White Hole Mechanics

 

Schwarzschild Solution for Black Holes

The Schwarzschild metric for a non-rotating black hole is given by:

$ds^2 = -\left(1 - \frac{2GM}{r}\right)c^2 dt^2 + \left(1 - \frac{2GM}{r}\right)^{-1} dr^2 + r^2 d\Omega^2$

where $G$ is the gravitational constant, $M$ is the mass of the black hole, $r$ is the radial coordinate, $c$ is the speed of light, and $d\Omega^2$ represents the angular part of the metric.

Schwarzschild Solution for White Holes

The white hole can be considered as a time-reversed black hole. Therefore, the metric remains the same, but the interpretation of the coordinates changes. For a white hole, the same Schwarzschild metric applies:

$ds^2 = -\left(1 - \frac{2GM}{r}\right)c^2 dt^2 + \left(1 - \frac{2GM}{r}\right)^{-1} dr^2 + r^2 d\Omega^2$

However, the physical interpretation changes:

• For a black hole, as $t$ increases, objects move towards $r = 0$ (the singularity).
• For a white hole, as $t$ increases, objects move away from $r = 0$.

Equations of Motion

For the geodesics (paths followed by particles and light) in the Schwarzschild metric, the equations of motion are derived from the Schwarzschild metric. In a white hole context, these equations describe motion away from the singularity. The equations are:

$\left(\frac{dr}{d\tau}\right)^2 = \left(E^2 - \left(1 - \frac{2GM}{r}\right)\left(1 + \frac{L^2}{r^2}\right)\right)$

where $\tau$ is the proper time, $E$ is the energy per unit mass of the particle, and $L$ is the angular momentum per unit mass.

Energy Conditions

White holes, like black holes, must satisfy the energy conditions in general relativity. The weak energy condition (WEC) states that for any timelike vector $v^\mu$, the energy-momentum tensor $T_{\mu\nu}$ must satisfy:

$T_{\mu\nu} v^\mu v^\nu \geq 0$

For a white hole, we would need to check if this condition holds throughout the spacetime.

Penrose Diagram

In the context of white holes, the Penrose diagram is often used to represent the causal structure. A Penrose diagram for a white hole is essentially a time-reversed version of the Penrose diagram for a black hole.

Final Equations

Given the time-reversed nature of white holes, the key equations remain the same as for black holes but with reversed temporal interpretations:

1. Schwarzschild Metric: $ds^2 = -\left(1 - \frac{2GM}{r}\right)c^2 dt^2 + \left(1 - \frac{2GM}{r}\right)^{-1} dr^2 + r^2 d\Omega^2$

2. Radial Geodesic Equation: $\left(\frac{dr}{d\tau}\right)^2 = \left(E^2 - \left(1 - \frac{2GM}{r}\right)\left(1 + \frac{L^2}{r^2}\right)\right)$

1. Metric for Rotating (Kerr) White Holes

For a rotating white hole, we use the Kerr metric, which describes the geometry around a rotating black hole. For a white hole, the metric remains the same but the physical interpretation changes.

$ds^2 = -\left(1 - \frac{2GMr}{\rho^2}\right)c^2 dt^2 - \frac{4GMar \sin^2\theta}{\rho^2} c\, dt\, d\phi + \frac{\rho^2}{\Delta} dr^2 + \rho^2 d\theta^2 + \left(r^2 + a^2 + \frac{2GMa^2r \sin^2\theta}{\rho^2}\right) \sin^2\theta d\phi^2$

where $\rho^2 = r^2 + a^2 \cos^2\theta$ $\Delta = r^2 - 2GMr + a^2$

2. Equations of Motion in the Kerr Metric

For the geodesics in the Kerr metric, the equations of motion for a white hole can be written as:

$\rho^2 \frac{dr}{d\tau} = \pm \sqrt{R}$ $\rho^2 \frac{d\theta}{d\tau} = \pm \sqrt{\Theta}$ $\rho^2 \frac{d\phi}{d\tau} = \frac{a}{\Delta}\left(E(r^2 + a^2) - La\right) + \left(L - aE \sin^2\theta\right)$ $\rho^2 \frac{dt}{d\tau} = \frac{r^2 + a^2}{\Delta}\left(E(r^2 + a^2) - La\right) + a\left(L - aE \sin^2\theta\right)$

where $R = \left(E(r^2 + a^2) - La\right)^2 - \Delta \left(r^2 + (L - aE)^2 + K\right)$ $\Theta = K - \left(a^2(E^2 - 1) + \frac{L^2}{\sin^2\theta}\right) \cos^2\theta$

Here, $E$ is the energy, $L$ is the angular momentum, and $K$ is the Carter constant.

