Taub-NUT space as countermeasure to almost anything - Einstein equation, classical mechanics, and differential equation

Misner, C. W.

English

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Taub-NUT Space as a Counterexample to Almost Anything 
by 
C. W. Misner 
TECHNICAL REPORT NO. 529 
~. 
November 19 65 
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ff 653 July 65 
UNIVERSITY OF MARYLAND 
DEPARTMENT OF PHYSICS A N D  ASTRONOMY 
COLLEGE PARK, MARYLAND 
A contributed lecture given at the American Mathematical Society's 1965 
Summer Seminar on Relativity and Astrophysics at Cornel1 University, 
August 1965. f 
https://ntrs.nasa.gov/search.jsp?R=19660007407 2020-03-16T23:07:54+00:00Z
Taub-NUT Space as a Coanterexample to Almost Anything* 
C.W. Misner 
University of Maryland 
Department of Physics and Astronomy 
College Park, Maryland 
* Supported in part by NASA Grant NsG 436 
1 
Taub-NUT Space as a Counterexample to Almost Anything 
C.W. Misner 
This lecture will discuss some of the peculiar properties of the 
metric 
2 2  2 2  ds2 = (t2 + & )(de + sin Qd8 
+ U(t)(2C)2(d$ + cosed8) 2 
where 
2 mt + C U(t) = -1 -f- 2 
t2 + k2 
This metric satisfied the empty-space Einstein equations 
R = O  
uv (3)  
and has been discovered by both of the prime exact-solution-finding 
methods mentioned by Kerr in his Lecture this morning. Taub (1951) 
discovered it in a systematic development of a class of metrics with 
high symetry. Later it was rediscovered by Newman, Unti, and Tamburino 
(1963) studying a class of algebraically special metrics. Actually, 
2 
Taub gave t h e  me t r i c  i n  a coord ina te  system covering only t h e  reg ion  where 
U ( t )  > 0 ("Taub space") i n  which t h e  t = cons t  hypersurfaces  are space-  
l i k e ,  whi le  Newman, Unt i ,  and Tamburino gave t h e  reg ion  where U ( t )  < 0 
("NUT space") i n  which the  $ - l ines  (t08 cons tan t )  are t i m e - l i k e .  
This Taub-NUT space has many unusual p r o p e r t i e s ,  some of which are 
a l s o  known i n  o t h e r  metr ics .  I w i l l  g ive  a s h o r t  l i s t  and then  d i scuss  
a few: 
- 1. Although t h e  Taub region U > 0 can  be i n t e r p r e t e d  as a 
cosmological s o l u t i o n  w i t h  homogeneous but  non- i so t ropic  space s e c t i o n s ,  
i t  evolves i n t o  NUT space which seems t o  have no reasonable  i n t e r p r e t a t i o n .  
- 2. The NUT reg ion  contains  c losed  t ime- l ike  l i n e s .  
- 3. The NUT reg ion  does r o t  con ta in  any decent  space- l ike  hyper- 
su r f aces ,  
- 4 .  Although t h e  curva ture  t enso r  vanishes  as one approaches i n f i n i t y  
uv-) L' i n  space - l ike  d i r e c t i o n s ,  asymptot ica l ly  r ec t angu la r  coordinates  (g 
do not  e x i s t .  
- 5 .  Taub space a l lows ,  besides  Eq. (1) another ,  i nequ iva len t ,  m a x i m a l  
a n a l y t i c  e x t  em ion. 
- 6 .  Taub-NUT space  i s  non-singular i n  a meaningful mathematical  sense  
3 
but i s  not geodesically complete. 
- 7.  There are closed geodesics (circles) on which one cannot extend 
the solution of the geodesic equation to infinite values of the 
path parameter. 
The first four points are discussed in my (1963) paper on NUT space 
and will not be considered further here. Points 6. and 7 .  are based on 
a study of the geodesics whi.ch Taub and I (1966) hope to finish writing 
soon. An example of inextendable closed geodesics (7 )  for a simpler 
metric is given in the paragraph containing Eq. (1) in my 1963 paper. 
The question of the non-uniqueness of analytic continuation for metrics 
(point 5) i.s also based on the paper by Taub and Misner (1966) and will 
occupy us for the reminder of this talk. 
Before disc-.msing ar.alytic cont.inuations we must discuss analyticity. 
