Contrary to their name, black holes in fact glow. According to Stephen Hawking’s famous result, black holes radiate energy and have a temperature. But where does this temperature come from?

In our research, we show that this strange heat of spacetime has a surprisingly topological origin. That is, it depends not just on the local properties of gravity, but on the global shape of spacetime itself. The Big Idea: Curvature Meets Topology.Every object-whether a donut or a sphere-has a property called the Euler characteristic. It's a number that tells you something deep about the object's shape. For example: A sphere has Euler characteristic 2, A donut (torus) has Euler characteristic 0. Our breakthrough is this: the temperature of spacetime horizons (like those around black holes or inside an expanding universe) is related to their Euler characteristic. We prove this using a powerful mathematical tool: the Chern-Gauss-Bonnet theorem, which links curvature and topology. But to apply it to spacetimes, we have to use a clever trick called Wick rotation--a way of turning time imaginary to smooth out the geometry. 


Black holes are examples of phenomena known as causal horizons - which are boundaries in spacetime across which events cannot influence each other due to the finite speed of light. These occur across many various spacetimes, and comprise an interesting intersection between gravity, quantum field theory, and thermodynamics. In the case of the Schwarzschild and de Sitter spacetimes, we find this connection can be characterised by the well-known Wick rotation (also known as analytic continuation) where $t \to i\tau$ and the resulting periodicity becomes $\tau \sim \tau + 2\pi \beta$. This periodicity in the Euclidean time $\tau$ is an indication of a non-trivial thermal state present within the spacetime - with temperature $T_H = 1/\beta$ - indicating a uniform radiation flux from the horizon. 

By applying the Wick rotation to these two well-studied phenomena:

1. 1. Schwarzschild Black Hole: A simple, non-rotating black hole. Its Euclidean version looks like a disc (D²) crossed with a sphere (S²). Its Euler characteristic is 2, and we show that the Hawking temperature is directly related to this number. 
2.  2. de Sitter Universe: A model of a universe filled with dark energy. Here too, Wick rotation gives us a compact space, and again the Euler characteristic determines the temperature-in this case, the Gibbons-Hawking temperature. 

This argument from Wick rotation places primary emphasis on the thermal properties of fields within spacetime. However, the analytic continuation $t \to i\tau$  and resulting angular compactification in $\tau$ is a non-trivial geometric operation that enforces global constraints on the topology of the manifold. For instance, in the case of de Sitter, this procedure maps the non-compact hyperbolic spacetime to the compact 4-sphere. The consequences of this from the geometric perspective are considerable: various integrals over scalars formed from the Riemann curvature are now guaranteed to converge (while before Wick rotation they automatically diverged in time $t$) to topological integers.

Fig.1 Under Wick rotation de Sitter space is transformed into an $S^4$ sphere  while  Black hole - into the product of disk and 2D sphere: $D^2\times S^2$. Both have Euler characteristic $\chi=2$}

Here we demonstrate that these considerations lead to a natural topological origin for horizon thermodynamics. For both Schwarzschild and de Sitter spacetimes, the temperature of the spacetime horizon is shown to be inversely proportional to the Euler characteristic $\chi$ (a topological integer) of the manifold after the Wick rotation and Euclidean time periodicity are performed. In addition, for de Sitter this framework explains topologically the origin of the discrepancy between the two physical temperatures discussed extensively by G. Volovik, A. Polyakov and others. Therefore, while the thermal field theory perspective is computationally powerful in quantifying horizon dynamics, the latter is in fact the consequence of a duality between the causal structure of a spacetime and the global topology of its analytical continuation. 

So why does it matter? This topological approach gives us: 

•  A unified framework to understand horizon temperatures.
•   A new geometrical insight into why time becomes periodic near a horizon.
•   A profound link between heat and shape-suggesting that spacetime itself may know its temperature because of its topology. 

In short, we show that temperature is not just about energy-it's about the shape of space itself.

Hughes J.C.M. and Kusmartsev F.V. 
JETP Letters 122, issue 4 (2025)