Пресс-релиз популярных книг

Авторы: 111 А Б В Г Д Е Ж З И Й К Л М Н О П Р С Т У Ф Х Ц Ч Ш Щ Э Ю Я

Книги: 164 А Б В Г Д Е Ж З И Й К Л М Н О П Р С Т У Ф Х Ц Ч Ш Щ Э Ю Я

На сайте 111 авторов, 92 книг, 72 статей, 5913 глав.

3.1 Reissner-Nordstrom

Back

Consider the Einstein-Maxwell action

S =

16_G

d4xp􀀀g [R 􀀀 F__F__ ] ;

􀀀

R = R __

(3.1)

The unusual normalization of the Maxwell term means that the magnitude

of the Coulomb force between point charges Q1;Q2 at separation r (large)

in at space is

GjQ1Q2j

r2 (`geometrized' units of charge) (3.2)

The source-free Einstein-Maxwell equations are

G__ = 2

F__F _

_ 􀀀

g__F__F__

(3.3)

D_F__ = 0 (3.4)

They have the spherically-symmetric Reissner-Nordstrom (RN) solution (which

generalizes Schwarzschild)

ds2 = 􀀀

1 􀀀

dt2 +

dr2

1 􀀀 2Mr + Q2

_ + r2d2 (3.5)

A =

dt (Maxwell 1-form potential F = dA) (3.6)

The parameter Q is clearly the electric charge.

The RN metric can be written as

ds2 = 􀀀

r2 dt2 +

dr2 + r2d2 (3.7)

where

_ = r2 􀀀 2Mr + Q2 = (r 􀀀 r+) (r 􀀀 r􀀀) (3.8)

where r_ are not necessarily real

r_ = M _

M2 􀀀 Q2 (3.9)

There are therefore 3 cases to consider:

i) M < jQj

_ has no real roots so there is no horizon and the singularity at r = 0

is naked.

This case is similar to M < 0 Schwarzschild. According to the cosmic

censorship hypothesis this case could not occur in gravitational

collapse. As con_rmation, consider a shell of matter of charge Q and

radius R in Newtonian gravity but incorporating

a) Equivalence of inertial mass M with total energy, from special relativity.

b) Equivalence of inertial and gravitational mass from general relativity.

|M{tozta}l

total energy

= |M{z0}

" rest mass energy

GQ2

| {Rz }

Coulomb energy

􀀀

GM2

| {Rz }

grav. binding energy

(M=total mass)

(3.10)

This is a quadratic equation for M. The solution with M ! M0 as

R ! 1 is

M(R) =

h􀀀

R2 + 4GM0R + 4G2Q2_1=2

􀀀 R

(3.11)

The shell will only undergo gravitational collapse i_ M decreases with

decreasing R (so allowing K.E. to increase). Now

M0 =

􀀀

M2 􀀀 Q2

2MGR + R2 (3.12)

so collapse occurs only if M > jQj as expected.

Now consider M(R) as R ! 0.

M 􀀀! jQj independent of M0 (3.13)

So GR resolves the in_nite self-energy problem of point particles in

classical EM. A point particle becomes an extreme (M = jQj) RN

black hole (case (iii) below).

Remark The electron has M _ jQj (at least when probed at distances

_ GM=c2) because the gravitational attraction is negligible

compared to the Coulomb repulsion. But the electron is intrinsically

quantum mechanical, since its Compton wavelength _ Schwarzschild

radius. Clearly the applicability of GR requires

Compton wavelength

Schwarzschild radius

= ~=Mc

MG=c2 = ~c

M2G _ 1 (3.14)

i.e.

M _

_1=2

_ MP (Planck mass) (3.15)

This is satis_ed by any macroscopic object but not by elementary

particles.

More generally the domains of applicability of classical physics QFT

and GR are illustrated in the following diagram.

Classical

Physics

.....................................

...........

..........

...................

..........

.........

.....................................

..........................................................................................

....................................

. ..... ..... ..... .... ..... ..... ..... .............................................................

QFT

Strings?

M=MP

RMc

= 1

c2R

= 1

_1=2

ii) M > jQj

_ vanishes at r = r+ and r = r􀀀 real, so metric is singular there,

but these are coordinate singularities. To see this we proceed as for

r = 2M in Schwarzschild. De_ne r_ by

dr_ =

dr =

_ dr

1 􀀀 2M

r + Q2

_ (3.16)

) r_ = r +

2_+

jr 􀀀 r+j

2_􀀀

jr 􀀀 r􀀀j

r􀀀

+(c3o.n17st)

where

__ =

(r_ 􀀀 r_)

2r2_

(3.18)

We then introduce the radial null coordinates u; v as before

v = t + r_; u = t 􀀀 r_ (3.19)

The RN metric in ingoing Eddington-Finkelstein coordinates (v; r; _; _)

ds2 = 􀀀

r2 dv2 + 2dv dr + r2d2 (3.20)

which is non-singular everywhere except at r = 0. Hence the _ = 0

singularities of RN were coordinate singularities. The hypersurfaces of

constant r are null when grr = _=r2 = 0, i.e. when _ = 0, so r = r_

are null hypersurfaces, N_.

