂ ∂= − ∇⎢ ⎥∂ ∂⎣ ⎦

[ ]2

2

1 . .4

g gc

o

υ γ υ γ υγ

= − +

=

5. Properties and identification of velocity dependent four potential with electromagnetic four vector potential

a) From equation (11) moving mass point induces an extra potential over

static mass at the point of observation so from equation (16) Aμ , the velocity dependent four potential is just a four dimensional flow of gravitational extra potential.

b) If moving mass point comes to rest instantly then there must be a disappearance of extra potential into certain types of energy according to mass energy conservation [1]. In this situation the disappearance of velocity dependent extra potential might be transformed as electromagnetic energy .So velocity dependant four potential likely to be the electromagnetic four potential.

c) The electromagnetic four potential associate with a charged particle are 1 , A Ur

λ λ∝ and also the induced velocity dependent four potential due to

moving mass point from (16) 1A Ur

μ μ∝ .

Both of them are long ranged rank one tensor. d) From Lorenz gauge the four divergence of electromagnetic four

potential vanish, [6] also (15) shows same property for the induce velocity dependent four potential.

e) Here ( )Aμ is purely rank one tensor so it doesn’t contribute to the gravitational field which actually characterized over second rank metric. All these above physically means that when mass is moving uniformly there will be a flow of extra potential from the observation point with the same velocity as the moving mass point, and the induced gravitational extra potential will be transformed into some kind of four potential similar to electromagnetic four potential. Here let it to be postulated that the induced four potential is nothing but the electromagnetic four potential. Also it is obvious as the relative

346 S. Biswas velocity between source and observer vanishes the induced electromagnetic potential vanish instantly. 6. Representation of gravitational electromagnetic field As electromagnetic field tensors follow the general covariance formula in four dimensional spaces, our choice of electromagnetic tensor regarding the velocity dependent extra potential should be followed by the same.

Since the Maxwell’s electromagnetic field tensor v[F ]μ [2] in Relativistic

electrodynamics expressed as

vFv

v

A Aμμ

μχ χ⎡ ⎤∂ ∂

= −⎢ ⎥∂ ∂⎣ ⎦ (17)

The tensor associated with velocity dependent gravitational extra potential using equation given as [10]

( . ) ( . )v vg gvF U Uμ μ

μ δφ δφχ χ

⎡ ⎤∂ ∂= −⎢ ⎥∂ ∂⎣ ⎦

(18)

For uniform velocity the four velocities vU ’s are independent on .uχ Thus

( ) ( )v vg gvF U Uμ μ

μ δφ δφχ χ

⎡ ⎤∂ ∂= −⎢ ⎥∂ ∂⎣ ⎦

(19)

Using the above equation, the component

( ) ( )jo jg gjF U Uο

οδφ δφχ χ

⎡ ⎤∂ ∂= −⎢ ⎥∂ ∂⎣ ⎦

(20)

Here jχ represents space coordinates and οχ (=ct) represent time coordinate. The first partial differentiation in the square bracket of must be carried over constant present time (t0) and the second partial differentiation with respect to time must be carried over constant present position ( ).α

οχ At present time separation between moving point source and observation point is

space like [3], because, 2( ) 0d α αο οχ χ∑ − ≠

And 2 22( ) 0,c d t tο− = (21)

As, 2t tο= , at present time.

Now ,cdtUd

ο ττ

= is proper time

Using equation (14), at constant present time, (22)

Theory of dynamic gravitational electromagnetism 347

2( ) 0cd t tUd

ο ο

τ−

= =

Moving source υ Points of observation Present position (Present position and time) (0,0,0, 2τ ) [ , ]o ox tα Retarded position (0,0,0, 1τ ) Fig. 1 In fig.1, the retarded and present position of source in terms of proper coordinates are given by (0,0,0, 1τ ) and (0,0,0, 2τ .) Now proper time interval for the moving point source

2 1

2 1

2 1

( )1 ( )

d d

d t t

τ τ ττ τ τ

γ

= −= −

= −

Then, 2 1( )d d t tγ τ = − But present time related with retarded time [6], 1

rct tο = +

1( ) drd t tcο − =

Or, 2 1( ) drd t tc

− = [here present time 2 ]t tο =

Hence drdc

τγ

= (23)

Now ( )j j

j dUd

οχ χτ−

=

Then at constant present position, 0jd οχ =

oψ

rψ

ro

r

o

348 S. Biswas

Or j j

j d dU cd dχ χγτ τ

= = (24)

At present position implying (22), (23) and equation (24) in equation (20), and

using [ ]j j

rυ χ υ∂

∂ = we have

2

20 1

4jj

cF g υ υ= (25)

Equating the tensor component with the electromagnetic tensor components [2] derived from the Maxwell’s electromagnetic relations. 0 [in S.I units]jj E

cF = − 0 [ C.G.S units]j jF E in= − (26) Where

1 2 3

13 2

23 1

32 1

0

0[ ]

