Observer, Acceleration, and Cosmological Horizons

Observer, acceleration, and cosmological horizons are all boundaries of restricted causal access, but the restriction is supplied by different data. A Rindler horizon depends on an eternally accelerated worldline; a cosmological particle horizon depends on the finite conformal age of the past; a cosmological event horizon depends on the entire future expansion. None is identified merely by a large redshift, a coordinate singularity, or the Hubble radius. The safest comparison begins with the causal set whose boundary is being taken.

Required background. We use timelike congruences and spatial projection to distinguish one observer from a cosmological congruence, and hypersurface normals and tangential projectors to interpret smooth null frontiers.

Helpful background. Pseudo-Riemannian metrics and signature fix the light cones, affine and variational geodesics support the null-ray calculations, Causality and Global Structure gives the global context, and Exact Solutions of Einstein’s Equations supplies Minkowski and de Sitter benchmarks.

Observer horizons from complete histories

For a future-inextendible timelike worldline $\gamma$, the region that can ever influence the observer is $I^-(\gamma)$. When this past set is not all of spacetime, its boundary

$$\mathcal H^+[\gamma]=\partial I^-(\gamma)$$

is a future observer horizon. It is invariantly defined once the complete worldline is fixed, but it is observer dependent because a different worldline can have a different causal past. The time-dual boundary $\partial I^+(\gamma)$ is a past observer horizon when $\gamma$ is also past-inextendible.

In Minkowski coordinates $(T,X)$, an eternally uniformly accelerated observer follows

$$T=a^{-1}\sinh(a\tau), \qquad X=a^{-1}\cosh(a\tau).$$

Its causal past is bounded by $T=X$, and its causal future is bounded by $T=-X$. These are the future and past Rindler horizons. The spacetime curvature vanishes everywhere, and an inertial observer crosses either null plane without encountering a geometric defect. What creates the restricted access is the accelerated worldline’s asymptotic behavior. The calculation is given in Ellis and Uzan 2016, §2.2, pp. 4–5 and treated from the accelerated-frame viewpoint on Uniform Acceleration and Rindler Wedges.

“Eternally” is load-bearing. If the rocket stops accelerating and becomes inertial, its later causal past can include events beyond the former Rindler boundary. A coordinate chart that retains only the right wedge may still end at the null planes, but the observer no longer has a causal horizon there. Thus a chart boundary and an acceleration horizon can coincide without being synonymous.

Particle and event horizons in cosmology

Consider a spatially homogeneous and isotropic spacetime

$$ds^2=-dt^2+a^2(t)\left[d\chi^2+f_K^2(\chi)d\Omega_2^2\right],$$

with cosmic time $t$, scale factor $a(t)$, and comoving fundamental observers. Radial null curves obey $d\chi=\pm dt/a(t)$. If the model begins at $t_i$, the radial conformal reach relevant to the particle horizon at time $t_0$ is

$$\chi_{\rm PH}(t_0)=\int_{t_i}^{t_0}\frac{dt}{a(t)},$$

provided the integral is finite. It marks the limiting comoving worldline that could already have influenced the observer by $t_0$. It is a past-history concept and changes with observation time.

If the model has future endpoint $t_f$, possibly $+\infty$, the radial conformal reach relevant to a cosmological event horizon is

$$\chi_{\rm EH}(t_0)=\int_{t_0}^{t_f}\frac{dt}{a(t)},$$

again only when the integral converges. An inward radial photon emitted at $t_0$ from comoving radius $\chi\lt\chi_{\rm EH}(t_0)$ reaches the observer at a finite time; the limiting ray from $\chi=\chi_{\rm EH}(t_0)$ approaches $\chi=0$ only as $t\to t_f$ (possibly $t_f=+\infty$), while one emitted from $\chi\gt\chi_{\rm EH}(t_0)$ never arrives, provided those radii lie before any compact-slice cut locus. This is a future-history concept: no finite observation time determines it without a model for the entire future. The integral criteria and their causal interpretation are derived in Ellis and Uzan 2016, §§4.1–4.2, pp. 8–10.

These integrals first give conformal causal reaches, not automatically nonempty horizon cross-sections. On a simply connected noncompact constant-curvature spatial section, a positive finite reach has a nonempty frontier. On the unit $S^3$, however, $0\leq\chi\leq\pi$: a reach below $\pi$ gives the usual spherical frontier, equality degenerates at the antipode, and a reach greater than $\pi$ covers the entire slice. For a compact quotient, the corresponding statement uses the spatial distance function, its cut locus, and the quotient’s diameter. Compact topology can therefore remove a particle-horizon cross-section at late times and can leave an early slice without an event-horizon cross-section even when one appears later as the remaining conformal time shrinks. The compact-domain radius criterion is discussed in Ellis and Uzan 2016, §6.1, pp. 16–17.