3. Thermodynamics and Hawking Radiation

In black hole thermodynamics, the temperature $T$ and entropy $S$ are given by:

$T = \frac{\hbar c^3}{8 \pi G M k_B}$ $S = \frac{k_B A}{4 l_p^2}$

where $A$ is the area of the event horizon, $k_B$ is the Boltzmann constant, $l_p$ is the Planck length, and $\hbar$ is the reduced Planck constant.

For white holes, if they emit radiation similar to black holes, we can use analogous expressions. However, the interpretation changes as white holes would theoretically emit everything within them.

4. Radiation Spectrum

The radiation spectrum for a white hole can be considered as the inverse of Hawking radiation:

$\frac{dN}{d\omega} = \frac{\Gamma(\omega)}{e^{\omega / T} - 1}$

where $\Gamma(\omega)$ is the greybody factor, which accounts for the probability of radiation escaping the potential barrier.

5. Energy Conditions and Stability

Energy conditions must be satisfied for the white hole's existence. The dominant energy condition (DEC) states:

$T^{\mu\nu} v_\mu v_\nu \geq 0 \quad \text{and} \quad T^{\mu\nu} v_\mu \text{ is non-spacelike}$

For a white hole, we would also analyze the stability against perturbations using modified versions of the Teukolsky equation for perturbations in the Kerr metric:

$\left[\frac{(r^2 + a^2)^2}{\Delta} - a^2 \sin^2\theta\right] \frac{\partial^2 \psi}{\partial t^2} + \frac{4Mar}{\Delta} \frac{\partial^2 \psi}{\partial t \partial \phi} + \left[\frac{a^2}{\Delta} - \frac{1}{\sin^2\theta}\right] \frac{\partial^2 \psi}{\partial \phi^2} - \Delta^{-s} \frac{\partial}{\partial r}\left(\Delta^{s+1} \frac{\partial \psi}{\partial r}\right) - \frac{1}{\sin\theta} \frac{\partial}{\partial \theta}\left(\sin\theta \frac{\partial \psi}{\partial \theta}\right) - 2s \left[\frac{a(r-M)}{\Delta} + \frac{i\cos\theta}{\sin^2\theta}\right] \frac{\partial \psi}{\partial \phi} - 2s \left[\frac{M(r^2 - a^2)}{\Delta} - r - i a \cos\theta\right] \frac{\partial \psi}{\partial t} + (s^2 \cot^2\theta - s) \psi = 0$

Here, $\psi$ is the perturbation field, and $s$ is the spin weight of the field.

6. Quantum Field Theory in Curved Spacetime (Hawking Radiation for White Holes)

In black holes, quantum field theory predicts Hawking radiation. For a white hole, the time-reversed scenario might involve a form of "inverse Hawking radiation," where the white hole emits energy, potentially leading to negative energy states inside the event horizon.

The typical equation for particle production in curved spacetime is:

$\langle 0 | T_{\mu\nu} | 0 \rangle = \frac{\hbar}{960 \pi^2 r^4}$

For white holes, the expectation value $\langle 0 | T_{\mu\nu} | 0 \rangle$ would be associated with outgoing radiation. The energy flux at infinity can be modeled similarly, but with the outgoing flux being a function of decreasing radius $r$:

$\frac{dE}{dt} = \frac{\hbar c^6}{15360 \pi G^2 M^2}$

7. Information Paradox and Entanglement Entropy

One of the key issues with black holes is the information paradox: whether information that falls into a black hole is lost forever. For a white hole, the reverse scenario can be considered: how information is ejected from the singularity.

Entanglement entropy $S_{\text{ent}}$ for fields outside a black hole is given by:

$S_{\text{ent}} = \frac{A}{4 l_p^2}$

For a white hole, the entropy associated with outgoing information would theoretically decrease:

$S_{\text{ent}}(t) = S_{\text{initial}} e^{-kt}$

where $k$ is a decay constant depending on the rate of information ejection.