Proposition The null hypersurfaces N_ of RN are Killing horizons of the Killing vector

_eld k = @=@v (the extension of @=@t in RN coordinates) with surface gravities __.

Proof The normals to N_ are

l_ = f_

grr @

+ gvr @

_____

= f_

(3.21)

(note grr = 0 on N_ and gvr = 1) for some arbitrary functions f_

which we can choose s.t. l_Dl_

_ = 0 (tangent to an a_nely parameterized

geodesic) so

= f􀀀1

_ l_ (3.22)

which shows that N_ are Killing horizons of @

@v (This is Killing because

in EF coordinates the metric is v-independent). We can interpret the

LHS of this equation as a derivative w.r.t the group parameter, and

the RHS as a derivative w.r.t the a_ne parameter. Now

(k _ Dk)r = 􀀀r

vv = 􀀀

grrgvv;r = 0 on N_ (3.23)

(k _ Dk)vjr=r_

= 􀀀v

vv = 􀀀

gvrgvv;r =

2r2

____

r=r_

(3.24)

2r2_

(r_ 􀀀 r_) on N_ (3.25)

= __ (3.26)

:_: k _ Dk_ = __k_ (3.27)

Since k = @=@t in static coordinates we have k2 ! 􀀀1 as r ! 1. So

we identify __ as the surface gravities of N_.

Each of the Killing horizons N_ will have a bifurcation 2-sphere in the

neighborhood of which we can introduce the KS-type coordinates

U_ = 􀀀e􀀀__u; V _ = e__v (3.28)

For the + sign we have

ds2 = 􀀀

r+r􀀀

e􀀀2_+r

r􀀀

r 􀀀 r􀀀

_􀀀 􀀀1

dU+ dV + + r2d2 (3.29)

where r (U+; V +) is determined implicitly by

U+V + = 􀀀e2_+r

r 􀀀 r+

r 􀀀 r􀀀

r􀀀

__+=_􀀀

(3.30)

This metric covers four regions of the maximal analytic extension of

RN,

......... .......... .......... .......... ......... ....... ....... ....... ........ ....... ........ ...... ......... ......... .......... .......... ..

................... .. .. .. .. .... .. ... .

.. .. .. .. .... ... .. ....................

.. . .. ... ... ... .... ...................

. . . . . . . . . . . . . . . . . . . ....... ...

. .. . .. . .. . .. .. .. ................

. . . . . . . . . . .............

..................... .. .. .. .... ... .

.. .. .. .. .. . . . .. . . . . . . . . .

................... .. ... . .. .. ... ...

.. . . .. .. .. . . . . . . .. . . . . . . . . . .

.. .. .. .. . .. .. .. .. ....................

............ . . . . . . . . . .. . . . . . .

................................... . . . . . . . . . . . . . . . . . . . .

.................................................................................................... ..........................................................................

..............................................................................................................................................................................

IV I

III

U+ V +

r = r+

singular

in these

coordinates

U+ = V + = 0

r = r􀀀 r = constant

hypersurfaces

r < r+

r > r+

bifurcation

2-sphere

These coordinates do not cover r _ r􀀀 because of the coordinate

singularity at r = r􀀀 (and U+V + is complex for r < r􀀀), but r =

r􀀀 and a similar four regions are covered by the (U􀀀V 􀀀) KS-type

coordinates to this case (Exercise).

ds2 = 􀀀

r+r􀀀

􀀀

e􀀀2_􀀀r

r+ 􀀀 r

__􀀀 _+ 􀀀1

dU􀀀 dV 􀀀 + r2d(3.231)

U􀀀V 􀀀 = 􀀀e􀀀2_􀀀r

r􀀀 􀀀 r

r􀀀

r+ 􀀀 r

__􀀀=_+

(3.32)

This metric covers four regions around U􀀀 = V 􀀀 = 0.

....

......

........

..........

........

......

....

..................... .. .. ... .. .. .. .. .

.. ..... .. .. .. .. .. ....................

... .

.. .. ..

.. ... .

.. .. .. ..

.. .. .. .. ..

.. .. .. ..

.. .. ..

... .

..................... .. .. ... .. .. .. .. .

.. .... .... .... ..........................................................

..................................... ................................................................................................

. .. ... .. ... .... .. . ...................

...................................................................................................................................

.....................................

........................................................................................

.....................................

.................................................... ...... ...... ..... .... ..... .... ..... .

.... ......

..........

.... ......

..........

