---
abstract: 'We study the consequences of nonzero neutrino masses for black holes evaporating by the emission of Hawking radiation. We find that the evolution of small, hot, black holes may be unaffected (if neutrinos are Majorana particles), or may show an increase in neutrino luminosity and a decrease in lifetime by up to a factor of 1.85 (if neutrinos are Dirac particles). However, for sufficiently large (e.g., stellar mass) black holes, neutrino emission is largely or entirely suppressed, resulting in a decrease in emitted power and an increase in lifetime by up to a factor of 7.5.'
author:
- Daniel Bambeck
- 'William A. Hiscock'
date: 'June 8, 2005'
title: Effects of nonzero neutrino masses on black hole evaporation
---

Since the discovery that black holes emit particles with a thermal spectrum [@Hawking], it has been understood that black holes will have a finite lifetime due to this emission. Most analyses of black hole evaporation have relied on the detailed studies completed by Page [@Page1; @Page2] of the flux of particles of differing spins and masses emitted by Schwarzschild and Kerr black holes.

The general picture that emerges is that any black hole in isolation will lose any initial charge and angular momentum in the early stages of evaporation, asymptotically approaching a Schwarzschild-like state long before reaching microscopic size. Specifically, any charged black hole with mass less than approximately $10^5 M_{\odot}$ will rapidly lose any initial electric charge [@Gibb; @HW]. Furthermore, unless there are a large number of undiscovered massless scalar fields in nature [@TCH], black holes will quickly spin down, with their specific angular momentum $a=J/M$ decreasing at roughly twice the rate their mass decreases [@Page2]. Hence, any black hole is expected to asymptotically approach a Schwarzschild state, nonrotating and uncharged, well before it completely evaporates.

The subsequent evaporation of the Schwarzschild-like black hole was treated by Page [@Page1]. He found that 81% of the power emitted by a hole with $ M >> 10^{17}$ g is in the form of neutrinos, assuming two species (electron and muon) of massless neutrinos (87% of the power would be in neutrinos with three massless species). Today there is strong evidence that the three known species of neutrino have nonzero masses. This evidence comes from atmospheric [@Fuk1; @Fuk2; @Yasuda] and solar neutrino experiments that imply neutrino oscillations are taking place [@BKS]. In order for neutrinos to oscillate among species, they must have mass. The details of the oscillations fix the differences of the squares of the masses of the neutrinos, but not their separate individual values. Oscillations occur because the weak interaction eigenstates are not identical to the mass eigenstates.

The purpose of this Letter is to examine the effect of nonzero neutrino masses on the evolution of evaporating black holes. Since so much of the emitted power for large black holes would be massless neutrinos, noteworthy quantitative, if not qualitative, differences in black hole evolution might be expected when neutrino masses are taken into account.

We take a conservative viewpoint concerning unknown neutrino physics, assuming, in particular, that there are only three species of neutrino (i. e. , that no additional “sterile” neutrino species exist that do not participate in charged or neutral current weak interactions). This assumption is favored by the atmospheric neutrino oscillation data [@Fuk3]. We utilize current experimental results to bound the possible changes in black hole evaporation caused by nonzero neutrino masses within these assumptions. Additional refinement is inevitable as the newly recognized richness of neutrino physics is further explored.

The atmospheric data, with the assumption of three neutrino species, is best fit by [@SKC] $$\Delta m_{13}^2 \simeq 2.4 \times 10^{-3} {\rm eV}^2 \; \; ,
\label{deltam2atm}$$ while the solar neutrino oscillations combined with KamLAND experimental data yield for the second squared mass difference [@KamLAND]: $$\Delta m_{12}^2 \simeq 8 \times 10^{-5} {\rm eV}^2 \, \, ,
\label{deltam2slr}$$ where (1,2,3) label the neutrino mass eigenstates.

These experimental results are compatible with two possible spectra for the neutrino mass eigenstates, with either two closely grouped low mass neutrinos and one higher (often termed the “normal” hierarchy), or vice versa (the “inverted” hierarchy). The specific (as opposed to relative) values of the mass eigenvalues are far less certain than the new knowledge of the squared differences from neutrino oscillations. The lowest possible set of masses is achieved by setting the lowest mass eigenvalue to be infinitesimally larger than zero. The maximum possible values for the neutrino masses are constrained by a recent cosmological bound obtained from the $2$ Degree Field Galaxy Redshift Survey [@Elgaroy], $$m_\nu^{\rm tot} < 1.8 \,{\rm  eV} \, \, ,
\label{masstot}$$ where $m_\nu^{\rm tot}$ is the sum of the neutrino mass eigenvalues, and assuming currently accepted values for the Hubble constant and total matter density [@Spergel].