The two horizons are logically independent. A big-bang model can have a particle horizon but no future event horizon; an eternally accelerating model can have an event horizon but no particle horizon; some models have both or neither. The word “cosmological” therefore does not select one universal surface.

De Sitter horizons and the Hubble scale

In the spatially flat de Sitter slicing, $a(t)=e^{Ht}$ with constant $H\gt0$. The future integral gives

$$\chi_{\rm EH}(t_0)=\frac{e^{-Ht_0}}{H}, \qquad a(t_0)\chi_{\rm EH}(t_0)=\frac1H.$$

For this special model, the physical event-horizon radius equals the Hubble radius. The equality is not a definition. In a general Friedmann–Lemaître spacetime, $H^{-1}$ is an instantaneous expansion length; light can cross it, and it need not coincide with a causal boundary. Calling every $H^{-1}$ a “horizon” obscures the integral test.

For example, in spatially flat power-law expansion $a(t)=a_* (t/t_*)^p$ with $p\gt1$, direct integration gives a proper event-horizon radius $R_{\rm EH}=t/(p-1)$, while $H^{-1}=t/p$. Their ratio is $p/(p-1)\gt1$. An inward photon has proper distance $R=a\chi$ and obeys $\dot R=HR-1$. Thus a photon initially between $H^{-1}$ and $R_{\rm EH}$ first increases its proper distance from us even though it is directed inward and eventually arrives. The event-horizon integral determines arrival; the sign of $\dot R$ at emission does not.

Global de Sitter spacetime has spacelike conformal infinities. A complete geodesic observer nevertheless has a causal diamond bounded by past and future cosmological horizons. Different observers select different diamonds, related by de Sitter isometries. The static coordinate patch displays one such diamond, but its coordinate boundary is meaningful because it coincides with the observer-relative causal boundary—not because the coefficient of $dt^2$ vanishes.

A cosmological particle horizon is also distinct from a visual horizon. The latter depends on the opacity of a messenger, such as the photon last-scattering surface; neutrinos and gravitational waves have different visibility limits. Causal contact and present observability are therefore not interchangeable.

Comparison with black-hole and quasi-local horizons

An asymptotically flat black-hole event horizon is defined relative to a specified $\mathscr I^+$, usually representing a whole asymptotic family of escaping observers. A Rindler horizon is relative to an accelerated worldline or wedge, and a de Sitter cosmological horizon is relative to a timelike observer in a spacetime with spacelike future infinity. All can be achronal null boundaries, but the target sets in their causal-past constructions differ. Their similar local geometry does not erase that global distinction.

Apparent horizons and trapping horizons belong to another category. They are defined from null expansions on codimension-two surfaces and can depend on a foliation. They may be timelike or spacelike rather than null. The expansion data are developed on Trapped and Marginal Surfaces, while the broader physical taxonomy belongs to What Is a Black Hole?.

No horizon listed here is automatically a thermodynamic system. Assigning temperature or entropy requires a quantum state, a generator normalization, a stationary or controlled dynamical regime, and a specified gravitational theory. Causal access is the common geometric starting point, not the complete physics.

Exercises

For a spatially flat model with $a(t)=a_0t^p$ on $0\lt t\lt\infty$ and $p\gt0$, decide when a comoving observer has a particle horizon and when it has a cosmological event horizon.

Solution

Up to the positive constant $a_0^{-1}$,

$$\chi_{\rm PH}(t_0)=\int_0^{t_0}t^{-p}dt$$

converges exactly for $p\lt1$. Hence the model has a particle horizon for $0\lt p\lt1$. The future integral

$$\chi_{\rm EH}(t_0)=\int_{t_0}^{\infty}t^{-p}dt$$

converges exactly for $p\gt1$, giving a cosmological event horizon during accelerated power-law expansion. Multiplying by $a(t_0)$ gives $R_{\rm PH}=t_0/(1-p)$ when $0\lt p\lt1$ and $R_{\rm EH}=t_0/(p-1)$ when $p\gt1$. At $p=1$ both integrals diverge logarithmically, so there is neither horizon. The two tests inspect opposite endpoints of conformal time and cannot be replaced by evaluating $H(t_0)=p/t_0$. Here $a_0$ carries the dimensions needed for the chosen scale-factor convention; the equivalent form $a_*(t/t_*)^p$ makes the dimensionless power explicit.

References

• Ellis, George F. R., and Jean-Philippe Uzan. “Causal Structures in Cosmology.” Comptes Rendus Physique 16, 928–947 (2015; arXiv version 2016). arXiv.
• Minguzzi, Ettore. “Lorentzian Causality Theory.” Living Reviews in Relativity 22, article 3 (2019). DOI.