A function of severai real variables f(xy .... z) is analytic at 
xoyo .-.. z 
non-zero radius of convergence. A function f(P) on a manifold M is 
analytic if it can represent as an analytic function f(x ,x , ... x ) 
of the coordinates in the various coordinate patches defining the analytic 
structure of the manifold (see Auslander and MacKenzie 1963 for the 
definition of manifolds and of differentiable structure). Tensors are 
analytic if their components are analytic functions of the coordinates 
is a metric only if in each coordinate patch. A symmetric tensor 
it has the proper signature (-+++); in particular g = det g cannot 
if it has a power series expansion about that point with a 
0 
1 2  4 
gUV 
UV 
4 
vanish. The metric of Eq. (1) fails to satisfy the signature requirement 
of 8 = 0,n since in these t$88 coordinates one has 
We will therefore try to interpret Eq. (1) as defining the metric on a 
coordinate patch where tJt88 have the ranges 
-a < t c + 
0 < $ < 4 n  
(5) 
Analyticity of the metric in this range is obvious, as is the choice of 
limits on t and 8. It remains to be seen whether there exists any 
larger manifold containing this coordinate patch on which the metric can 
be analytic. We will study some simpler examples before returning to this 
question. 
As a very simple example consider the metric 
ds2 = dt 2 + d6 2 
on a coordinate patch -03 t 0 3 ,  0 c 8 .e 2n. (Fig. 1). We are accustomed 
to interpret this metric as a cylinder R x S'. Here R means the real 
line -m < t c O D ,  and S' is the 1-sphere or circle (base of the cylinder). 
I shall not attempt to explain how one looks at a metric on a single 
coordinate patch, such as Eq. (1) or Eq. (6), and tries to guess what the 
5 
full manifold containing this patch should be. 
how to verify a guess. I guess that the metric 
But we shall now consider 
L 
0 
dxL + dy 
ds2 = 0 
L x + yL 
which is clearly analytic in the region 0 e x  2 2  + y < + QP might be 
related to the metric of Eq. ( 6 ) .  The coordinate transformation 
t x = e case 
t y = e sin0 
(7) 
shows indeed that these metrics are equivalent on the region 
(which is the largest region to which this correspondence between 
0 # 0 
xy 
and te coordinates can be 
to be preferred since it is 
xy plane. One may begin to 
extended in a one-to-one way) but Eq. (7) is 
analytic also on this positive x-axis of the 
worry about the "singularity" of x 2 2  + y 0
in Eq. (7); since it corresponds to t + -00 in Eq. ( 6 )  it clearly is 
- not a singularity. A coordinate independent way of stating this is to 
note that every geodesic which approaches the boundary of the manifold 
(x + y + 0 or m) has infinite length. 2 2  
Some slightly more complicated metrics to serve as further examples 
are 
= dt 2 + de 2 + sin 2 0d8 ' 2  I ds 
= -(2 1 + cos0)dt 2 + de 2 + sin 2 2  0d8 
ds I1 3 
. 
6 
c 
= dt 2 + e 2t de 2 + sin 2 2  ed4) 
111 ds 
The first metric here obviously represents a higher dimensional cylinder 
2 2 R x S which is just a linear (R) stack of ordinary spheres (S ), while 
the other two metrics represent some deformations of this first hyper- 
cylinder. 
