International Journal of Astronomy and Astrophysics, 2012, 2, 174-179
http://dx.doi.org/10.4236/ijaa.2012.23021 Published Online September 2012 (http://www.SciRP.org/journal/ijaa)
Kruskal Dynamics for Radial Geodesics
Abhas Mitra1,2
1Theoretical Astrophysics Section, Bhabha Atomic Research Centre, Mumbai, India
2Homi Bhabha National Institute, Mumbai, India
Email: amitra@barc.gov.in
Received February 25, 2012; revised March 28, 2012; accepted April 24, 2012
ABSTRACT
The total spacetime manifold for a Schwarzschild black hole (BH) is believed to be described by the Kruskal coordi-
nates and , where r and t are the conventional Schwarzschild radial and time coordinates re-
spectively. The relationship between r and t for a test particle moving along a radial or non-radial geodesic is well
known. Similarly, the expression for the vacuum Schwarzschild derivative for a geodesic, in terms of the constants of
motion, is well known. However, the same is not true for the Kruskal coordinates; and, we derive here the expression
for the Kruskal derivative for a radial geodesic in terms of the constants of motion. In particular, it is seen that the value
of

=,uurt

=,vvrt
dd =1uv ) is regular on the Event Horizon of the Black Hole. The regular nature of the Kruskal derivative is in
sharp contrast with the Schwarzschild derivative, dd =tr
, at the Event Horizon. We also explicitly obtain the value
of the Kruskal coordinates on the Event Horizon as a function of the constant of motion for a test particle on a radial
geodesic. The physical implications of this result will be discussed elsewhere.
Keywords: Black Hole; Kruskal Coordinates; Spacetime Singularity
1. Introduction
It is known for more than 90 years that the region exterior
to a point mass or the event horizon () of a
Schwarzschild Black Hole (BH) can be described by the
vacuum Schwarzschild metric[1,2]:
>=2
g
rr m
22 22
d= dddd
tt rr
sgtgrg g
 
