Kruskal Dynamics For Radial Geodesics. I

The total spacetime manifold for a Schwarzschild black hole (BH) is described by the Kruskal coordinates u=u(r,t) and v=v(r,t), where r and t are the conventional Schwarzschild radial and time coordinates respectively. The relationship between r and t for a test particle moving on 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. Here, we, first, explicitly obtain the value of the Kruskal coordinates on the Event Horizon as a function of the constant of motion (E=energy per unit rest mass) for a test particle on a radial geodesic by directly using the r-t relationship as obtained by Chandrasekhar and also by Misner, Thorne and Wheeler. It is found that u_H and v_H are finite for E<1. And then, 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 |du/dv| (= 1) 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, |dt/dr| =infty, at the Event Horizon.

The coordinate time t too has a physical significance as the proper time of a distant inertial observer S ∞ . At r = 2m, g rr blows up and as r < 2m, the g tt and g rr suddenly exchange their signatures though the signatures of g θθ and g φφ remain unchanged. The detail dynamics of a "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 Relativity (GTR). 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 dr/dt or dt/dr. And the fact that dt/dr = −∞ blows up at the Event Horizon restricts the utility of the Schwarzschild dynamics below the EH. This tantamounts to the well known fact that the vacuum Schwarzschild metric fails to describe the spacetime inside r ≤ 2m.
On the other hand, we learnt in 1960 that both the exterior and the interior regions of a BH may be described by a one-piece coordinate system suggested by Kruskal and Szekeres [3,4]. Though, in the intervening 39 years hundreds of articles have been written on Kruskal coordinates, and, most of the treatises on GTR too regularrly deliberate upon the original work of Kruskal 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. As a first step towards this direction, in this paper, we would derive expressions for the Kruskal derivative du/dv for a radial geodesic. For the sake of completeness, we 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 (u H and v H ) in terms of r and t. We shall show here that u H and v H are always non-zero and finite in general. More importantly, we shall explicitly show that unlike the Schwarzschild derivative, the Kruskal derivates are regular on the EH in accordance with the singularity free nature of the Kruskal coordinates.

II. KRUSKAL COORDINATES
For the region exterior to the EH (Sectors I & III), the Kruskal coordinates are defined as follows: where Here the plus sign corresponds to "our universe" while the 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), and is a time reversed mirror image of "our universe". And for the region interior to the horizon (Sectors II & IV), we have where f 2 (r) = ± 1 − r 2m In terms of u and v, the metric for the entire spacetime is The metric coefficients are regular everywhere except at the intrinsic singularity r = 0, as is expected. Since afterall the Kruskal coordinates are defined using r and t, for a proper understanding of the Kruskal dynamics, it is necessary to recall the inter-relationship between the Schwarzschild coordinates for a geodesic.

A. Inter Relation Between Schwarzschild Coordinates
For a test particle on a radial geodesic, the angular momentum 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: 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 motion of the particle is determined by (see Chandrasekhar, pp. 98) [5] dr ds and Clearly as r → 2m, dt/dr → −∞. On the other hand, we do not expect such irregular behaviour for the Kruskal derivative.
Here note that if the particle is released from rest (dr/ds = 0) at r = r i at t = 0, from Eq. (8), it is seen that [5] or, It is convenient to introduce a (cylic) parameter η through Obviously, η = 0 when r = r i and at the EH, we have Now after some manipulation, Chandrasekhar arrived at the following Eq. involving t and η [5]: This Eq. can be integrated to find the exact relation between t and r for a radial geodesic (actually, even for non-radial geodesic this Eq. would hold good): The above Eq. may also be written without introducing η H and E explicitly: (see pp. 824 of ref. [1] or pp. 343 of ref. [2]): We find from Eqs. (16-17) that, as r → 2m from Sector I, the logarithmic term blows up and t → ∞, which is a well known result. Further Kruskal coordinates envisage that approach to the EH from the Sectors III & IV corresponds to t = −∞.

