Cosmology · Conditional futures of the Universe

Theoretical black-hole evaporation and its limits

Follow a finite semiclassical mass-loss scaling and compare it with incoming energy without extrapolating to an unknown quantum endpoint.

Preparation: ratios, powers of ten and ordinary arithmetic. Physical quantities and notation are introduced as they are needed.

Back to the chapter

After reading, you can

Describe Hawking radiation as a semiclassical prediction with a stated regime.
Calculate mass, temperature and power ratios in a supplied finite model.
Separate isolated evaporation from competing accretion and endpoint questions.

A quantum-field prediction on a classical background

Hawking’s calculation treats quantum fields in the spacetime of gravitational collapse and predicts outgoing radiation with a thermal relationship to the black hole’s surface gravity.

For an uncharged nonrotating Schwarzschild black hole, the associated temperature is T_H = ℏc³/(8πGk_BM). Its inverse dependence on mass is a theoretical relation within that description.

Here it is used as a semiclassical prediction, not as a directly measured emission signal from an astrophysical black hole. Matter heated outside a black hole and radiation from an accretion flow are different physical mechanisms.

The following finite model assumes a large, slowly evolving black hole. It does not describe a near-Planck-mass endpoint, resolve information questions or assign a final remnant state.

Define the approximation before integrating it

Supply a constant positive coefficient A, in kg³/s, and an isolated mass-loss law dM/dt = −A/M². A is fixed only over the displayed interval, representing a chosen fixed-channel scaling rather than an exact greybody luminosity.

Start at mass M₀ and stop at M₀/2. Require even this final mass to be much larger than the Planck mass, with fractional mass change over a light-crossing time small throughout the interval.

Set μ = M/M₀ and t_* = M₀³/(3A). With u = (t−t₀)/t_*, integration gives μ = (1−u)¹ᐟ³ for 0 ≤ u ≤ 7/8.

Define T₀ = ℏc³/(8πGk_BM₀) and P₀ = c²A/M₀². Then T_H/T₀ = 1/μ and the supplied outgoing power P/P₀ = 1/μ². Temperature has kelvin units and power watts.

Finite isolated scaling; the calculation stops at half mass while the semiclassical hierarchy is still assumed valid
μ = M/M₀	u	T_H/T₀	P/P₀	Net rate at j = 0	Net rate at j = 0.5	Net rate at j = 2
1	0	1	1	−1	−0.5	1
0.8	0.488	1.25	1.5625	−1.5625	−1.0625	0.4375
0.5	0.875	2	4	−4	−3.5	−2

Emission need not imply net mass loss

For a separate instantaneous comparison, supply an incoming mass-equivalent rate J ≥ 0 and define j = JM₀²/A. The normalized net rate is (dM/dt)/(A/M₀²) = j−μ⁻².

At μ = 1, outgoing emission contributes −1 in these rate units. An incoming j = 2 gives a net +1: the mass increases at that instant despite the specified outgoing radiation.

At μ = 1/2, the outgoing term is −4, so j = 2 instead gives net −2. The sign depends on both the current mass and the incoming contribution.

The j columns are instantaneous rate comparisons at the displayed masses. The isolated u(μ) trajectory is not a solution with nonzero incoming J; solving that changing environment requires a new evolution calculation.

Optional mathematical path: integrate only within the valid interval

Multiplying the isolated law by M² gives M²dM = −A dt. Integration yields M³ = M₀³−3A(t−t₀), from which the finite μ expression follows.

At M = M₀/2, one has u = 1−1/8 = 7/8. The temperature doubles and the supplied power quadruples. The emitted energy over that isolated interval is the mass decrease times c², namely M₀c²/2.

A sufficient slowly evolving condition for this toy is 2GA/[c³(M₀/2)²] ≪ 1, together with M₀/2 much larger than the Planck mass. Both are assumptions of the selected hierarchy, not measurements.