2
 (1)
where

=1 2
tt
g
mr,

1
=12
rr
g
mr
 ,2
=
g
r

,
and 22
=sin
gr

(). Here, we are working
with a spacetime signature of
==1Gc
1, 1, 1, 1
 and r has a
distinct physical significance as the invariant area radius;
i.e. 2
=4π
A
r
=0A
2m
=2rm
indicates invariant/scalar area of sym-
metric 2-surfaces. Clearly, then, by definition, the point
mass having is at , and hence, the scalar r
should continue to be a space-like coordinate even for
. The coordinate time t too has a physical sig-
nificance as the proper time of a distant inertial observer
. At , rr
=0r
r
S
blows up and as , the tt
<2rm
g
and rr
suddenly exchange their signatures and the
metric acquires a signature 1, 1, 1, 1
. Such an in-
version of signature appears to be unphysical because (1)
by, definition, r is spacelike and (2) by principle of
equivalance, there must be a local inertial frame where
the metric is Lorenzian with the original signature
. Further though, the signature of 1, 1, 1, 1 rr
Horizon, =2rm, the signatures of
would change if there would be region beneath the Event
g
and
g
remain un. Despite such fundaml phys
inconsistencies, in the black hole (BH) paradigm, it is
believed that the EH is merely a coordinate singularity
because the Kretschmann scalar is finite there. But it can
easily be seen that, 4
EH
changedenta ical
K
m
[3], and the idea that
E
H
K
is finite presume integration constant
ring in the vacuum Schwarzschild solution =2m
es that th
appea
,
is finite. However, it has been shown that,
=2 >0m though
is indeed finite for an object with radius
ch as the Sun or a neutron star, =2 =0m>2
b
rm, su
utral point particle with =0
b
r [4
order to progress, we ignore this scientific fact and
continue to work in the artificial and incorrect mathe-
matical paradigm of BHs.
The detail dynamics of a
for a ne-7]. Yet, in
“test particle” in the vacuum
external spacetime is well known for a very long time
and discussion on it is contained in practically every text
book or monograpgh on classical General Theory of Re-
lativity (GTR) [1,2]. One of the key aspects for studying
the kinematics of a test particle is the knowledge about
the relevant derivative of the spatial coordinate with the
temporal one. For instance for any geodesic having
angular momentum or not, one knows the details about
the behaviour of the Schwarzschild derivative ddrt or
ddtr. And the fact that dd=tr blows the up at
C
opyright © 2012 SciRes. IJAA
A. MITRA 175
Event utility of
othe beenlaimed since 1960 that
bo
Horizon restricts thethe Schwarzschild
dynamics below the EH. This tantamounts to the well
known fact that the vacuum Schwarzschild metric fails to
describe the spacetime inside 2rm. Some authors,
accordingly, swap the nomenclar and t inside
the EH. But this is inconsistent because the location of
the (1) EH is still denoted by =2rm and not by
=2tm and (2) by initial definitionoint mass re-
=0r and not at =0t.
On ther hand, it has c
tures
, the p
sides at
th the exterior and the interior regions of a BH may be
described by a one-piece coordinate system suggested by
Kruskal and Szekeres [8,9]. In this case too, the nomen-
clatures of r and t remain unchanged for <2rm
unlike what is done b some authors in a desperat
to uphold the BH paradigm. Though, in the intervening
42 years hundreds of articles have been written on
Kruskal coordinates, and most of the treatises on GTR
too regularly deliberate upon the original work of Kru-
skal and Szekers, the fact remains that sufficient effort
has not been made to study the kinematics of a test
particle in terms of the Kruskal coordinates so that one
could have a better insight and appreciation of the
kinematics inside the EH. Accordingly, here, we would
derive the expressions for the Kruskal derivative
ye move
dduv
for a radial geodesic. For the sake of completene
shall start from the usual description about the Kruskal
coordinates and first derive the exact expression for the
value of the Kruskal coordinates on the EH (
ss, we
H
u and
H
v) in terms of r and t. We shall show here t hat
H
u
and
H
v are alwas non-zero and finite in general. M
importantly, we shall explicitly show that unlike the
Schwarzschild derivative, the Kruskal derivates are re-
gular on the EH. Apparently this might be in accor-
dance with the singularity free nature of the Kruskal
coordinates. However, in a later paper, we shall show
that this is, ironically, not the case.
y ore
2. Kruskal Coordinates
are defined by pre-
crepen
First note that, Kruskal coordinates
suming that the integration constant appearing in the
vacuum Schwarzschild solution is non-zero in variance
with the actual result that, even though for an object with
>2rm (such as the Sun) >0m, for a point mass
[3-7]. Yet, in this papeignore this basic dis-
cy. For the region exterior to the EH (Sectors 1 &
3), the Kruskal coordinates are defined as follows [1,2]:
=0m r, we


1
1
=cosh;
4
=sinh; 2
4
t

12
4
1=1
2
rm
r
fr e
m



 (3)
Here the plus sign corresponds to “our universe” while
th
r
e negative sign corresponds to the “other universe” [1,
2]. The “other universe” is a legitimate mathematical
solution of the Schwarzschild problem irrespective of its
observational reality (under the assumption >0m), and
is a time reversed mirror image of “our universe”.
And for the region interior to the horizon (Sectos 2 &
4), we have


2
2
=sinh;
4
=cosh; 2
4
t
ufrm
t
vfrr m
m
(4)
where

12
4
2=1
2
rm
r
f
re
m



 (5)
In terms of u and v, the metric for the entire spacetime
is


3
222
22 2
2
32
d=d d
dd
sin
rm
m2
s
evu
r
r


(6)
The metric coefficients are regular everywhere (under
th
ng
zschild
For n a radial geodesic, the angular mo-
e assumption >0m) except at the intrinsic singularity
=0r. Since aftee Kruskal coordinates are defined
usi r and t, for a proper understanding of the Kruskal
dynamics, it is necessary to recall the inter-relationship
between the Kruskal and Schwarzschild coordinates for a
radial geodesic.
2.1. Inter Relation between Schwar
rall th
Coordinates
a test particle o
mentum is zero, and there is only one conserved quantity,
the energy of the particle (per unit rest mass), E, as
measured by a distant inertial observer:

d12
t
Emr
ds (7)
where s is the proper time. For a massless particle like a
photon, we have =E
, otherwise E is finite. For a
radial geodesic, the mn of the particle is determined
by (see Chandrasekhar, p. 98) [10]
otio

