Proper time and length in Schwarzschild geometry
PProper time and length in Schwarzschildgeometry
O. Brauer , H. A. Camargo , and M. Socolovsky Facultad de Ciencias, Universidad Nacional Autónoma deMéxico, Circuito Exterior, Ciudad Universitaria, 04510México D. F., México Instituto de Ciencias Nucleares, Universidad Nacional Autónomade México, Circuito Exterior, Ciudad Universitaria, 04510México D. F., México
Abstract:
We study proper time ( τ ) intervals for observers at rest in theuniverse ( U ) and anti-universe ( ¯ U ) sectors of the Kruskal-Schwarzschild eternalspacetime of mass M , and proper lengths ( ρ ) in the black hole (BH) and whitehole (WH) sectors. The fact that in asymptotically flat regions, coordinate time t at infinity is proper time, leads to a past directed Kruskal time T in ¯ U . Inthe BH and WH sectors maximal proper lengths coincide with maximal propertime intervals, πM , in these regions, i.e. with the proper time of radial freefalling (ejection) to (from) the singularity starting (ending) from (at) rest atthe horizon. Keywords:
Proper time; proper length; Schwarzschild spacetime; Kruskaldiagram.
PACS (2014): 04.70.Bw, 04.70.-s U and ¯ U All our analysis will be based on the Kruskal diagram corresponding to Schwarzschildspacetime i.e. in its maximal analytical extension [1]. This is illustrated in Fig-ure 1. We use geometric units G N = c = 1 .In region I = U (universe), r > M , the square of the proper time is givenby dτ = (1 − Mr ) dt − dr − Mr − r d Ω (1) a r X i v : . [ phy s i c s . g e n - ph ] J un here M is the mass and d Ω = dθ + sin θdϕ . For r → ∞ spacetime is flatand t is the proper time. For an observer at rest at r > M , dτ = (1 − Mr ) dt (2) and therefore ∆ τ U = (cid:114) − Mr ∆ t, ∆ t = t − t . (3) I.e. ∆ τ U = ∆ τ U ( M ; r , ∆ t ) ; so, ∆ τ U → as r → M (time “does not pass"for light) and ∆ τ U → ∆ t as r → ∞ . The maximum value of ∆ τ U is + ∞ sincethis is the maximum value of ∆ t .To determine the corresponding ∆ τ ¯ U (by symmetry it should equal ∆ τ U ) wehave to use the expression for dτ in terms of the Kruskal variables T and R ,valid in the four regions U , BH , ¯ U , and W H [2]: dτ = 4 × Mr e − r M ( dT − dR − r d Ω ) , (4) with r implicitly given in terms of T and R by M ( R − T ) = ( r M − e r M . (5) In region I (cid:48) = ¯ U (anti-universe) the relation between T and R with t and r isgiven by T = − M (cid:114) r M − e r M Sh ( t M ) ∈ ( −∞ , + ∞ ) , (6) and R = − M (cid:114) r M − e r M Ch ( t M ) ∈ ( −∞ , . (7) For constant r , dT = ∂T∂t dt = − (cid:114) r M − e r M Ch ( t M ) dt,dR = ∂R∂t dt = − (cid:114) r M − e r M Sh ( t M ) dt and therefore dT − dR = (( ∂T∂t ) − ( ∂R∂t ) ) dt = 14 ( r M − e r M dt . (8) Then, dτ U ( r ) = (1 − Mr ) dt and so dτ ¯ U ( r ) = ± (cid:114) − Mr dt. (9) dτ ¯ U ( r ) < for dt > or viceversa, dτ ¯ U ( r ) > for dt < . No of these results is admisible, since both τ (at finite distances)and t (at infinity) are proper times, and any proper time must always be futuredirected [3]. Then, the plus sign has to be chosen in (9), and therefore ∆ τ ¯ U ( r ) = (cid:114) − Mr ∆ t = ∆ τ U ( r ) . (10) But then T < T i.e. Kruskal time decreases : ∆ T ( r ) = T − T = − M (cid:114) r M − e r M ( Sh ( t M ) − Sh ( t M ) < since t > t . The fact that T is past directed in ¯ U , indicates that T is not aphysical time in ¯ U , but only a coordinate (though global) in Kruskal spacetime,with no intrinsic physical meaning, at least with respect to the eternal blackhole. Figure 1: Proper time and length in Kruskal diagram.3 Proper lengths in BH and W H
In region II = BH (black hole), < r < M , the square of the interval is givenby ds = dr Mr − − ( 2 Mr − dt − r d Ω (12) which, at fixed θ , ϕ , and time t can be interpreted as the elementary properlength dρ along dr : dρ BH = dr (cid:113) Mr − , (13) independent of t . Integrating this expression between r and r gives ∆ ρ BH ( r , r ) = (cid:90) r r dr (cid:114) r M − r = 2 M (cid:90) x x dx (cid:114) x − x (14) with x = r M and x i = r i M , i = 1 , . Using (cid:90) dx (cid:114) x − x = − (cid:112) x (1 − x ) + arctg ( (cid:112) x (1 − x )1 − x ) + const. (15) one obtains, in particular for the limits r → + and r → (2 M ) − , ∆ ρ BH (0 , M ) = 2 π × π πM. (16) So, the maximal proper length in BH coincides with the proper time of radialfree falling to the future singularity at r = 0 of a massive test particle, startingat rest from the future horizon [4].By symmetry, the same result should hold in region II (cid:48) = W H (white hole),but now the maximal proper length should coincide with the proper time ofradial free ejection from the past singularity at r = 0 of a massive test particle,ending at rest at the past horizon. In fact, in II (cid:48) the relation between T and R with t and r is given by T = − M (cid:114) − r M e r M Ch ( t M ) ∈ ( −∞ , , (17) and R = − M (cid:114) − r M e r M Sh ( t M ) ∈ ( −∞ , + ∞ ) . (18) For constant t , dT = ∂T∂r dr = r M e r M (cid:112) − r M Ch ( t M ) dt, R = ∂R∂r dr = r M e r M (cid:112) − r M Sh ( t M ) dr, and using again (4) with dθ = dϕ = 0 , one obtains dρ W H = dρ BH (19) which, after integration, leads to the same results (14) and (16), but for ∆ ρ W H . It is believed that, probably, eternal black holes do not exist in nature, andthat only black holes resulting from gravitational collapse (and also primordialblack holes produced in the very early universe) exist [5],[6]. For black holesproduced in gravitational collapse, T behaves as a physical time coordinate since,as proper ( τ ) and coordinate ( t ) times, it also is future directed. Nevertheless,eternal black holes are solutions of Einstein equations, and for them, as shownin section 1, T looses physical character in region ¯ U . Acknowledgment
This work was partially supported by the project PAPIIT IN105413, DGAPA,UNAM.
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