An additional issue that arises once neutrinos are recognized to be massive particles is whether the neutrino and antineutrino are distinct particles. It may well be that neutrinos carry no conserved quantum number that would distinguish particle from antiparticle; if not, then neutrinos are Majorana particles. On the other hand, if there is, say, a conserved lepton number, then neutrinos are Dirac particles and neutrinos and antineutrinos are distinct particles. In the massive Dirac case, neutrinos and antineutrinos can each occur in two spin states, yielding twice as many states as are available for massless neutrinos. If they are Majorana particles, then the neutrino is its own antiparticle, lepton number is not conserved, and there are only two states available, as in the massless case. The Dirac or Majorana nature of the neutrino is presently one of the most intriguing questions in neutrino physics. In this Letter, we will describe the effects on black hole evaporation of massive neutrinos of both types.

There are thus three key questions to which the answers are not yet known, even within the conservative assumption of only three neutrino species. First, what is the mass of the lightest neutrino? Second, is the small (solar) mass splitting at high mass or low mass? Third, are neutrinos Dirac or Majorana particles?

There are also, however, three crucial facts that are now established by experiment and observation. First, neutrinos are not massless particles. Second, even assuming the neutrino mass spectrum favors low masses (the “normal” hierarchy), two of the three neutrinos have masses of order at least $10^{-2}$ eV. Third, based on cosmological observations, the heaviest neutrino has a mass of at most about $2$ eV.

The new physics resulting from massive neutrinos can have significant effects on black hole evaporation both for small, hot black holes, and for large, cold black holes. We address three questions in this Letter. First, how does the existence of massive neutrinos change the predicted initial mass of a black hole that is evaporating to zero mass today? Second, if massive neutrinos are Dirac particles, by what factor can the lifetime of an initially small, hot black hole be reduced? Third, for large, cold black holes of astrophysical mass, by what factor will their evaporation lifetimes be increased by massive neutrinos?

Following Page [@Page1], we assume black holes emit particles in each mode independently. Therefore, the emission rate (and emitted power) of each species of particle is directly proportional to the number of modes allowed to that particle. The constants of proportionality depend on the spin and total angular momentum of the particle, and were determined numerically by Page.

Because emission in all modes is independent, our work depends primarily on counting available modes, without need to re-run Page’s code. There are three ways in which our counting differs from that of Page: First, at the time of his paper, only two species of neutrino, electron and muon, were known, whereas now there are known to be three. Secondly, emission in some or all neutrino species may be suppressed, due to their mass. Third, neutrinos may have more spin states available than previously thought.

The power emitted by an evaporating Schwarzschild black hole (and hence, the rate of mass loss) is proportional to the inverse square of the mass, $${dM}/{dt} = -a/{M^2}
\label{massdiffeq}$$ implying a lifetime of $$T = {M_0^3}/{3a}
\label{lifetimeeq}$$ where $M$ is the mass of the hole at a given time, $M_0$ is the initial mass, $a$ is a constant of proportionality which depends on the particle species being emitted by the hole, and Planck units are used. From Page’s work [@Page1], $$a = 4.091 \times 10^{-5} n + 3.755 \times 10^{-5}
\label{adef}$$ where $n$ is equal to the number of modes, considering particle species and spin states, available in spin-1/2 particles, and the final constant term is due to radiation in massless photons and gravitons.

Over its lifetime, as a hole shrinks and becomes hotter, the constant $a$ will increase in an almost stepwise fashion (hence the justification in treating it to be constant over substantial periods of time), as the temperature of the hole passes thresholds corresponding to the masses of various particles. However, since a black hole spends most of its lifetime very near its initial mass, it is a reasonable approximation to use the initial mass to determine which species are available to be emitted. As long as the initial temperature is not exceedingly close to a mass threshold, the lifetime calculated using the initial value of $a$ is a very good approximation to the precise value, as originally noted by Page [@Page1].

The first question we wish to address is what the initial mass of a primordial black hole would be that is evaporating completely (reaching zero mass) at the present time. It is easy to show that such a black hole would be initially small enough, and therefore hot enough, to emit all neutrinos, as well as electrons and positrons, and of course massless photons and gravitons. If neutrinos were massless and had definite helicity, as was previously believed, then there would be six neutrino states available (neutrinos and antineutrinos in each of the three types), as well as four electron states (electrons and positrons, with two spin states each) for a total of ten states available in spin-1/2 particles. Assuming the black hole is the same age as the universe, $1.37 \times 10^{10}$ years [@Spergel], using Eqs.(\[lifetimeeq\]) and (\[adef\]), this yields an initial mass of $2.21 \times 10^{19}$ times the Planck mass, or $4.80
\times 10^{14} {\rm g}$. If neutrinos are massive Majorana particles, then the number of neutrino modes available is the same as for the case of massless neutrinos, and therefore the initial mass is the same, since the initial black hole is more than hot enough to produce all three types of massive neutrino.