8. Gravitational Waves from White Holes

Gravitational waves emitted by a white hole can be described using perturbation theory in the context of the Einstein field equations. In the case of black holes, these waves are often studied in the context of mergers or ringdowns. For white holes, the reverse process might involve an “anti-ringdown” or a build-up of waves.

The Einstein field equations are:

$G_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}$

For small perturbations $h_{\mu\nu}$ on a flat spacetime background, the linearized field equations are:

$\Box h_{\mu\nu} - \eta_{\mu\nu} \Box h = - \frac{16\pi G}{c^4} T_{\mu\nu}$

In a white hole context, these perturbations could be associated with waves traveling outward from the horizon, with the potential energy decreasing over time:

$\Box h_{\mu\nu}^{WH} = \frac{16\pi G}{c^4} \frac{T_{\mu\nu}}{r^2}$

9. Modified Penrose Inequality

The Penrose inequality relates the total mass $M$ of a spacetime to the area $A$ of a marginally outer trapped surface (such as an event horizon). For a black hole, it is:

$M \geq \sqrt{\frac{A}{16\pi}}$

For white holes, the inequality might reverse, indicating a maximum energy that can escape:

$M \leq \sqrt{\frac{A}{16\pi}}$

This suggests a limit on the energy emitted by the white hole as it evolves.

10. Bekenstein-Hawking Entropy (Revised)

The Bekenstein-Hawking entropy for black holes is given by:

$S = \frac{k_B c^3 A}{4 G \hbar}$

For white holes, if we consider entropy reduction or reverse entropy flow, a modified version could be:

$S_{\text{WH}}(t) = \frac{k_B c^3 A}{4 G \hbar} \cdot f(t)$

where $f(t)$ is a time-dependent function that decreases over time, reflecting the outgoing nature of radiation and information.

11. Modified Kerr-Newman Metric for Charged White Holes

The Kerr-Newman metric describes a charged, rotating black hole:

$ds^2 = -\left(\frac{\Delta - a^2 \sin^2 \theta}{\rho^2}\right)c^2 dt^2 + \frac{\rho^2}{\Delta} dr^2 + \rho^2 d\theta^2 + \left(\frac{(r^2 + a^2)^2 - \Delta a^2 \sin^2 \theta}{\rho^2}\right) \sin^2 \theta d\phi^2 - \frac{2a \sin^2 \theta (r^2 + a^2 - \Delta)}{\rho^2} c dt d\phi$

with

$\Delta = r^2 - 2GMr + a^2 + \frac{Q^2 G}{c^4}$

For a white hole, a modified interpretation might involve outgoing charged particles and radiation, so we use:

$\Delta_{\text{WH}} = r^2 + 2GMr + a^2 + \frac{Q^2 G}{c^4}$

This suggests a time-reversed evolution, where the $2GMr$ term leads to expansion rather than contraction.

12. White Hole Cosmology

In cosmology, white holes could be hypothesized as contributing to certain cosmological phenomena. A modified Friedmann equation incorporating a white hole term might look like:

$\left(\frac{\dot{a}}{a}\right)^2 = \frac{8\pi G}{3} \rho - \frac{kc^2}{a^2} + \frac{\Lambda c^2}{3} + \frac{H_{\text{WH}}(t)}{a^3}$

where $H_{\text{WH}}(t)$ is a function representing the influence of white hole energy density over time.

13. Quantum Gravity Corrections (Loop Quantum Gravity)

In Loop Quantum Gravity (LQG), the area operator is quantized, and the area spectrum is discrete. This might affect white hole dynamics, leading to a modified area spectrum:

$A_j = 8\pi \gamma l_p^2 \sqrt{j(j+1)}$

where $\gamma$ is the Barbero-Immirzi parameter, and $j$ is a quantum number. For a white hole, the evolution of the horizon area might involve a step-wise reduction:

$A_j(t) = 8\pi \gamma l_p^2 \sqrt{j(j+1)} - \delta(t)$

where $\delta(t)$ represents the quantum decrement in area as the white hole evolves.