2
d=1
rEm
ufr m
t
vfrrm
m
(2)
where
2
dr
s (8)
and
d=
d12
tE
s
mr (9)
Copyright © 2012 SciRes. IJAA
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176
so that


1
2
12
d=
d12
Emr
t
rEmr
 (10)
Clearly as 2rm, ddtr. The latter relation
dd=0rt mcating that the r has a minimum at
2mmum of r i
ind
ay be indi
e mi. But since thnis =0r, it may be
icating that, for a point mass 2=0m [3-6]. However,
we shall ignore this indication in this paper.
Here note that if the particle is released from rest
(dd=0rs ) at =i
rr at =0t, from Equation (8), it is
seen that [10]

2=1 2i
Emr (11)
or,

1
2
2=1
i
rm E (12)
It is convenient to introduce a (cyclic) parameter
through

 
22
2
2
2
=2
=2
cos cos
1i
mr
E

(13)
Obviously,
=1cos
i
r
r
=0
when and at the EH, we have =i
rr
==2arcsin;=2
HEr m

(14)
ome ion
at the following Equation involving and
Now after smanipulat, Chandrasekhar arrived
t
[10]:

 
12 4
22
cos 2
d=
d22 2
cos cos
i
r
tEm



 (15)
H
This Equation can be integrated to find the
relation between t and r for a radial geodesic (actually, even
for
exact
non-radial geodesic this Equation would hold good):




32
2
1
=sin1
222
i
r
tEE
mm
tan2 tan2
ln tan2 tan2
H
H
 






(16)
The above Equation may also be written without intro-
ducing







H
and explicitly: (see p. 824 of ref.[1] or p.
34
E
3 of ref. [2]):



12 12
12
21 1
1sin
24
ii
ii
rm rr
rr
mm
12 12
211
=ln
2
ii
rm rr
t
m




 




(17)
We find from Equations (16) and (17) that, as
from Sector 1, the logarithmic term blows up and
, which is a well known result. Further Kruskal
envisage that approach to the EH
s III & IV corresponds to
zon
lation
 
2rm
t
coordinates
Sector
2.2. Kruskal
In
from the
=t .
Coordinates on the Event Hori
Sectors 1 & 3, Kruskal coordinates obey the re
=coth4
ut
vm
(18)
And since 2rm correspondts to , at the
have EH, we
=1; =
2 a
2
H
urm
(19
H
v)
other hand, in Sectors On the nd 4, we see
=tanh4
vm
(20)
and as r
ut
, we are led to
2r
m, the same
limit, m
that
2
(19).
t
In the sameEquation
we find
and t,
22 2
1
=exp
HH t
uv f
2m (21)
ght appear that since
we wo have
to
ain the actual
It mi
2f
he EH
values of
 
= 2=0m fm on
12
. Ae has
the EH, uld==0
HH
uv. But this is
incorrect because the temporal part of u and v tends
to blow up much more rapidly on tnd on
carefully obt
H
u and
H
v by
g out appropriate lim
so we introduce a new
workin
To do
its
variable
2
2
=2
i
zr E (22)
and, let, in the vicinity of the EH,
1=
1
mE
2=1 ;0rm
(23)
so
that
22
i
fm
H, by retai
(24)
Then, in the vicinity of the Ening terms first
order in
, we can rewrite Equation (17) as
12 12
12 12
14
=ln
14
i
i
r
zz
tmz
r
mzz

12
sin
21 4
ii
rr
mm
2
mz









(25)


As 0
, the logarithmic term in the above ex-
on becomes pressi
Copyright © 2012 SciRes. IJAA
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
11 8
4
=ln ln
11 4
i
ii
rmz
mz
At rr
mz


(26)
Then, using Equations (24) and (26), we find


22
1
8
exp= 412= 4
i
i
emz
f
AemreE
r
 (27)
Now considering the other terms in the expression for
2tm in Equation (25), we find that, in this limit,


22
12
==412
exp 1sin
24
HHi
ii
HHH
uv e mr
rr
mm



 

 


 
 



