Hawking radiation and the trans-Planckian problem
The theoretical discovery of Hawking radiation is regarded as one of the important milestones of modern theoretical physics, since it brings thermodynamics, quantum mechanics, and general relativity into contact within a common framework. Stephen Hawking’s derivation shows that, in the semiclassical picture, black holes possess thermal radiation. At the same time, this derivation contains a well-known conceptual difficulty: the so-called trans-Planckian problem.
In the standard continuum, tracing later observable radiation modes backwards leads to extremely high frequencies near the event horizon. This is not a directly experimentally accessible statement, but a consequence of the idealised mathematical backward evolution of the modes. Nonstandard analysis can serve here as a precise language for treating such contributions not as indeterminate infinities, but as exactly specified hyperreal quantities, and for separating them structurally from finite remainder terms.
The trans-Planckian problem in the continuum
If one formally traces a thermal radiation quantum, registered far away from a black hole, backwards in time, its origin approaches the event horizon. Owing to the gravitational frequency shift, that is, the gravitational redshift, the same mode is described near the horizon by ever higher frequencies.
In standard analysis, this frequency may tend to infinity under idealised backward evolution. Formally, this means that the corresponding mode enters a regime in which its wavelength lies below the Planck scale. Yet precisely there it is no longer guaranteed that the semiclassical approximation consisting of classical spacetime and quantum field theory remains valid without modification.
The trans-Planckian problem is therefore not an immediate experimental contradiction, but an indication that the derivation depends sensitively on how the extremely short-wavelength degrees of freedom are treated.
The hyperfinite lattice and the constant \(\omega\)
In the nonstandard analytical modelling considered here, there is no unbounded continuum in which frequencies can escalate without limit. The spacetime structure at the event horizon is described by a fine, but strictly hyperfinite, lattice. The maximum frequency is exactly specified and bounded by the midfinite constant \(\omega\).
In addition, it is assumed that vacuum modes at the Planck boundary no longer couple unchanged to the macroscopic curvature of spacetime. For this purpose, a smooth damping function \(g(x)\) is introduced. For macroscopic frequencies it is approximately \(1\); at the upper boundary \(\omega\) it falls smoothly to zero. Thus \(g(0)=1\) and \(g(\omega)=0\).
In this model, the energy density of the vacuum state at the horizon is written as a hyperfinite sum over discrete frequencies \(n\):
\[ E_{\mathrm{horizon}} \propto {\LARGE{\textbf{+}}}_{n=0}^{\omega} n \cdot g(n). \]This representation is to be understood as a model kernel. It does not replace the full derivation of Hawking radiation via field modes, Bogoliubov coefficients, and the comparison of ingoing and outgoing vacuum states.
Structural deflation at the horizon
To separate the finite, physically relevant contribution from the background structure at the event horizon, the hyperreal Euler-Maclaurin formula is applied to the target function \(f(x) = x \cdot g(x)\):
\( {\LARGE{\textbf{+}}}_{\check{q}=0}^{\check{r}}f(\check{q}) = {\uparrow}_0^{\check{r}}{f\left(x\right){\downarrow}x} +\check{f}(\check{r}) +\check{f}(0)\)\( +{\LARGE{\textbf{+}}}_{\check{m}=1}^{\check{n}-1}{\widetilde{m!}}B_m \left({}^{\acute{m}}f(\check{r})-{}^{\acute{m}}f(0)\right) +\mathcal{O} \left( {\widetilde{n!}}B_n \left({}^{\acute{n}}f(\check{r})-{}^{\acute{n}}f(0)\right) \right). \)
Here too, the problem separates into three essential structural components:
The integral: the hyperreal pole
The integral term \(P_{\mathrm{vacuum}} = {\uparrow}_{0}^{\omega} x \cdot g(x) {\downarrow}x\) represents the dominant background contribution of the vacuum model directly at the event horizon. It is the part associated with the trans-Planckian problem in the standard continuum. In hyperreal space, however, this term is not an undefined infinity, but an exactly specified pole. By itself, it does not describe emitted Hawking radiation, but rather the not directly measurable background structure of the model.
The upper boundary at \(\omega\)
Since the spacetime lattice is cut off at the constant \(\omega\), and since \(g(\omega)=0\) is assumed together with the relevant derivatives, the corresponding boundary terms at the maximum frequency vanish. Thus \(f(\omega)=0\), \({}^1 f(\omega)=0\), \({}^2 f(\omega)=0\), and so on. The added term \(\check{f}(\omega)\) is also zero.
In this modelling, the trans-Planckian problem is thereby regularised: the highest-energy boundary modes are not treated as physically freely extendable modes, but are removed from the measurable remainder by the cut-off.
The lower boundary at 0: the regular remainder
Since \(f(0) = 0\), the isolated term \(\check{f}(0)\) also vanishes. The low-frequency, macroscopic modes at the lower boundary determine the remaining term through the derivative terms:
\[ R_{\mathrm{thermal}} = {\LARGE{\textbf{+}}}_{\check{m}=1}^{\check{n}-1} \widetilde{m!} B_m \left( 0 – {}^{\acute{m}}f(0) \right). \]This boundary term is to be understood as the regular, macroscopic remainder of the deflated mode structure. In a complete physical derivation it must be combined with the usual mode analysis of Hawking radiation, in particular with the thermal occupation number arising from the mixing of positive and negative frequencies.
Conclusion of the derivation
The trans-Planckian problem arises when the semiclassical calculation is traced back into frequency ranges in which the underlying continuum description itself becomes questionable.
The nonstandard analytical perspective offers a precise way of reorganising this situation: the extreme frequencies at the horizon are collected in a dominant hyperreal integral \(P_{\mathrm{vacuum}}\). This is not interpreted as a physically directly radiating component, but as a background structure of the model.
The actual Hawking radiation is then not obtained from undefined trans-Planckian infinities, but from a finite, regular remainder term that remains after the algebraic separation of the hyperreal pole. Hawking’s result is therefore not replaced, but reorganised in a nonstandard analytical language: the thermal radiation remains the macroscopic result, while the trans-Planckian contribution appears as an exactly specified background pole.
© 2026 by Boris Haase