The black hole energy and the energy of the other universe

Abstract

Frames adapted to Kruskal and Novikov coordinate systems are used to compute the energy of the Schwarzschild black/white hole. It turns out that no regularization procedure is necessary if the right hypersurface of simultaneity is used. The importance of using a hypersurface that is consistent with the observers who “measure” the energy is discussed. The effect that the wormhole throat may have on the total energy inside this hypersurface is also discussed; the result shows that this energy becomes the famous $Mc^2$ only when the throat pinches off completely. The energy of the other universe is calculated. It is shown that, because of reflections at the time and the radial axes of the Kruskal and Novikov frames with respect to the Schwarzschild frame, it is not possible to obtain the energy in both universes “simultaneously.”

Appendices

Appendix

The Levi-Civita spin connection of the Kruskal frame

Here, we use the approach developed in [10, 17] to compute the superpotential associated with . In this approach, one uses a certain coordinate system and at the same time the directions of another coordinate system; for instance, we can use a spherical coordinate system with the frame pointing toward the directions defined by x, y and z. The calculations will be easier to handle in this approach.

From the spherical coordinate system in a flat manifold, we can define the vector components

(26)

When the spacetime is curved, we can still use (26) to simplify the expressions by defining the new components , , , , which allows us to write the frame as

(27)

Note that we can raise and lower indices easily: . It should also be clear that ${\hat{t}}^\mu {\hat{t}}_\mu ={\hat{t}}^a{\hat{t}}_a=1$, ${\hat{r}}^\mu {\hat{r}}_\mu ={\hat{r}}^a{\hat{r}}_a=-1$ and so on.

Comparing Eq. (8) with (27), we find that

$$\begin{aligned} {\hat{t}}^\mu= & {} \frac{\delta ^\mu _0}{F},\quad {\hat{r}}^\mu =\frac{\delta ^\mu _1}{F},\quad {\hat{\theta }}^\mu =\frac{\delta ^\mu _2}{r},\quad {\hat{\phi }}^\mu =\frac{\delta ^\mu _3}{r\sin \theta }, \end{aligned}$$

(28)

$$\begin{aligned} {\hat{t}}_\mu= & {} F\delta _\mu ^0,\quad {\hat{r}}_\mu =-F\delta _\mu ^1,\quad {\hat{\theta }}_\mu =-r\delta _\mu ^2,\quad {\hat{\phi }}_\mu =-r\sin \theta \delta _\mu ^3. \end{aligned}$$

(29)

where we have used the metric (5) in the last expression.

To calculate the spin connection, we need first the commutation coefficients of . Since ${\hat{r}}^a$, ${\hat{\theta }}^a$, and ${\hat{\phi }}^a$ depend only on $\theta$ and $\phi$, and we are using the coordinates $x^\mu =(T,X,\theta ,\phi )$, we can calculate them by using (for more details, see section VII of [10])

(30)

(31)

(32)

(33)

(34)

To calculate the component (31), we apply to ${\hat{t}}_\mu$ given by (29). The result is , where $F'\equiv \partial _X F$. From (29), one easily sees that $\delta ^2_{[\mu }{\hat{\theta }}_{\nu ]}=\delta ^3_{[\mu }{\hat{\phi }}_{\nu ]}=0$. So, the component can be obtained by applying to ${\hat{r}}_\nu$. Using (29), we find that , where ${\dot{F}}\equiv \partial _T F$. From the expression of ${\hat{r}}_\mu$ in (29), we obtain $2\delta ^2_{[\mu }{\hat{r}}_{\nu ]}=2F\delta ^1_{[\mu }\delta ^2_{\nu ]}$ and $2\partial _{[\mu }{\hat{\theta }}_{\nu ]}=-2\left( {\dot{r}}\delta ^0_{[\mu }\delta ^2_{\nu ]}+r' \delta ^1_{[\mu }\delta ^2_{\nu ]} \right)$. Thus, we have . A similar procedure for (34) will lead us to . Now, we use (29) to eliminate the deltas in these expressions. This gives

(35)

(36)

(37)

Substituting Eqs. (35)–(37) into Eq. (30), and using the fact that ${\dot{F}}/F^2=\left( 1+2m/r\right) FT/(16m^2)$ and $F'/F^2=-\left( 1+2m/r\right) FX/(16m^2)$, we arrive at

(38)

where we have used ${\dot{r}}=-F^2T/(4m)$, $r'=F^2X/(4m)$ , and $W_+\equiv 1+2m/r$.

To calculate the coefficients of the Levi-Civita connection in the basis , we use Eqs. (3) and (38). In this calculation, we only need to focus on the permutation of the elements $\{{\hat{t}},{\hat{r}},{\hat{\theta }},{\hat{\phi }}\}$. For instance, the term ${\hat{t}}_{[c}{\hat{r}}^{a]}{\hat{t}}_b+{\hat{t}}_{[b}{\hat{r}}^{a]}{\hat{t}}_c-{\hat{t}}_{[b}{\hat{r}}_{c]}{\hat{t}}^a$ can be simplified to $-2{\hat{t}}_b{\hat{t}}^{[a}{\hat{r}}_{c]}$. Doing the same for the other terms, we obtain

(39)

From this connection, we can compute all quantities of interest, regardless of the nature of the indices. For instance, if we want , we can simply change a to $\lambda$ and use (28). (Note that .).

From (39), (2), (1), and (28), we obtain , where we have used the fact that $T^{\lambda a}=0$ (there is no matter field). Finally, integrating in the region indicated by a horizontal line on the right side of Fig. 2, and using $e=32m^3\exp [-r/(2m)]r\sin \theta$, we obtain (9).