Although the algebraic formula has a formal zero-mass limit at u = 1, the calculation is stopped earlier. Continuing the formula to that limit would not establish the physics of a final quantum regime.

A supplied uncharged nonrotating large, slowly evolving Schwarzschild hole obeys isolated dM/dt = −A/M², with fixed A in kg³/s only from M₀ to M₀/2. Assume the final mass remains much larger than Planck mass and 2GA/[c³(M₀/2)²] is much less than one. Dimensionless time u = (t−t₀)/t_* runs from 0 to 7/8, where t_* = M₀³/(3A). Three zero-based panels use distinct ordinates: mass μ = M/M₀ from 0 to 1.1; temperature T_H/T₀ from 0 to 2.2; outgoing power P/P₀ from 0 to 4.4. Solid blue circles show μ = (1−u)^(1/3), dashed orange squares show 1/μ, and dotted green diamonds show 1/μ². All stop at 7/8. Samples μ = 1, 0.8 and 0.5 occur at u = 0, 0.488 and 0.875, with temperature factors 1, 1.25 and 2 and power factors 1, 1.5625 and 4. T₀ = ℏc³/(8πGk_BM₀) is in kelvin and P₀ = c²A/M₀² in watts. A separate native table compares instantaneous physical mass rates in units A/M₀². For incoming j = JM₀²/A equal to 0, 0.5 and 2, the rates j−μ^−2 are (−1,−0.5,+1), (−1.5625,−1.0625,+0.4375) and (−4,−3.5,−2) at the three masses. These are not incoming-energy trajectories. The normalized derivative dμ/du is one third of this physical rate ordinate. The finite emitted energy is M₀c²/2. No exact luminosity calibration, actual lifetime, direct observed emission, information resolution or quantum endpoint is asserted. Original Ark teaching model.

Separate the mechanism from a cosmic timeline

Actual outgoing spectra and power depend on absorption properties, particle species and the black hole’s state. Ambient radiation and material accretion can oppose mass loss, with their own spectrum and dynamics.

A supplied J is not the flux from a specified thermal bath. Determining that balance requires an absorption calculation and an environmental model rather than identifying an ambient temperature with a mass rate.

This example assigns no astrophysical evaporation time, direct-emission detection, final disappearance or solution of the information problem. These conclusions do not follow from a finite inverse-mass scaling.

In remote-future scenarios, state when net evaporation is assumed to dominate, where the semiclassical approximation applies and where the calculation ends. That preserves the distinction between a predicted mechanism and an established endpoint.

Worked reasoning

Stop at half the starting mass

What are the time, temperature and power ratios when the isolated toy reaches M₀/2?

Use μ³ = 1−u with μ = 1/2, giving u = 7/8.
The inverse-mass temperature law gives T_H/T₀ = 2.
The supplied inverse-mass-squared power law gives P/P₀ = 4.

At u = 7/8, temperature is twice and power four times the reference value; the model stops while the mass is still assumed large.

Try the reasoning

At μ = 1 and j = 2, does the black hole lose mass in the instantaneous comparison?
Show the answer and reasoning
No: the net normalized rate is +1.
The incoming +2 exceeds the outgoing −1.
Can the isolated μ = (1−u)¹ᐟ³ curve also be used as the trajectory for j = 2?
Show the answer and reasoning
No.
Adding incoming mass changes the differential equation and therefore the trajectory.
Does the formal μ = 0 algebra establish the final quantum state?
Show the answer and reasoning
No.
The finite semiclassical approximation was deliberately stopped at μ = 1/2 and does not describe a near-Planck endpoint.

What to carry forward

Hawking emission is used here as a semiclassical theoretical mechanism.
Net mass evolution depends on incoming energy as well as outgoing radiation.
A finite large-mass scaling does not determine a final quantum endpoint or an actual cosmic date.

References

Sources supporting the explanations, physical distinctions and examples in this lesson.