14. Higher-Dimensional White Holes (String Theory/Brane World)

In the context of string theory and higher-dimensional spacetimes, white holes could be extended to higher dimensions. For example, in a 5-dimensional spacetime (often called the "bulk" in brane world scenarios), the metric might take a form analogous to the Schwarzschild solution in higher dimensions:

Higher-Dimensional Schwarzschild Metric:

For a $(n+1)$-dimensional spacetime, the Schwarzschild metric can be generalized as:

$ds^2 = -\left(1 - \frac{2GM}{r^{n-2}}\right)c^2 dt^2 + \left(1 - \frac{2GM}{r^{n-2}}\right)^{-1} dr^2 + r^2 d\Omega_{n-1}^2$

For a white hole in higher dimensions:

$ds^2 = -\left(1 - \frac{2GM}{r^{n-2}}\right)^{-1} dr^2 + \left(1 - \frac{2GM}{r^{n-2}}\right)c^2 dt^2 + r^2 d\Omega_{n-1}^2$

This could lead to different behaviors for white holes in higher dimensions, possibly altering their stability, information flow, and interaction with lower-dimensional branes.

15. Modified General Relativity (f(R) Gravity)

In modified theories of gravity, such as $f(R)$ gravity, the Einstein-Hilbert action is modified by a function $f(R)$ of the Ricci scalar $R$:

$S = \frac{1}{2\kappa} \int d^4x \sqrt{-g} f(R) + S_{\text{matter}}$

The corresponding field equations are:

$f'(R) R_{\mu\nu} - \frac{1}{2} f(R) g_{\mu\nu} - \nabla_\mu \nabla_\nu f'(R) + g_{\mu\nu} \Box f'(R) = \kappa T_{\mu\nu}$

For white holes in $f(R)$ gravity, the function $f(R)$ could introduce corrections that affect the event horizon's structure, stability, and radiation:

$R_{\mu\nu} = \frac{\kappa}{f'(R)} \left(T_{\mu\nu} + \frac{1}{2} g_{\mu\nu} \left(f(R) - R f'(R)\right)\right) + \frac{1}{f'(R)} \left(\nabla_\mu \nabla_\nu - g_{\mu\nu} \Box\right) f'(R)$

This may result in a more complex white hole geometry that could potentially lead to new observable effects, such as altered gravitational lensing or changes in emitted radiation spectra.

16. Quantum Tunneling and White Holes

Quantum tunneling in black holes is a concept where particles can tunnel through the event horizon, leading to Hawking radiation. For white holes, the concept would be reversed, where particles might tunnel out of the singularity, contributing to the hypothetical "white hole radiation."

Tunneling Rate:

The tunneling rate $\Gamma$ for a particle can be expressed as:

$\Gamma \propto \exp\left(-\frac{2 \pi E}{\hbar \kappa}\right)$

where $E$ is the energy of the particle and $\kappa$ is the surface gravity. For a white hole:

$\Gamma_{\text{WH}} \propto \exp\left(\frac{2 \pi E}{\hbar \kappa}\right)$

This suggests an exponential increase in the probability of particle emission, potentially leading to a faster rate of mass and energy ejection compared to black hole evaporation.

17. Hypothetical Cosmological Implications (White Holes as Big Bangs)

One of the more speculative ideas is that the Big Bang itself could be interpreted as a white hole event. This idea is based on the similarity between the mathematical descriptions of the Big Bang and a white hole explosion.

Friedmann-Lemaître-Robertson-Walker (FLRW) Metric with White Hole Terms:

The FLRW metric describes a homogeneous and isotropic universe:

$ds^2 = -c^2 dt^2 + a(t)^2 \left(\frac{dr^2}{1 - kr^2} + r^2 d\Omega^2 \right)$

For a white hole scenario, the scale factor $a(t)$ might include a term representing white hole-driven expansion:

$\left(\frac{\dot{a}}{a}\right)^2 = \frac{8\pi G}{3} \rho + \frac{C_{\text{WH}}}{a^3} - \frac{kc^2}{a^2} + \frac{\Lambda c^2}{3}$

where $C_{\text{WH}}$ represents a constant related to the energy density of the white hole, suggesting that early universe expansion could be driven by a white hole-like event.

