Concerning our cosmic event horizon, an interesting question arises from the seemingly innocent statement, "My cosmic event horizon is not your cosmic event horizon." Ie: Since 'you' and 'I' are not in the same place and the universe is isotropic around each one of us, and we're separated by some distance $r$, then in principle, our two CEHs are also separate (offset) from one another by some distance, on the order of $r$. This implies two distinct CEHs... each of which is sending Hawking radiation (HR) into our universe. The question then is, are these two distinct HRs additive? If so, into how many ways can our ~de Sitter universe be divided up into $N$ distinct 'cells', each of which is operationally at the center of its own physically distinguishable CEH? Where each such CEH is generating its own additive HR?
1 Answers
No, it is not additive. The reason is the origin of these sorts of effects. For simplicity, I'll treat our Universe as being exactly de Sitter.
Firstly, I should notice that the thermal radiation on de Sitter spacetime created by means of quantum field theoretic effects is not due to the Hawking effect (which occurs in a collapsing star spacetime), but rather due to the related Unruh effect, which occurs in spacetimes with a geometric structure known as a "bifurcate Killing horizon". These two effects are fairly often mistaken, especially because one can derive the Unruh effect in Schwarzschild spacetime (a non-rotating, uncharged black hole) in a manner way simpler than one can derive the Hawking effect. Here are the main differences:
- the Unruh effect concerns spacetimes with a notion of time-translation symmetry and predicts that some observer sees a thermal spectrum of particles all around them. For a black hole, this means there's also particles coming in from infinity. Rotating black holes do not exhibit the Unruh effect due to technical reasons.
- the Hawking effect concerns the spacetime of a star that collapses to a black hole. In this case, the thermal particles come only from the black hole. Rotating black holes do exhibit the Hawking effect.
Non-Technical Version
The thermal spectrum depends on the observer. The temperature coming from each horizon is not observer-independent, but rather is the temperature measured by the observer with that horizon. Since each horizon gives rise to a notion of temperature in a different sense (each one corresponding to a particular observer), they don't add up. The reason is because the temperature is due to emission of particles by the horizon, and particles are not an observer-independent concept (I discussed the reason behind this in a not-so-technical way in this answer).
In short, the temperatures don't add up because they are different notions of temperature. Each CEH leads to the notion of temperature associated to the observer that has that particular CEH.
Technical Version
I'll first state the effect in more technical terms, then discuss it more physically, without all the technicalities. The short statement is that if an observer's worldline follows the orbit of a Killing vector field generating such a horizon and the quantum field is in a symmetric state with respect to the isometry generated by this vector field, then the observer experiences the state as a thermal state.
Let me write this is a different way. Suppose you have a symmetry in your spacetime. For example, in Minkowski spacetime, we have boosts in the $x$ direction, as an example. Consider the transformation on the spacetime when you boost it with some velocity $v$. Each point is mapped to some different point. The way this happens in Minkowski spacetime is that, as we continuously vary $v$, some points describe a timelike trajetory (those with $|x| > |t|$). some describe a spacelike trajectory ($|x| < |t|$), and some describe a null trajectory ($|x|=|t|$). This sort of behaviour is, roughly, what we call a bifurcate Killing horizon. The Killing horizons are the surfaces in which the trajectories are null ($x = \pm t$), and the "bifurcate" is because it bifurcates into two horizons.
Suppose now one observer is following one of these trajectories. This is the case, for example, for accelerating observers. What the Unruh effect predicts then is that if the quantum field is in a state invariant under these trajectories (for example, the Minkowski vacuum), then such an observer will see this state as a thermal state. Hence, accelerated observers see a thermal bath in the Minkowski vacuum, which inertial observers see as having no particles at all, and consider to be at vanishing temperature. This is known as the Unruh effect in flat spacetime.
Notice that different observers have different notions of what is going on. It is important to point out that they disagree on the temperature and they disagree on whether there are particles on the spacetime, but they do agree on the quantum state of the field. There is nothing paradoxical about this, because quantum field theory is a theory of fields, not of particles. It simply means that what is measured by an inertial thermometer is not the same thing measured by an accelerated thermometer. This answer of mine discusses more about the relativity of the notion of particle in Quantum Field Theory in Curved Spacetimes.
Now, de Sitter spacetime. In this situation, for every timelike geodesic we can find a Killing vector field with those properties. In other words, for any free falling observer, we have such a construction like the one I just mentioned in Minkowski spacetime. Hence, all inertial observers in de Sitter spacetime experience the field (if it is in a symmetric state) as being in a thermal state, with temperature given by $T = \frac{1}{2\pi}\sqrt{\frac{\Lambda}{3}}$. These effects are not additive, because the temperature each observer measures is due to their own notion of particle. While I have a CEH and you have another one, we also have different notions of particles. What you call a particle and what I call a particle are different things, and hence each of us only perceives as temperature those particles that we see due to our own CEH.
For more information on these topics, I recommend Wald's Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics, which discusses the Unruh effect in de Sitter spacetime on Sec. 5.3. There is also a seminal paper by Gibbons and Hawking on the matter. I quote a phrase from their abstract:
The derivation of these results involves abandoning the idea that particles should be defined in an observer-independent manner.
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