B. Kruskal Coordinates on the Event Horizon
In Sectors I & III, Kruskal coordinates obey the relation And since r → 2m corresponds to t → ±∞, at the EH, we have On the other hand, in Sectors II and IV, we see and as r → 2m, t → ±∞, we are led to the same Eq. (19). In the same limit, r → 2m and t → ±∞, we find that It might appear that since f 1 (2m) = f 2 (2m) = 0 on the EH, we would have u H = ±v H = 0. But this is incorrect because the temporal part of u and v tend to blow up much more rapidly on the EH. And one has to carefully obtain the actual values of u H and v H by working out appropriate limits To do so we introduce a new variable and, let, in the vicinity of the EH, so that Then, in the vicinity of the EH, by retaining terms first order in ǫ, we can rewrite Eq.(17) as As ǫ → 0, the logarithmic term in the above expression becomes Then, using Eqs. (24) and (26), we find Now considering the other terms in the expression for t/2m in Eq.(25), we find that, in this limit, In terms of E, we have One would have u H = v H = 0 if E = 0 or, if the test particle is injected from rest right at the EH. Clearly, this is unphysical, and thus we see that u H and v H are non-zero. Further, for a finite value of r i /2m or for E < 1, they are finite too. The finiteness of u and v at the EH is physically appealing because u and v are expected to be completely regular at the EH. However for r i /2m = ∞ or E = 1, we find u 2 H = v 2 H = ∞. On the other hand, since r = r i at t = 0, by using the definition of u and v, we find that the initial values of and v 2 = v 2 i = 0 (31)

III. KRUSKAL DERIVATIVE: A DIRECT APPROACH
Having shown that u H and v H are, in general, non-zero, we are now in a position to evaluate the Kruskal derivative, the key ingredient for studying the Kruskal dynamics for a radial geodesic. We first confine ourselves to Sector I. By differentiating f 1 (r) (Eq. [3]) with r we obtain Then by directly differentiating Eq.(2) by r, we find that irrespective of the sign of df 1 /dr, we will have Interestingly, in all the sectors, we obtain the same functional form of du/dr. Using Eqs. (2) and (4) in the foregoing Eq., we see that On the other hand by differentiating Eqs. (4) and (5) valid in all the sectors. Similarly, we obtain the ultimate functional form of dv/dr which is valid for all the sectors: And, the general value of du/dv in any Sector is obtained by dividing Eq.(37) with (38): A. Kruskal Derivative at the Event Horizon Since u and v are expected to be differentiable smooth continuous (singularity free) functions everywhere except at r = 0, and also since the "other universe" is a mirror image of "our universe", we expect that the value of du/dv for any given r must be the same, except for a probable difference in the signature, in both the universes. The meaningful way to find the value of du/dr at the EH will be to concentrate on the Sectors II & IV for which u H The Eq. (39) for the Kruskal redivative, however, tends to yield a "0/0" form at r = 2m for Sectors I & III having u H = v H . But as mentioned above, we expect this 0/0 form to acquire the value du/dv = +1 because these Scetors are the mirror images of Sectors II & IV. Otherwise the whole idea of having an extended time symmetric Schwarzschild manifold would be inconsistent. Thus, in general, we must have The fact that we must have du/dv = +1 for the Sectors I & III can be reconfirmed in the limiting case of u 2 H = v 2 H = ∞ for E = 1 or u H = v H = 0 for the (unphysical case) E = 0 directly by using L' Hospital's theorem.
Note that, by this rule, we can write,

IV. A DIFFERENT ROUTE
It may be of some interest to rederive the limiting value of du/dv by using other generic relationships between u and v. As before, to avoid 0/0 forms, we work with Sectors III & IV. In particular, in Sector III, we have By differentiating this equation w.r.t. v, we obtain By recalling that sinh(t/4m) = v/f 1 , we rewrite the above Eq. as Now, from Eqs. (10) and (38), note that And the limiting value of And since f 1 (2m) = 0, we find from Eq. (47) that Or, Similarly, for the sake of overall consistency, in Sectors, I & III, we must have du/dv = +1 at r = 2m.

V. A DIFFERENT CONSIDERATION
Actually we could have obtained the above derived unique result in a relatively simpler manner by differentiating the Global Eq.
Then, by using this above Eq. in (53), we find so that

VI. CONCLUSIONS
The Kruskal coordinates were found way back in 1960, and in the present paper, we have worked out some aspects of the kinematics of a test particle following a radial Kruskal geodesic. To attain this we used, for the first time, the precise value of u H and v H as a function of the initial conditions of the problem r i , m or E. It is clearly found that u 2 H = v 2 H is non-zero in general. We then proceeded to obtain the expression for the Kruskal derivative in terms of m, E and r. We found that the Kruskal derivative is regular at the EH unlike the Scharzschild deivative(s) where dt/dr = −∞ at the EH.
In particular du/dv = +1 at the EH if we consider the "other universe"whose existence is suggested by the full Kruskal manifold, and which is a time reversed version of "our universe". But, if we move to the "our universe", the expected value of du/dv = 1 at the EH.
The regular nature of the Kruskal derivative is in keeping with the notion that Kruskal coorinates are free of singularities at the EH.
In a subsequent paper, we shall find out other important features of the Kruskal dynamics vis-a-vis the well known Schwarzschild dynamics.