18. Anti-de Sitter (AdS) White Holes and Holography

In AdS/CFT correspondence, black holes in an Anti-de Sitter space are dual to thermal states in a conformal field theory. For white holes, this duality might imply a reverse process, where information flows out of the AdS space.

AdS White Hole Metric:

The AdS white hole metric can be written as:

$ds^2 = -\left(1 + \frac{r^2}{L^2} - \frac{2GM}{r}\right) c^2 dt^2 + \left(1 + \frac{r^2}{L^2} - \frac{2GM}{r}\right)^{-1} dr^2 + r^2 d\Omega^2$

In the dual CFT, this might correspond to an information outflow, possibly providing insights into resolving the information paradox:

$Z_{\text{CFT}} = \text{Tr} \exp\left(-\beta H_{\text{CFT}} + \alpha Q_{\text{WH}}\right)$

where $\alpha$ is a parameter representing the white hole’s influence on the boundary theory.

19. Thermodynamic Identity Modifications

For white holes, the first law of thermodynamics might be modified to reflect the reverse nature of the system:

$dE = -T dS + P dV$

This suggests that as a white hole radiates, its internal energy decreases (hence the negative sign), possibly leading to the release of negative entropy and a corresponding increase in the universe's overall entropy.

20. Quantum Gravity and Discrete Spacetime

In approaches like Loop Quantum Gravity, spacetime itself is quantized. This could lead to a scenario where white holes are fundamentally discrete objects, leading to a modified area quantization law:

$A_{\text{WH}} = 8 \pi \gamma l_p^2 \sqrt{j(j+1)}$

where the area decreases step-wise as information and radiation are ejected:

$\Delta A_{\text{WH}} = - 8 \pi \gamma l_p^2 \sqrt{j(j+1)} \Delta n$

Here, $\Delta n$ represents a quantum decrement in the area as the white hole evolves.

21. Wormholes and White Holes

White holes are often theorized to be connected to black holes via a wormhole (Einstein-Rosen bridge). The throat of the wormhole could exhibit different properties when considering the time-reversed nature of white holes.

Wormhole Metric:

The Morris-Thorne wormhole metric is given by:

$ds^2 = -c^2 dt^2 + \frac{dr^2}{1 - \frac{b(r)}{r}} + r^2 d\Omega^2$

where $b(r)$ is the shape function. For a white hole:

$b_{\text{WH}}(r) = b_0 + \delta b(t)$

This suggests that the wormhole throat could evolve over time, potentially leading to a shrinking or expanding throat depending on the dynamics of the connected black and white hole pair.

22. Hypothetical Interaction with Dark Matter

If white holes exist, they might interact with dark matter differently than black holes. One possibility is that white holes could emit dark matter particles, leading to modified equations for dark matter density around them:

Dark Matter Density Profile:

The Navarro-Frenk-White (NFW) profile is commonly used to describe dark matter halos:

$\rho_{\text{DM}}(r) = \frac{\rho_0}{\frac{r}{r_s} \left(1 + \frac{r}{r_s}\right)^2}$

For a white hole, the dark matter density might include a term representing repulsion or ejection:

$\rho_{\text{DM,WH}}(r) = \frac{\rho_0}{\frac{r}{r_s} \left(1 + \frac{r}{r_s}\right)^2} + \rho_{\text{eject}}(r)$

where $\rho_{\text{eject}}(r)$ represents the additional dark matter density due to ejected particles.

23. Quantum Cosmology and White Holes

In quantum cosmology, the wave function of the universe is described by the Wheeler-DeWitt equation. For white holes, a modified equation might incorporate a term representing the reverse evolution of a collapsing universe:

Wheeler-DeWitt Equation (Modified):

$\left(-\frac{\hbar^2}{2m_p^2} \frac{\delta^2}{\delta h_{ij}^2} + \mathcal{V}[h_{ij}]\right) \Psi[h_{ij}] = 0$

For a white hole-driven universe:

$\left(\frac{\hbar^2}{2m_p^2} \frac{\delta^2}{\delta h_{ij}^2} + \mathcal{V}[h_{ij}] + \Lambda_{\text{WH}}(t)\right) \Psi[h_{ij}] = 0$

where $\Lambda_{\text{WH}}(t)$ represents a time-dependent term associated with white hole expansion effects.