Past light cones and the events we can observe

A received signal selects an emission event

An observation happens at a particular observer event: here and at a specified time. Light received then connects that event to earlier emission events along past-directed light paths. A sky image therefore organizes information from those emission events, rather than directly displaying an entire simultaneous spatial slice.

The past light cone is the boundary traced by those null paths in a transparent ideal description. Events inside the causal past can also influence the observer through slower signals or earlier-arriving information; lying exactly on the light cone is a more specific statement.

Real observations add emission, scattering, absorption, foreground and detector limitations. A geometric causal path alone does not guarantee that a particular photon source can be detected. The calculation here uses test light with no absorption to isolate the geometry.

The familiar special-relativistic cone can be adapted to an expanding background when the metric and coordinates are stated. A straight-looking light line is a coordinate representation whose slope must be interpreted using the chosen axes.

Keep cosmic and conformal time separate

Choose an ideal noncompact spatially flat radiation-only general-relativistic background. Let T be a positive time unit, u = t/T > 0 and a = √u. The observation event O is at u = 1 and at the spatial origin.

Use a length-valued radial comoving coordinate r and its dimensionless form x = r/(cT). Along a radial null path, dr/dt = ±c/a. The sign distinguishes the two radial directions in this one-dimensional diagram through the observer.

Define dimensionless conformal time η by dη = du/a(u). For this supplied history choose η = 2√u, using zero as a formal lower limiting value. Then radial light paths obey dx/dη = ±1.

Equal numerical spacing on the x and η axes makes these null lines appear at 45 degrees. Cosmic time is instead u = η²/4, so the same paths are not generally 45-degree lines on an x-versus-u plot. Conformal time is a useful coordinate, not the proper time recorded by the observer’s clock.

The same emission time does not mean the same arrival time

At cosmic time u_e = 1/4, conformal time is η_e = 1. Consider three emitters at fixed comoving positions x = 1/2, 1 and 3/2. Each sends a radial photon directly toward the origin at that emission time.

An inward photon traversing a comoving interval x requires a conformal-time interval x in these normalized coordinates. Its arrival time therefore satisfies η_arrival = 1 + x and u_arrival = (1 + x)²/4.

The x = 1/2 signal arrives at u = 9/16, before O. The x = 1 signal arrives at u = 1, exactly at O. The x = 3/2 signal arrives at u = 25/16, after O. Only the middle emission event lies on O’s past light cone.

The first emission can influence O because its information can already have arrived and been retained. It is not the event whose directly emitted photon is arriving exactly at O. This distinction separates the causal past from its null boundary.

A cone relates different cosmic-time slices

O has η = 2. Its past radial null boundary is |x| = 2 − η = 2(1 − √u) for the past interval 0 < u ≤ 1. At u = 1/4, the boundary lies at |x| = 1, matching the middle signal.

At u = 1/16 it lies at |x| = 1.5, while at u = 9/16 it lies at |x| = 0.5. Different positions on the cone correspond to different emission times. A horizontal same-time slice intersects the cone but is not the whole cone.

The full sky includes directions as well as this radial coordinate. The two-sided radial diagram is a geometric teaching section, not a picture of all galaxies or an assertion that space has only one dimension.

The u → 0 endpoint is a formal extrapolation of the ideal background. The diagram does not verify an origin state or remove the physical limits of a classical model at extreme conditions.

Keep causal and practical visibility limits distinct

A past light cone is a set of spacetime events associated with reception at one observer event. A particle-horizon radius, introduced next, describes a different construction on a chosen spatial slice, involving the available past history.

Photon transparency adds another limitation. Scattering can reprocess radiation before it reaches us, while earlier processes can still leave later observable traces. A visibility window should not be treated as identical to every causal boundary.

Local light propagation remains null according to the metric. A coordinate speed such as dr/dt = c/a is not an ordinary speed measured with one local inertial ruler and clock over a distant interval.

When interpreting a light-cone plot, read the observer event, coordinate definitions, axis units, propagation assumptions and domain. Those choices determine what a line, intersection or boundary actually establishes.

Worked reasoning

Try the reasoning

1. Must a radial light path appear at 45 degrees when cosmic time u is plotted against comoving x?

   Show the answer and reasoning

   No.

   The simple unit slope applies to the supplied conformal η coordinate with matched axis units.

2. Does the x = 1/2 emission at u = 1/4 lie on O’s past light cone?

   Show the answer and reasoning

   No. Its direct photon arrives earlier, at u = 9/16.

   The event can influence O through information already received, but it is inside the causal past rather than on the specific null boundary.

3. Does a sky image received at O directly show all distant objects on O’s simultaneous cosmic-time slice?

   Show the answer and reasoning

   No.

   It connects O to earlier emission events along light paths.

What to carry forward

• An observer event’s past light cone is a spacetime construction, not a simultaneous spatial snapshot.
• Conformal coordinates can give radial null lines unit slope without making conformal time a physical clock.
• Causal reach, photon transparency and practical detection supply different limits.