first two metrics are analytic, the third is not even continuous over the 
whole cylinder R x S . Analyticity of the first metric (by which we 
now mean the existence of an analytic extension of it to cover all of 
R x S ) is demonstrated by writing 
What is not so obvious to the unpractised eye is that while the 
2 
2 
ds 2 1  = T(dx 2 + dy2 + dz 2 ) 
I r  
with 
2 r2 s x2 + y2 + z 
2 which is obviously analytic for 
Eq. (9) by the transformation 
0 < r < +a, and which corresponds to 
t x = e sinecosfi 
t y = e sinesin@ 
t 
z = e cose 
and write 2 ds I1 Use the same transformation on 
c 
dsII 2 = ds12 + -(cos0 1 - 1)dt 2 3 
Then since cos0 = z/r is analytic for r2 > 0 and 
1 
r 
dt = y(xdy + ydy + zdz) 
is an analytic differential form in this region, the x y z  components of 
ds 
is verified from 
will also be analytic functions. The condition of proper signature I1 
in xyz coordinates leads to no interesting extension 2 111 But studying ds  
for we have 
2 2 + (e2t - l)d0 = ds 2 111 I ds 
and from the transformation (14) one sees that 
dz z xdx + ydy 
r (x + Y )  
- (x2 + y 
2 de = - r 2 2 2 %  
is not an analytic differential form on the ,z-axis where x/(x 2 + y 2 %  ) 
for instance is not a continuous function. We could then ask whether some 
coordinate transformation different from Eq. (14) might not lead to a form 
8 
which allowed some analytic extension beyond the region to 2 
Of dsIII 
-Q) e t e + 01, 0 < 8 n, 0 e 8 < 27 where Eq. (11) deomonstrates 
analyticity. But a computation of the curvature tensor (see Appendix A 
of my 1963 paper for rapid computational techniques which here gave 
R starting from Eq. (11) in twelve minutes work) shows that 
UUolB 
t R. . ~ ~ j  - % R ~= 2(e sine)-‘ 
1 J  
The lines of infinite curvature at 8 = 0 and 8 = are therefore 
natural boundaries to this space; since the metric puts these lines 
arbitrarily close to regular points of the space we say that this space 
is intrinsically singular. 
An analytic extension of the line element of Eq. (1) is obtained by 
the coordinate transformation 
This transformation can also be written 
.. 
where q is the quaternion 
. 
9 
q = w + i.x + j y  + kz 
2 
and i j k  are t h e  imaginary quaternion u n i t s  which obey k = -1, 
i j  = - j i  = k ,  etc. This t r ans fo rmt i .on  shows t h a t  $e8 are the  "Euler 
angle"  coord ina tes  on S which are fami1ia.r i n  d i scuss ions  of t h e  
r o t a t i o n  group SO(3). (See, f o r  i n s t a n c e ,  Corben and S t e h l e  1960, 
Appendix IV. ) 
3 
I n  o rde r  t o  w r i t e  t h e  Taub-NL'T me t r i c  ii. mxyz coord iqa tes  i t  i s  
convenient f i r s t  t o  de f ine  s e v e r a l  d i f f e r e n t i a l  forms 
(21) 
2. 
These d i f f e r e n t i a l  f o r m  a r e  obvio-;sly analyt ic .  or.. t h e  reg ion  0 141 <a, 
and from t h e  fol lowing equat ion M e  see t h a t  the componer-ts of the transformed 
m e t r i c  a r e  a lso 
This eqciation can aLso be w r i t t e n  
ds 2 = g #UV 
(111 
vhere 
0 
= dt 
and 
10 
(24) 
In this form it is easy to verify the signature requirement on the metric 
is satisfied, for we need only verify the signature of the g matrix 
in Eq. (26) and check that ths basis differential froms of Eq. (25) are 
Llv 
linearly independent. This lin.ear independence is clear when one notes 
th.at the coefficients in Eqs. (21) form non-zero orthogonal vec tors  Fr. t h e  
standard flat Eudjdean metric on wxyz space. The signature-of g in 
Eq. (26) is obvious when U(t) = 0, and cannot change since 
LlV 
g = d e t g  = - 1 # 0  (27) uu 
2 It can be shown that themetric of Eq. (23) on the domain 0 c Iql c + OD 
of wxyz space is maximal in the sense that it cannot be identified with 
a coordinate patch on an even larger connected manifold. Taub and I (1966) 
have shown this by studying its geodesics and verifying that every geodesic 
arc which approaches the boundaries ( lqI2 + 0 or Iq12 4 a) is infinitely 
long as measured by any affine path parameter for the geodesic equation. 
In contrast, 
of, a coordinate patch can outer it with finite, even arbitrarily small, 
changes in the path parameter. 
geodesics which start from points outside, but on the boundary 
Let us now turn to another question of analytic continuation, its 
uniqueness. We may and do now choose to consider Eq. (23) on the region 
0 e Is\" e a 
which is the region 
as giving one maximal analytic extension of Taub space, 
where t and t2 are the two zeros of U(t>. Since analytic continuation 
of functions on the real line is a unique process, we might expect this also 
to he true for analytic metric manifolds. 
