Virtual black hole

Inquantum gravity,avirtual black holeis a hypotheticalmicro black holethat exists temporarily as a result of aquantum fluctuationofspacetime.It is an example ofquantum foamand is thegravitationalanalog of the virtualelectron–positronpairs found inquantum electrodynamics.Theoretical arguments suggest that virtual black holes should have mass on the order of thePlanck mass,lifetime around thePlanck time,and occur with a number density of approximately one perPlanck volume.

The emergence of virtualblack holesat thePlanck scaleis a consequence of the uncertainty relation

$\Delta R_{\mu }\Delta x_{\mu }\geq \ell _{P}^{2}={\frac {\hbar G}{c^{3}}}$

where$R_{\mu }$is the radius of curvature of spacetime small domain,$x_{\mu }$is the coordinate of the small domain,$\ell _{P}$is thePlanck length,$\hbar$is thereduced Planck constant,$G$is theNewtonian constant of gravitation,and$c$is thespeed of light.These uncertainty relations are another form of Heisenberg'suncertainty principleat thePlanck scale.

Proof

Indeed, these uncertainty relations can be obtained on the basis ofEinstein's equations $G_{\mu \nu }+\Lambda g_{\mu \nu }={8\pi G \over c^{4}}T_{\mu \nu }$ where$G_{\mu \nu }=R_{\mu \nu }-{R \over 2}g_{\mu \nu }$is theEinstein tensor,which combines theRicci tensor,thescalar curvatureand themetric tensor;$\Lambda$is thecosmological constant;а$T_{\mu \nu }$is the energy-momentum tensor of matter;$\pi$is the mathematical constantpi;$c$is thespeed of light;and$G$is theNewtonian constant of gravitation. Einstein suggested that physical space is Riemannian, i.e. curved and therefore put Riemannian geometry at the basis of the theory of gravity. A small region of Riemannian space is close to flat space. For any tensor field$N_{\mu \nu...}$,we may call$N_{\mu \nu...}{\sqrt {-g}}$a tensor density, where$g$is thedeterminantof themetric tensor$g_{\mu \nu }$.The integral$\int N_{\mu \nu...}{\sqrt {-g}}\,d^{4}x$is a tensor if the domain of integration is small. It is not a tensor if the domain of integration is not small, because it then consists of a sum of tensors located at different points and it does not transform in any simple way under a transformation of coordinates.Here we consider only small domains. This is also true for the integration over the three-dimensionalhypersurface$S^{\nu }$. Thus, theEinstein field equationsfor a small spacetime domain can be integrated by the three-dimensionalhypersurface$S^{\nu }$.Have ${\frac {1}{4\pi }}\int \left(G_{\mu \nu }+\Lambda g_{\mu \nu }\right){\sqrt {-g}}\,dS^{\nu }={2G \over c^{4}}\int T_{\mu \nu }{\sqrt {-g}}\,dS^{\nu }$ Since integrable space-timedomainis small, we obtain the tensor equation $R_{\mu }={\frac {2G}{c^{3}}}P_{\mu }$ where$P_{\mu }={\frac {1}{c}}\int T_{\mu \nu }{\sqrt {-g}}\,dS^{\nu }$is the component of the4-momentumof matter,$R_{\mu }={\frac {1}{4\pi }}\int \left(G_{\mu \nu }+\Lambda g_{\mu \nu }\right){\sqrt {-g}}\,dS^{\nu }$is the component of the radius of curvature small domain. The resulting tensor equation can be rewritten in another form. Since$P_{\mu }=mc\,U_{\mu }$then $R_{\mu }={\frac {2G}{c^{3}}}mc\,U_{\mu }=r_{s}\,U_{\mu }$ where$r_{s}$is theSchwarzschild radius,$U_{\mu }$is the 4-speed,$m$is the gravitational mass. This record reveals the physical meaning of the$R_{\mu }$values as components of the gravitational radius$r_{\text{s}}$. In a small area of space-time is almost flat and this equation can be written in theoperatorform ${\hat {R}}_{\mu }={\frac {2G}{c^{3}}}{\hat {P}}_{\mu }={\frac {2G}{c^{3}}}(-i\hbar ){\frac {\partial }{\partial \,x^{\mu }}}=-2i\,\ell _{P}^{2}{\frac {\partial }{\partial \,x^{\mu }}}$ or The basic equation of quantum gravity $-2i\ell _{P}^{2}{\frac {\partial }{\partial x^{\mu }}}|\Psi (x_{\mu })\rangle ={\hat {R}}_{\mu }|\Psi (x_{\mu })\rangle$ Then the commutator of operators${\hat {R}}_{\mu }$and${\hat {x}}_{\mu }$is $[{\hat {R}}_{\mu },{\hat {x}}_{\mu }]=-2i\ell _{P}^{2}$ From here follow the specified uncertainty relations $\Delta R_{\mu }\Delta x_{\mu }\geq \ell _{P}^{2}$ Substituting the values of$R_{\mu }={\frac {2G}{c^{3}}}m\,c\,U_{\mu }$and$\ell _{P}^{2}={\frac {\hbar \,G}{c^{3}}}$ and reducing identical constants from two sides, we get Heisenberg'suncertainty principle $\Delta P_{\mu }\Delta x_{\mu }=\Delta (mc\,U_{\mu })\Delta x_{\mu }\geq {\frac {\hbar }{2}}$ In the particular case of a static spherically symmetric field and static distribution of matter$U_{0}=1,U_{i}=0\,(i=1,2,3)$and have remained $\Delta R_{0}\Delta x_{0}=\Delta r_{\text{s}}\Delta r\geq \ell _{P}^{2}$ where$r_{\text{s}}$is theSchwarzschild radius,$r$is the radial coordinate. Here$R_{0}=r_{\text{s}}$and$x_{0}=c\,t=r$,since the matter moves with velocity of light in the Planck scale. Last uncertainty relation allows make us some estimates of the equations ofgeneral relativityat thePlanck scale.For example, the equation for theinvariant interval$dS$в in theSchwarzschild solutionhas the form $dS^{2}=\left(1-{\frac {r_{\text{s}}}{r}}\right)c^{2}dt^{2}-{\frac {dr^{2}}{1-{r_{\text{s}}}/{r}}}-r^{2}(d\Omega ^{2}+\sin ^{2}\Omega d\varphi ^{2})$ Substitute according to the uncertainty relations$r_{\text{s}}\approx \ell _{P}^{2}/r$.We obtain $dS^{2}\approx \left(1-{\frac {\ell _{P}^{2}}{r^{2}}}\right)c^{2}dt^{2}-{\frac {dr^{2}}{1-{\ell _{P}^{2}}/{r^{2}}}}-r^{2}(d\Omega ^{2}+\sin ^{2}\Omega d\varphi ^{2})$ It is seen that at thePlanck scale$r=\ell _{P}$space-time metric is bounded below by thePlanck length(division by zero appears), and on this scale, there are real and virtual Planckian black holes. Similar estimates can be made in other equations ofgeneral relativity.For example, analysis of theHamilton–Jacobi equationfor a centrally symmetric gravitational field in spaces of different dimensions (with help of the resulting uncertainty relation) indicates a preference (energy profitability) for three-dimensional space for the emergence of virtual black holes (quantum foam,the basis of the "fabric" of the Universe.).This may have predetermined the three-dimensionality of the observed space. Prescribed above uncertainty relation valid for strong gravitational fields, as in any sufficiently small domain of a strong field space-time is essentially flat.

If virtual black holes exist, they provide a mechanism forproton decay.This is because when a black hole's mass increases via mass falling into the hole, and is theorized to decrease whenHawking radiationis emitted from the hole, the elementary particles emitted are, in general, not the same as those that fell in. Therefore, if two of aproton's constituentquarksfall into a virtual black hole, it is possible for anantiquarkand aleptonto emerge, thus violating conservation ofbaryon number.

The existence of virtual black holes aggravates theblack hole information loss paradox,as any physical process may potentially be disrupted by interaction with a virtual black hole.

See also

• Quantum foam
• Virtual particle
• Quantum tunnelling