(28)
In terms of E, we have
22 2
2
sin
==4exp
21 E

e would have ==0
HH
uv if =0E or, if t
HH
HH H
uv eE E


(29)
On he test
particle is injected
is unphysical, and t see that
from rest
hus we
right at the EH. Clearly, this
H
u and
H
v are
non-zero. Further,te value for a fini of 2
i
rm
and v
v
Howe
or for
<1E
th
ex
for
, they are finite at
e EH is physically apb and are
r. ver
e too. The finit
pealing
regula
ness of
ecause u
at the E
u
Hpected to be completely
2=
i
rm or =1E, we find22
HH
uv==
.
usinOn the other ha
definition of and , we find that the iitial values of
nd, since =rr
i at =0t, byg the
u vn

2
22
2
== 21=
1
ii E
uu rmE
(30)
and
22
==0
i
vv 31)
3. Kruska l Derivative: A Direct Approach
Ha
(
ving shown that
H
u and
H
v
sic.
rentiating
aregeneral, non-
zero, we are now in a position to evaluate the Kruskal
derivative, thkey inredient for studying the Kruskal
, in
e
dynamics for a raodeWe first con
selves to Sector 1.diffe
g
dial ge
By
fine our-

1
f
r (Equation
[3]) with r we obtain

4
1
d= 2
d24
rm
fre rm
rmm
we
12
1
(32)
Then by directly differentiating Equation (2) by r,
find that irrespective of the sign of 1
dd
f
r, we will have
11
d
dd
=coshsinh
dd 44 4d
ff
ut tt
rrmm mr
(33)
Interestingly, in all the sectors, we obn the same
functional form of
tai
ddur. Using Equations (2) and (4)
in the foregoing Equation, we see that

1
dd
=12
d
t
r
(34)
d4 4
uu v
mr
rm m

On the other hand by differentiating Equations (4) and
(5), we find that

412
2
d= 21
d24
rm
fre rm
rmm
an
(35)
d
22
d
dd
=sinhc
dd 44
ff
ut tt
rr mm
osh
4dmr
(36)
And by using Equations (4) and (35) into the
Equation, we obtain the same expression (34) for
foregoing
ddur
in
e expression for
Sectors 2 & 4. Further, using Equation (10) in (34), we
obtain the ultimat

1
2
12
d=
d4 12
mr
uv
u
rm Em





E
r
valid in all the sectors. Similarly, we obtain the ultimate
functional form of
(37)
ddvr which is valid for
sectors:
all the

1
2
12
d=
d4 12
mr
vuE
v
rm Emr





(38)
dduv
37) with
And, the general value ofin any Sector is
obtained by dividing Equation ((38):
212
d=
d
Emr
uuE
v
 (39)
212
vE
u
vEm
r

3.1. Kruskal Derivative at the Event Horizon
Since u and v are expected to be differentiable smooth
co
also since
universe” is a mirror im “our universe”,
hat the value of
ntinuous (singularity free) functions everywhere except
at =0r (under the assumption >0m), and
the “other
we expect t
age of
dduv for a
bable di
he m
ny given r must
e, except for a pro
in both the universes. Teaningful way to
fin value of
be the sam
signature,
d the
fference in the
ddur at the EH o concentrate
onSectors 2 & 4 for which
will be t
=
the
H
H
uv
2
d==1; =2
H
u
uuv rm

(
d2
H
vv
uv
The Equation (39) for the Kruskal redivative, however,
tends to yield a “0/0” form at =rmfor Sectors 1 & 3
having =
40)
2
H
H
uv. But as mentioned above, we expect
this 0/0 form to acquire the valuedd=1uv because
Copyright © 2012 SciRes. IJAA
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178
these Scetors are the mirror images of Sectors 2 & 4.
Otherwise the whole idea of having an extended time
symmetric Schwild manifold would be incon-
sistent. Thus, in general, we must
arzsch
have
dd=1; =2uvr m
despite the 0/0 forms in some sectors. The fact that we
m
(41)
ust have dd=1uv for the Sectors 1 & 3 can be
reconfirmed in the limiting case of 22
==
HH
uv
for
=1E o=0
H for the (unphysical case) =0E
directly by using L’ Hospital’s theore
Note that, by this rule, we can write,
 
r
m
=
H
uv
.
 
