ELECTRODYNA..~ICS IN ABSOLUTE SPACE

ONE OF THE ORANGE AIV PREPRINT SERIES IN NUCLEAR, ATOMIC & THEORETICAL ASTROPHYSICS

2.2 ELECTRODYNA..~ICS IN ABSOLUTE SPACE

Here G is Newton's gravitational constant.

For the special case of a rotating Kerr black hole of mass M and angular momentum per unit mass a - L /Mc, and a Boyer-Lindquist spatial coordinate system: ^H

/:::, - r2 -

^2GMr^I^c²^{+ a}²

a w = 2aGMr/cA

_ c 2

A [

6

^1/2^-2_⁽^p²

6) e~

⁺^-2_^(p²

6)

^eAJ

2^p3 ll

a

^r ^A ^-^r

a

⁸ ^A ^{- 8}

with

!:r -

(6112

/p) afar and

:e =

(l/p) a;ae unit vectors

4 2 H

At.91, K=c (r+-GM/c )/2GMr+, r2 3

c a/2GMr+ n = e~

e

-r

f

(r+ +a cos ² ² ^{2 1/2}8) d8 = (r++a) ² ^{2 1/2}E[8, a (r+ +

^I

² a 2 1/2 )

J

near .91 0

with E - (elliptic integral of the second kind) ;

a const. x exp(-Kt/c) ~ r- r+ = const. x exp[-c ³t(r+ -GM/c ²)/GMr+) for freely falling observers near .J{ •

(2.16)

Our absolute-space formalism is also applicable to stationary, axially synnnetric black holes whose spatial metric yjk' lapse function a, and ZAMO angular velocity w are modified away from Kerr by the gravitational effects of surrounding matter.

'V E 411 Pe (2.17a)

'V B 0 (2.17b)

'V x (aB) 4naj/c + (l/c)

[E

+w£ E (E • 'Vw)m] (2.17c)

- - m-

'V x (C!E) -(1/c)[B+w.:EB -

m- (B · 'Vw)m] (2.17d)

Here £m~ is the "Lie derivative" of E along the toroidal Killing vector m

.:Sn~

:::

·

~)~

-

(E • 'V)m (2.18)

Notice that the unfamiliar expression in square brackets

(2.19)

is just the "Lie-type" time derivative, moving with the ZAMOs (dx/dt = wm), of the electric field. When the electric and magnetic fields are stationary and axisylllliletric, the terms

E,

_{_} £-_{m _}E,

B,

_{_} ^and:£_m^Bwill vanish. Notice also that the lapse function

a

is introduced to convert the curl and current terms in (2.17) over to the same "per unit global time t" basis as the time derivative terms.

The fields aE and aB, which we shall meet extensively below, are the electric and magnetic fields measured by ZAMOs, if the ZAMOs use global time t rather than ZAMO proper time T in computing the rate of change of momenta:

(d~/dt)Lorentz

force

a

(dp/dT)

The four Maxwell equations (2.17) can be reexpressed in integral form (equations I,4.1- I,4.4)

f ~. ~

⁴¹¹

f

^Pe^dV (Gauss's law for E) (2.20a)

a

7r 7r

f ~ . d~

⁰ (Gauss's law for B) (2.20b)

a?r

f

^{a(B-l vx}_- _c_- _-^E)·di= .!_i__ _- _{c dt}

f

^E•_- ^{d[ +}_- ⁴⁷¹_c

f

^et(j-p_- _e-^{v) · d[}

a.A( t)

f a(~+~~

~)

^·di

a..A<

..A(t) ..A(t)

1 d _c_dt

f

^{B. d[}

A( t)

(Ampere's law)

(Faraday's law) .

(2.20c)

(2.20d)

Here 'II is any 3-dimensional volume entirely outside the horizon and a'/f is its 2-dimensional boundary; dV = d(proper volume) and

d[ = (outward pointing ~;:~_:: ;:~~~~ to a'/f)d(proper area). Also .A(t) is any 2-dimensional surface entirely outside the horizon and aA(t) is its boundary curve; d£ = d(£:~e:: ~~:_::~~::along boundary curve); and the orientations of d[ and d£ must be chosen in accordance with the right-hand rule. Finally,

~ is the physical velocity

v [

d<p:~P:: ~.:~:)]

dT relative to ZAMOs

(2.2la)

of a point on .A(t) or a.A(t), relative to and as measured by the ZAMOs. (Thus

~.