.... ......

..........

...................

..........

V 􀀀

V VI

r = constant r􀀀 < r < r+

r = r􀀀

r = constant 0 < r < r􀀀

curvature singularity at r = 0

U􀀀

III'

Region II is the same as the region II covered by the (U+; V +) coordinates.

The other regions are new. Regions V and VI contain the

curvature singularity at r = 0, which is timelike because the normal

to r = constant is spacelike for _ > 0, e.g. in r < r􀀀.

We know that region II of the diagram is connected to an exterior

spacetime in the past (regions I, III, and IV), by time-reversal invariance,

region III' must be connected to another exterior region (isometric

regions I', II', and IV').

... .. ...... .. .... ....................

.............................................................

.................................. ................................................................

...................... .... .... ... ....

............................................................................................. r = constant r􀀀 < r < r+

I' another branch of r = r+

II'

IV'

III'

Regions I' and IV' are new asymptotically at `exterior' spacetimes.

Continuing in this manner we can _nd an in_nite sequence of them.

Internal In_nities

Consider a path of constant r; _; _ in any region for which _ < 0, e.g.

region II. In ingoing EF coordinates

ds2 = 􀀀

r2 dv2 (3.33)

= j_j

r2 dv2 since are considering _ < 0 by hypothe(s3is.34)

Since ds2 > 0 the path is spacelike. The distance along it from v = 0

to v = 􀀀1 (i.e. to V + = 0 or V 􀀀 = 0) is

s =

Z 0

􀀀1

j_j1=2

dv = j_j1=2

Z 0

􀀀1

dv since r is constant(3.35)

= 1 (3.36)

So there is an `internal' spatial in_nity behind the r = r+ horizon.

(Note that one can still reach V _ = 0 in _nite proper time on a timelike

path, so the null hypersurfaces V _ = 0 are part of the spacetime).

If all points at 1, external and internal, are brought to _nite a_ne

parameter by a conformal transformation, one _nds the following CP

diagram, which can be in_nitely extended in both directions:

................................................................................................................................................................................................................................................................................................................................................................................................ ..... ...... ...... ...... ..... ...... ...... ..... ...... ...... ...... ..... ...... ...... ..... ...... ...... ...... ..... ...... ...... ..... ...... ...... ...... ..... ...... ...... ..... ...... ...... ...... ..... ...... ...... ...... ..... ...... ...... ..... ...... ...... ...... ..... ...... ...... ..... ...... ...... ...... ..... ...... ...... ..... ...... ...... ...... ..... ...... ...... ..... ...... ...... ...... ..... ...... ...... ..... ...... ...... ...... ..... ...... ...... ..... ...... ...... ...... ..... ...... ...... ..... ...... ...... ...... ..... ...... ...... ..... ...... ...... ...... ..... ...... ...... ..... ...... ...... ...... ..... ...... ...... ...... ..... ...... ...... ..... ...... ...... ...... ..... ...... ...... ..... ...... ...... ...... ..... ...... ...... ..... ...... ...... ...... ..... ...... ...... ..... ...... ...... ...... ..... ...... ...............................................................................................................................................................................................................................................................................................................................................................................................

........ ....... ....... ....... ....... ........ ........ ....... ....... ....... ....... .................. .......... .......... .........

.....................................

................ . . . . . . . . .. . . . . . . . . .

...... ...... ...... .................

. .. ... .. . . . . . . . . .. . . . . .

.....................................

.. .. .. . . . . .. . . . . . . .

. . . . . . . . .. . . . . ..................................... ................ ... .. ... .. .. ... .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .

. .... ... .. ... .. .....................

..... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

....... ........ ......................

.. ..... .... .. .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . .

.....................................

.. .

.. .. .. .. .. .. .. .. . . .. . . . . . . . . . . . . . . . . . . . . . .

...... ....

..... .....

.... ......

.......... . ...... ...

...... ....

..... .....

.......... .......... . ...... ...

..... .....

.... ......

.......... .. ........ .....

.... ..... .

.......... . ...... ...

...... ....

..... .....

.......... .......... . ...... ...

..... .....

.... ......

.......... .......... . ....

...................................... . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . .

IV I

III

=􀀀

i00

III'

II'

IV'

curvature singularity

at r = 0

curvature

singularity

at r = 0

internal

spatial 1

path

with r constant,

_ constant,

_ constant

new asymptotically at

exterior spacetime

=+0

=􀀀0

V VI

radial null ingoing

geodesic hits singularity

r = r􀀀

event horizon at r = r+

Пресс-релиз популярных книг

3.1 Reissner-Nordstrom

Популярные книги

Популярные статьи

Навигация

Пресс-релиз популярных книг

3.1 Reissner-Nordstrom

Популярные книги

Популярные статьи

Навигация

3.1 Reissner-Nordstrom