0

0

uv

E E Ec c c

E B BcFE B BcE B Bc

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥− −⎢ ⎥

= ⎢ ⎥⎢ ⎥− −⎢ ⎥⎢ ⎥− −⎢ ⎥

⎣ ⎦

In S.I system (27)

Equation (25) and (26) gives

2 3

20

1 14 4E= Gm

c r cg υ υ υ

γυ∧

− = − [In S.I units]

[In C.G.S units] (28) Equation (28) is the required representation of electric field at the observation point due to a moving point mass. The field tensor component

( ) ( )2 2

2 21 1

4 4Fj i

i jij d d

g gd dc cχ χυ υ

γ τ γ τχ χφ φ∂ ∂

∂ ∂⎡ ⎤= −⎣ ⎦ (29)

Here also putting drcd γτ =

, and contravariantly ,g

iigφ

χ

∂

∂= − the gravitational field

component.

=

= But Fij Bκ= [in C.G.S and S.I units]

Theory of dynamic gravitational electromagnetism 349 Then the required relation for the magnetic field due to a moving point mass is  

(30)

Equation (23) represents the magnetic field at the observation point due to a moving point mass. 7. Reconstruction of relations in the unit systems of electric and magnetic field and Maxwell’s relations Table-I Relations between C.G.S and S.I unit systems

2

2 2 2

8 10

S.I C.G.S/ 10 /

. / 10 /3 10 / 3 10 /

m s cm sg m s g cm s

c m s c cm s

υ υ ××

= × = ×

Using these values in equation (28) and (30), I have come to the relation for both the electric and magnetic field in the two units systems 41

3E [1S.I= 10 . . ]C G S−⇒ × 4B [1 S.I =10 . . ]C G S⇒ The above relations give the first essence about the similarity with the practically observed relations between two unit systems (C.G.S and S.I) in case of electric and magnetic field. Except this the consistency of the Maxwell’s equation regarding equation (28) and (30) can be proved as the following. ( )21

4E = cg υ ν∇ × ∇ × −

2

20

10,4 ( )Gm

r tc gυτ

ν ν⎡ ⎤= − ∇ × − ∇ ×⎢ ⎥⎣ ⎦

2

( 0 )2

cg rr

υ

νο

∧−= × 0,as [ 0]r t ν∇ × =

2

2 ( )cr g vυο= − × (31)

And

350 S. Biswas 1

4( [ ]t t c gυ ν∂ ∂∂ ∂− = ×B )

1

4 [ ]c t gυ ν∂∂= × But [ ]( ) 0rt

νο

∂∂ =

2

2 ( )cr gυο ν= − × (32)

Equation (31) and (32) implies E= .t

∂∂∇× − B (33)

Also as v[F ]μ in (17) is anti symmetric, it is obvious B O∇⋅ = (34) 8. Explanation of the Earth-Magnetism The most recently accepted theory to explain Earth magnetism is Dynamo theory [8] of molten ion core. This theory requires circulation of currents but according to Lenz’s law the induced magnetic field must oppose the very cause of generating the current up till now the energy needed to sustain circulating current are not very well understood .Except this there are number of discrepancies in dynamo theory [11].Here we need to consider about highly massive bodies which will be moving with very high speed to have the significant magnetic field using the equation just have been developed. Then we have the scope to treat the Earth as an experimental object for calculating its magnetic field as the following. For simplicity let us consider the Earth is made of with two hemispherical lobes and the centre of mass each of the lobe is at distance ' 'r from the Earth’s center (Fig.2). 1

4' 'cB gυ ν= − × (35)

' 14 ( ' ) coscB g

υ

υ ν θ= − × (36)

' 1 ( ' )sin4HB g

cυ ν θ= − × (37)

For horizontal component '

HB total magnetic field along horizontal direction for two lobes is zero. Only the vertical component exists and added up. The total vertically down word field from fig.2

Theory of dynamic gravitational electromagnetism 351 Fig2 represents Polar magnetic field of Earth for two hemispherical lobes.

'2B Bυ=

1 ' cos2 c gυ ν θ= ×

2 214 ( )

1. . cos ,[ 1]GMc R rυ υ θ

γ+= → (38)

2 214 ( ) 2 2 2 2

. . . , cosr GMc R r

r rB rR r R r

ω ω θ+

⎡ ⎤= =⎢ ⎥

+ +⎣ ⎦ (39)

2 3 3

2 2 2

14 (1 ). (1 )1/ 2'

R GMcR R

ω αα α

=+ +

Putting r Rα=

2 3

2 3/2

14 (1 )

GMcω α

α=

+.