[

d(l'.E~l'.EC:E ~~~E~!:~EC;)]

uv +Wm =

- - dt relative to absolute space

(2.2lb)

is the velocity, per unit global time, relative to absolute space.) The Maxwell equations (2.17) imply the differential law of charge conservation (equation I,5.11)

ap e /at + wm _ • \IP _ e + \I· _ (aj) 0 (2.22) Note that a/at+wm · \I is the global time derivative along the ZAMO

trajectories (where Pe and j are measured), and the factor a converts j over from "charge per unit area per unit proper time T" to "charge per unit area per unit global time t." The integral formulation of this law of charge conservation is (equation I,4.5)

d dt

J

'Ir( t) p dV

J

a vc

^t)

(2.23)

The electric and magnetic fields can be derived from a scalar paten- tial Ao and a vector potential A (equations I,5.9)

(2.24a) - A· m

B 'V x A (2.24b)

When these expressions are used, the source-free Maxwell equations (2.17b,d) are automatically satisfied.

A test particle of charge q and rest massµ, moving through absolute space, obeys the equation of motion (equation I,5.12)

a-1

[a/at+ (av+ wm) · 'V)p

(2.25)

Here vis the particle's physical velocity relative to the ZAMOs (Eq. 2.21);

µf is its mass-energy, and pits momentum as measured by the ZAMOs (

v2)-l/2

r =

1 -

2

p (2.26)

a-1

[3/3t+ (av+wm) · 'V] is the ZAMO proper time derivative moving with the

particle; µf g is the "gravitational acceleration" of the hole on the particle;

-1[

J

a w(p·'il)m - C£·~)~w is the "frame-dragging" force of the hole's rotation on the particle; and q[~ + (':_/c) x ~] is the Lorentz force.

The ZAMOs will characterize the electromagnetic field and/or any continu- ous medium present by

E - (total mass-energy density, erg/cm3

, as measured by ZAMOs) S - (total flux of energy, erg/cm sec, as measured by ZAHOs) 2

W _ (stress tensor, dyne/cm, ² as measured by ZAMOs). (2.27) In terms of these we can also define densities and fluxes of "redshif ted energy" (also often called "energy-at-infinity"), and of "angular momentum about the hole's symmetry axis" (equations I,5.15 and I,5.16)

aE+ws · m/c ² ~E ^as⁺^{wW · m}

EE - - (2.28a)

S · m/c ² ~L ^W·m

EL - ^- (2.28b)

For the electromagnetic field

E =

s

(c/4rr) (Ex B) (2.29a)

w (2.29b)

(a/Brr) (E2 + B2

) + (w/4rrc) (Ex B) · m (2.30a)

~E

^(2.30b)

(l/4rrc) (Ex B) · m (2.3la)

~L ^(l/4^rr) [-(E·m)E (2.3lb)

For a perfect fluid with density of mass-energy p (erg/cm ) and pressure 3 p (dyne/cm ) as measured in its own rest frame, and with velocity v as 2 measured by the ZAMOs,

2 2 2

E =

r (

p + pv I c )

w

(l/c )(p+p)f ² ²~@:'. +

PJ

s

(p+p)fv ²

2 2 -1/2 f - ( 1 - v / c )

(2.32a) (2.32b)

The law of local energy balance as formulated by the ZAMOs is (equation I,5.13)

0 if all forms of energy and stress are included in E, ~' ~

-j E if only electromagnetic contributions are included (2.33a) in E, ~' W.

(The third term on the left side of the equation is caused by the shear of the ZAMO trajectories relative to each other.) The law of local momentum balance (force balance) as formulated by the ZAf-!Os is (equation I,5.14)

0 if all stress and energy are included in S and W

(2.33b) -c2

(p E+ (j/c) x B] if only electromagnetism is included.

e- - -

(The test-particle equation of motion (2.25) can be derived in the usual way from this general force-balance equation.) From the laws of energy balance and force balance one can derive differential and integral conservation laws for redshifted energy and for angular momentum (equations I,5.17 and I,5.18)

0 i f all included

if only electromagnetism,

-1 -1 ( )

o. (Cl/Clt + w~ · ~)£L + o. 'V • O.~L 0 i f all included

if only electromagnetism;

£E dV +

f o.(~E

£E~)

^•^dl: ⁰ ^{i f}^all^included

Cl'V( t)

f {o.

~

^·

~+aw[pe~+

^(1/c)^x

~] ·~}dV

^{i f}only electromagnetism, 'V(t)

£LdV+

f a(~L-£L~)·d~

<1/r(t)

f a[pe~+

^(l/c)^x

~]

^{· m dV}

'V ( t)