The meaning of relativity in spacetimes of constant curvature

Nuclear Physics B (Proc. Suppl.) 6 (1989) 381-383 North-Holland, Amsterdam

381

T H E M E A N I N G O F R E L A T I V I T Y IN S P A C E T I M E S O F C O N S T A N T C U R V A T U R E Han-Ying G U O Institute of Theoretical Physics, Academia Sinica, P.O. Box 2735, Beijing, China We show that Einstein's principles of the special relativity can be realized not only on the Minkowskian spacetime but also on the spacetimes of non-zero constant curvature.

It is well-established principles

to realize Einstein's

flat Minkowskian

space-time.

group plays an extremely realization.

that these principles,

the speed of light, space-times

+ AnirnjsxrxS~-2(x,x)]dxidx j ,

The Poincar~

in addition,

i.e. the principle of

where A is the constant curvature of ~ A and (nlj} i,J = 0 ..... 3 = diag (I,-I,-i,-I).

of invariance of

can also be realized on the

of constant curvature on the same

footing with the one on the Minkowskian

space-

the fractionally

linear transformations

anti-de Sitter group S0(3,2) Sitter group S0(4,1)

Under of the

[for A < O, the de

(for A > 0), and the

Poincar~ group ISO(3,1)

(for A = 0):

This is, in fact, an analogy with the

well-known

realization

Riemannian

and Lobachevskian

this talk, we present realization.

x i ~ x~i = ¢ 1 / 2 ( a , a ) ¢ - l ( a , x

of the Euclidean, geometries.

,

the main results on this

Dji = L i i + An k aiakLj[¢(a,a ) + ¢I/2(a,a)]-I

Riemannian

L = ( L ~ ) i , j = O. . . . . 3 c SO(3.1) , the condition invariant.

Fundamental theorem: 4-dimensional

space-times

There e x i s t t h r e e

the principle

I and the metric II are

The coordinate

systems

{x i} are

inertial and the geodesics of DA are (projective)

~ A of constant

which satisfy necessarily

sufficiently

{III)

and Lobachevskian

geometries.

and

straight

lines which describe the

inertial motions of free particles.

the principle of relativity and

The causality between two events A(a i) and B(b i) can be classified by the quantity

of invariance of the speed of

They hold the condition A2(a,b)=[~i3¢-l(a,b)

¢(x,x)=l

D

It should be easy to prove them

the Euclidean,

curvature,

) (xiai)D

In

if it would be kept in mind the analogies among

light.

(II)

important role in this

We have found {I-4},

relativity and the principle

time.

ds 2 = [~ij¢-l(x,x)

of the special relativity on the

i i - Anijx x >0,

i,j=O ..... 3.

A ~irnjs a and have the Beltrami-Minkowskian

metric

0920-5632/89/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

+

(I} r.s o ¢ -(a 'b} "l(a-b)i[a-bJ "" " " i

P

382

H.-E Guo

/ Relativity in spacetimes o f constant curvature

which is invariant under the transformation III.

x

o

~o

~ X

1/2,

= ~Z

,

ta, a;~

-1,

ta, x}x ° ,

A pair of points ACa i) and B(b i) are said

to be tlme-llke, space-llke or light- llke,

x~

~

x~~ =

~l/29a, a)~lca, x) Cx~

- a~)O~

,

according to A2(a,b) > O, < O, or = O, respectively.

~ + 1/2 -I o8 = R~~ + A ~ a a ~ a~R~[¢z(a'a) CZ (a,a)]

The proper interval between A(a i)

and B(b i} is the length of the geodesic segment AB, s(a,b) = I ds: AB

(R~)~,~=I,2, 3 e S0(3) where a

A

< o

Stime_llke(a,b)

Sspace_like(a,b)

IAl-1/2arctg

llxl 1/2^rth C¢1~1 IACa,b)l)

C¢l~lCa,b) l) = 0

A(a,b)

ilA{a,b) l

i

s a t i s f y the above conditions, and the

conditions, the metric, and the r a t i o

RZ(x) = cz(x,x)c-1(x.x)

are all invariant,

i.e. Z is transformed onto

itself. A-I/2Arth >o

C~l~(a,b)l

.FA-I/2arctg

Two events A(a i} and B(b £) are said to be

(wT~Ca,b)l)

simultaneous if they are on the same space-llke hypersurface Z and the proper distance element

In order to define the simultaneity and the

on Z is given by

proper distance, we consider the subgroup of dl2 = -dsZlz = R 2(x)d~ 2 .

space-like transformations which transform

0

space coordinates onto themselves. Such a Z may be called the simultaneous spaceTheorem:

In the ~)A' there exists a kind

like hypersurface. The proper time of an ideal clock located

of space-like hypersurfaces Z characterized by

at a point on Z can be calculated.

the following conditions

It turns

out to be x°~-I/2(x,x) = const. IAI -I/2 arc sin(~r~[~-i/2(x,x)xO), ~E(x,x) = I - Anc~sx=x ~ > 0 ,

A < 0 ,

=,~=1,2,3 T=

where (~}~8)~=i,2,3 = diag(-l,-l,-l}.

x

0

h=O

On the

Z, the metric of ~)A is reduced to

A -I/2 Arsh(v~cr-I/2(x,x)x°) ,

A > 0 .

dsZlz = ~-l(x,x)~Z(x,x)d~oz , This proper time may be chosen to parameterize the family of the simultaneous space-like d~Zo 12o = -[W~8~Z l(x'x) +

hypersurfaces,

Z z.

Then the metric on D A

gives rise to the metric on E : + k ~ f n ~ x ~ x ° ~ Z (X, X) ]dx~dx B.

Under the transformations of the space-like subgroup S0(3,1)

(for A O)

and ISO[3) (for A = 0):

,

T

H.-Y. Guo / Relativity in spacetimes of constant curvature

dsZlX

dz2-R2(T)d~ ,

=

hypersurfaces,

383

ZT, for A = 0 shows a plcture of

expandlng universe, whlch may play some role in R2(T)

= cos2Vr[X'FT

(A < 0)

, the very early stage of our universe.

=

z

(x

=

o)

,

= ch2~A T (X>0)

. I.

O. K. Lu, unpublished (1970); O. K. Lu, Z. L. Zhou and H. Y. Guo, Acta Physlca Slnlca 23, 225 (1970).

2.

H. Y. Guo, Kexue Tongbao 22, 487 (1977).

3.

Z. L. Zhou, J. S. Chen, P. HuanE, L. N. ZhanE, and H. Y. Guo, Sclentia Sinica 6, 588 (1979).

4.

G. Y. Li and H. Y. Guo, Acta Physica Sinica 3_!I, 1501 (1982).

Obviously, for the case of A = O, all these results are well-known in special relativity so that the cases of A ~ 0 may be taken as the special relativities of Rlemannlan and Lobachevskian version respectively.

In

addition, the family of simultaneous space-like