2 2 3

2 2 3/2

1 . . .4 (1 )

GM RR c

ω αα

=+

Hence 2 3

2 3/2

1 ( ) .4 [1 ]

g RBc

ω αα

=+

(40)

B

R g '

M/2 o r M/2

B ' H P

Fig.2

ω

θ

θ

v '

vB

352 S. Biswas Equation (40) is applicable for any large spherical rotating bodies. 9. Calculation of magnetic field for Earth and other terrestrial bodies The matter density is not homogeneous in the Earth. And .375r R= as usual for the homogeneous body The Earth has an average mantle density 3gm/cc and the average core density 13gm/cc [8] and taking the density as a linear function of distance r . ( ignoring the’ Weichert-Gutenberg’ [7] Discontinuity at R’=.55 R from the centre.) Introducing R=radius of the earth

And density, 10( ) (13 )rrR

ρ = −

For a elementary cone, centre of mass

2 2

0

2 2

0

( ) tan'

( ) tan

R

R

r r drX

r dr

ρ τ π θ

ρ τ π θ

∫=

∫

' .68X R= For the hemispherical part centre of mass will be situated at

1 ' .342

r X R= =

.34α = In S.I units for earth 29.8 /g m s= ,

83 10 /c m s= × ,36370 10R m= ×

2 /

86400rad sπω =

Using equation (40), the calculated value of magnetic field at the pole,

4.58 10B −= × Tesla shows a good agreement with the order of observed value [8]. With the help of equation (40) the calculated values of polar magnetic field for the planets and a typical Neutron star (Magnetar) are as following. Calculated and observed values for polar magnetic field of the terrestrial bodies [7]-[10]. Here for all the bodies assuming value of ‘α ’ to the same to the

earth and taking 3 2 5

52 3/2

1 ( ) 5.8 10 5.8 104 (1 )

g Rc c

α ωα

−−= ×

= ×+

Tesla

Theory of dynamic gravitational electromagnetism 353

Table-II

2

9 7

10

5 5

6

Terrestrial bodies (m/s ) ( ) ( / ) calculated (T) observed (T)Mercury .38 .3829 .069 .90 10 10Venus .904 .9499 .0041 7.81 10 No significant field

1 1 1 5.8 10 6 10Mars .37 .53 .97 5.68 10

g R m w rad sg R wg R w

Earth g R wg R w

− −

−

− −

−

××× ×× 6

11 9 9

10Neutron Star 10 .0031396 8640 4.26 10 10g R w

−

×

In Table-2, T stands for magnetic field in tesla. * The other planet and sun are excluded from the table as they are gaseous in nature and contribute in various ways to their magnetic field. In the above table only mercurial magnetic field is 100 times high enough in the nano order calculated value. This could be happened due to solar wind and proximity to suns magnetic field which somehow enhanced Mercurial magnetic field. The others also show good agreement with the order of observed value. Specially the Neutron star,-very highly dense with temperature 1011 Kelvin and mainly consist of neutron and few percentage of proton and electron, although as a whole globally electrically neutral forbids one to set up ion-core-dynamo theory of magnetism. The very high value of magnetic field of a neutron star thus can be explained over the theory of dynamic gravitational electromagnetism, because more massive object with higher angular velocity produces higher magnetic field. 11. Conclusion Here I wish to say that the electric and magnetic field are evolved due to dynamic change of gravitational field. And the expression (28) and (30) represents the nature of electric and magnetic field corresponding to the dynamical gravitational field according to the theory which has been developed. The expressions (33), (34), and table 2, have introduced in supporting the Theory of dynamic gravitational electromagnetism. From equation (39) we can

immediately have 3

1 ,BR

∝ i.e., the dipolar nature of planetary magnetic field.

Especially the expression (40) may be put to the test of the theory for large massive rotating spherical body. Acknowledgements For writing this article I am grateful to Dr. Abhijit De and Prof. Bikash Chakroborty for their suggestions.

354 S. Biswas References

[1] A. Einstein, The principle of Relativity Einstein. A collection of original papers as the special and general theory of Relativity. Dover Publication (1952),69-71, 133, 153 [2] David J. Griffiths. Introduction to electrodynamics, third edition. Printice hall of India (1999) 536, 537 [3] Herbert Goldstein, Classical Mechanics, 2nd edition, Narosa Publishing House (1997) 15, 300-302 [4] J.V.Narlikar Introduction to cosmology , second edn.(1993),61-65 [5] S. Weinberg, Gravitation and cosmology. John Wiley and son (Asia), 2004 chapter 3-7, 180-181

[6] Wolfgang K.H. Panefsky, Melba Phillips, Classical Electricity and Magnetism Dover publication,2ndedition ,158-159, 343-347, 466-467

[7] Earth magnetic field, wikipedia, the free encyclopedia-en. wikipedia. org/wiki/earth magneticfield cited on 2nd April 2010. [8] Mercury magnetic field, en.wikipedia.org/wiki/mercury-[planet] cited on 10th April 2010. [9]Mars magneticifield and magnetosphere www.ssc./gpp.ucla.edu/personne/russell/papers/mars-(1997)-cited on 6th April 2010. [10] Neutron-star-en.wikipedia.org/wiki/neutron-star-cited on 29th March 2010. [11] Source of earth magnetic field-mb_soft.com/public/tecto2.html cited on April 2010. Received: October, 2011