0,0 0,0
dd
=
limlim dd
uv uv
uur
vvr
  (42)
In any case, from Equation (19), we already know that
=1uv at . Then we can rearr=2rm ange the fore-
going Equation as
 
0,0
dd
=1
lim dd
uv
ur
vr
 (43)
or,
2
du=1
lim d
rm
v
(44)
4.
It may bome in
A Different Route
e of sterest to rederive the limiting value
of dduv by using other generic relationships between
u and v. As before, to avoid 0/0 forms, we work with
Sectors 3 & 4. In particular, in Sector 3, we have
=coth4
ut
vm
(45)
By differentiating this equation w.r.t. v, we obtain

2=
dvvv
 2
1d11 d
44d
sinh
uu t
mtm v
(46)
By recalling that

1
sinh 4=tm vf, we rewrite the
aboation as ve equ
2
1
d1d
=
d4d
f
uu t
vv mvv
(47)
Now, from Equations (10) and (38), note that
2
ddd 4
==
ttr mE
dd
d12
vrv vEmr uE(48)
And the limiting value of
d4 42
==
d
2
H
H
H
tmm
vuvuuu
rm

(49)
And since , we find from Equation (47)
that
m

12=0fm
d=0; =2
dvv
uu rm (50)
Or,
d
=2 S
vv
rm
d== =1;
ectors 34
H
H
uu u
u
(51)
Similarly, for the sake of overall consiste
Sectors, 1 & 3, we must have
ncy, in
dd=1uv at
5.nt Consi
ally we could have obtained the above derived
unique result in a relatively simpler ma
rentiating the Global Equation
=2rm.
A Differederation
Actu
nner by diffe-
22 2
=21
rm
uvrme (52)
w.r.t. v:
2
2
dd
2 2=
dd
4
rm
urr
uv e
vv
m
(53)
First let us note from Equation (38) that in Sectors 2 &
4, the limiting value of


412
dmmr
r
d
21
2=0;
H
vv
u
mmr
v
(54)
2rm
Then, by using this above Equation in (53), we find
d=; 2
d
HH
u
uvr
v
m
(55)
so that
d=
d
v
u
vv
=1;2
H
H
rm
(56)
6. Conclusions
The Kruskal coordinates were found w
and in the present paper, we have worked out some
aspects of the kinematics of a test particle following a
rahe
precise
ay back in 1960,
dial Kruskal geodesic. To attain this we used t
value of
H
u and
H
v
i
ras a function of the initial
conditions of or
found that
the problem
22
, m E. It is clearly
=
H
H
uv
eeded
is no ge
hen toe esion for the
Kruskal derivative in terms of and r. We found
that the Kruskal derivative is ree EH
Scharzschild deivative(s) wher
n-zer
obtai
o in
n th
neral.
xpresWe t procm,
gu
e
E
lar at th unlike the
ddtr= at the EH.
In particular dd=1uv
at the EH if we consider the
Copyright © 2012 SciRes. IJAA
A. MITRA
Copyright © 2012 SciRes. IJAA
179
ose existence is suggested by the full Kreitler, Ed., Focus on Black Hole Research, Nova, New
,” Advances in Space Re-
pp. 2917-2919.
“other universe” wh
York, 2006.
[4] A. Mitra, “On the Non-Occurrence of Type I X-Ray Bursts
from the Black Hole Candidates
search, Vol. 38, No. 12, 2006,
Kruskal manifold, and which is a time reversed version
of “our universe”. But, if we move to the “our universe”,
the expected value of dd=1uv
at the EH.
The apparent regular nature of the Kruskal derivative
may be seen to be in keeping with the notion that Kruskal
coodinates are free of singularities at the EH. However, in
a subsequen we will delve into this question, and
fin
doi:10.1016/j.asr.2006.02.074
[5] A. Mitra, “Quantum Information Paradox: Real or Ficti-
tious,” Pramana, Vol. 73, No. 3, 2009, pp. 615-620.
doi:10.1007/s
t paper,12043-009-0113-9
d out other important features of the Kruskal dynamics
vis-a-vis the well known Schwarzschild dynamics.
Finally, this manuscript is a revision of the Cornell Univ.
Electronic Preprint (arXiv: gr-qc/9909062) which has been
vie readers in t
s, Vol. 50,
.
ole,
ysics and Space Science,
[6] A. Mitra, “Comments on ‘The Euclidean Gravitational
Action as Black Hole Entropy, Singularities, and Space
Time Voids’,” Journal of Mathematical Physic
No. 4, 2009, Article ID: 042501
wed by manyhe past 13 years.
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