first introduce a new coordinate system on Taub space by the transformation 
1 
To exhibit non-uniqueness we 
in which we retain the old t@8 coordinates. The metric then becomes 
2 2  ds2 = (t2 + t2)(dQ2 + sin ed6 ) 
+u(t) (2&)2(d1T~ + cosedd) 2 
12 
which differs from Eq. (1) only by one minus sign and the capitalization of 
Although the transformation (29) is only regular in the region 
tl e t e t2, limited by logarithmic singularities at the zeros of U(t), 
the metric of Eq. (30) can clearly be continued to all values of t and 
its analyticity verified by a transformation analogous to Eq. (20) 
%tefk8,+iee%k\D Q = W + iX + jY + kZ = e 
Rather than continue with the details of this example, we consider 
a simpler case based on the 2-dimensional metric 
as2 = 2dgdt + tdQ2 (32) 
which can be interpreted (by assigning 
metric of signature (-,+) on the cylinder S x R. The most significant 
difference between this example and Taub-NUT space is that S x R is 
Q a period of , say, 2n) as a 
1 
1 
--a. lIuL ur,,ryly n;,m cnnnected and can  
the range --03 to SCO) while 
be covered by the plane R x R (assign Q 
S x R is simply connected and no range 3 
for JI larger than 0 to 4n is possible there. An inequivalent 
analytic extension of the region t > 0 in the metric (32) is obtained 
by first making the coordinate transformation 
Jr = - 24nt 
on the t > 0 region. A simple computation using 
(33) 
dg = d Q -  2t-ldt 
in Eq. (32) gives 
2 2 ds =: -2dUt + td9 
13 
(34) 
(35) 
The coordinate transformation (33) shows the equivalence of the metrics 
(32) and (35) on the regions t > 0. To show their inequivalence 
when each is considered to define a metric manifold on which t varies 
from -w  to - k ~  we look at the curve defined on the $t cylinder by 
g = o  
t = -x  (36) 
with -a < c + (I). Since (A) and t(X) are each analytic functions of 
X here this is an analytic curve. Its image on the t > 0 part of 
the W cylinder is also an analytic curve 
where 1 < 0. However the tangent vector to this curve is 
t v = dt/dh = -1 
14 
\Tf and becomes infinite (i.e. v + -a) as h -$ 0. Since the analytic 
structure of the cylinder is defined by taking d 9  to be an analytic 
differential, an analytic extension of this curve segment (-co c 1 c 0) 
would have to make d W d X  an analytic function at h = 0, which is 
impossible. Thus this curve cannot be extended in the Qt manifold, 
while it can be in the $t manifold, demonstrating the inequivalence of 
these two manifolds. 
The geodesics for the metric (32) are easily obtained. One finds 
that any geodesic segment from a finite point ( J r  t ) to a boundary (t = +co) 
is infinitely long in the sense of the affine path parameter, so this 
space is maximal. However not all geodesics can be extended to parameter 
values both iw and 1 -$ -OD. For instance 
0 0  
t = O  
= -24,nX 
(39) 
is a (null) geodesic, where 1 has a maximum range 0 < h <a. Consequently 
this space is not complete. 
A further curious property of the example of Eq. (32) which was dis- 
covered in a discussion with Bonner is that it is flat. This one may 
verify by starting from the flat metric 
2 2 2 ds = de - dT) = d([ + q)d([ - T)) 
. 
15 
and making the transformation 
under which the half plane 7 7 5 covers the t$ cylinder infinitely 
many times. 
16 
REFERENCES 
L. Amlander and R.E. MacKenzie (1963) 
Introduction to Differentiable Manifolds, (McGraw-Hill, New York) 
HOC. Corben and P, Stehle (1960) 
Classical Mechanics (2r.d edition) (Wiley & Sons, New York) 
C.W. Misner (1963) 
J. Math. Phys. 4, 924 - 937 
E. Newman, L. Tamburin0 and T. Unti (1963) 
J. Math. Phys. 4,  915 - 923 
A. Taub (1951) 
Ann. Math. 53, 472 
A. Taub and C.W. Misrrer (1966) 
(to be submitted to JETP) 
Fig. 1 Both the cylinders shown here are flat and can be represented 
away from the seam ( b  = 0 or 2n) by the metric of Eq. (6). 
Only (a) is a smooth (analytic) manifold, while (b) with its 
sharp edged seam does not inherit a useful class of differentiable 
functions from the Eiiclidean 3-space in which it is embedded. 


Taub-NUT space as a counterexample to almost anything Technical report no. 529

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