However, if neutrinos are massive Dirac particles, then there are a total of 12 neutrino states available (three types of neutrino and (distinct) antineutrino, with two spin states each), for a total of 16 spin-1/2 particle states. This results in a higher luminosity, so the initial mass must therefore be larger to last the same amount of time. Specifically, the initial mass must be $2.55 \times 10^{19}$ times the Planck mass, or $5.56 \times
10^{14} {\rm g}$, some 16% larger than the initial black hole mass for massless neutrinos.

In either case, the initial temperature of the black hole is of order 20 MeV, well above the masses of the electron and all neutrino species, so the initial assumption of emission in neutrinos and electrons is justified.

The possibility that massive neutrinos are Dirac particles leads to the second scenario we wish to address: the (naively surprising) result that massive neutrinos may cause small black holes to evaporate more quickly than if neutrinos were massless. The effect is most dramatic for black holes that are large and cool enough initially so that electrons and positrons are not copiously produced ($T << 0.511\; {\rm MeV}, M
>> 2 \times 10^{16} {\rm g}$), but are still small and hot enough so that all three types of neutrinos are sure to be produced initially ($T >> 2\; {\rm eV}, M << 5 \times 10^{21} {\rm g}$). In this case, the existence of additional neutrino states that the black hole can emit into can decrease the lifetime of such a hole by as much as 46%, a factor of 1.85.

The third situation in which neutrino masses can be important in black hole evaporation is for large black holes whose lifetimes greatly exceed the age of the universe. For such large black holes, a critical issue is whether they are initially hot enough to emit substantial numbers of neutrinos.

Since a black hole radiates thermally with a temperature proportional to its surface gravity, emission of particles with a rest mass higher than the surface temperature will be exponentially suppressed. In particular, this implies that a hole can only emit significant quantities of particles with rest mass less than the temperature of the hole, or, in Planck (natural) units, $m << 1/8\pi M$, where $m$ is the mass of the particle emitted. For a stellar mass black hole of three solar masses, this temperature corresponds to $1.8 \times 10^{-12} {\rm eV}$.

However, the smallest possible neutrino mass splitting is $\Delta
m^2 \simeq 8 \times 10^{-5} {\rm eV}^2$ , which implies that at least two neutrinos have masses greater than $10^{-2} {\rm eV}$, and certainly all three [*could*]{} have masses greater than $10^{-12} {\rm eV}$. This leads, then, to three possibilities which must be considered for an astrophysical mass black hole: No neutrinos initially emitted, one type of Majorana neutrino initially emitted, or one type of Dirac neutrino initially emitted. Again, since any black hole spends most of its lifetime near its initial mass, only the number of species emitted at the initial temperature is relevant to the approximate lifetime.

In the first case, the lightest neutrino has a mass greater than the temperature of the hole, and therefore no neutrinos (or any other spin-1/2 particles) will be initially emitted. Since no neutrinos are emitted, it does not matter whether they are Majorana or Dirac particles. This leads to a lifetime of $8877
M_0^3$, or $1.157 \times 10^{67} ({M_0}/{M_{\odot}})^3$ years.

In the second case, we assume the neutrino is a Majorana particle, and that the lightest neutrino has a mass less than the initial temperature of the black hole. In this case, there are two spin states of that light neutrino that will be initially emitted. The lifetime of the hole would then be $2792 M_0^3$, or $3.638 \times
10^{66} ({M_0}/{M_{\odot}})^3$ years.

In the third case, we assume the neutrino is a Dirac particle, and that the lightest neutrino has a mass less than the initial temperature of the black hole. In this case, there are four states available: Neutrino or antineutrino, with two spin states each. The lifetime is then $1657 M_0^3$, or $2.159 \times 10^{66}
({M_0}/{M_{\odot}})^3$ years.

For comparison, if neutrinos are considered to be massless particles with definite helicity, then there are six neutrino states (particle and antiparticle, of each of three types), that will be emitted initially, and the life-span is therefore $1178
M_0^3$, or $1.535 \times 10^{66} ({M_0}/{M_{\odot}})^3$ years.

Thus, in comparison with the massless neutrino case, the existence of neutrino mass may increase the evaporation lifetime of large, astrophysical mass black holes by as much as a factor of $7.5$.

This work was supported in part by NSF Grant No. PHY-0098787.

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