Cosmology: an expanding universe

Follow three widely separated galaxies carried by an expanding background. If every separation grows by the same fraction during the same time, one function can describe that common change: the scale factor. We first study this smooth background, then use it to calculate the light received from a distant source. Individual galaxies and other departures from uniformity require additional structure.

Homogeneity says no spatial location is special in the background model. Isotropy says no spatial direction is special for its fundamental observers. Neither assumption requires time independence. The universe is allowed to evolve while treating every background location equivalently.

19.1 A metric for uniform expansion#

Attach spatial labels to the background observers so their labels stay fixed during expansion. Such coordinates are called comoving. Use the chart $(t,\chi,\theta,\phi)$, with $t$ in seconds and the spatial labels dimensionless. The common scaling and spatial symmetries lead to

where $a(t)$ has units of length and $k\in\{-1,0,+1\}$ is dimensionless. This is the Friedmann–Lemaître–Robertson–Walker (FLRW) metric. For $k=+1$, the displayed radial chart has $0\le\chi<1$ and does not cover the whole closed spatial geometry; Section 19.7 gives a radial coordinate that continues past this patch. The angular coordinates have their usual pole limitations.

At fixed time, isotropy makes the sectional curvature the same for every spatial two-plane, and homogeneity makes it the same at each point. Call that value $K_s=k/a^2$. In an orthonormal spatial frame the curvature therefore has the form ${}^{(3)}R_{ijkl}=K_s(\delta_{ik}\delta_{jl}-\delta_{il}\delta_{jk})$. Contracting gives ${}^{(3)}R_{ij}=2K_s\delta_{ij}$, and a second contraction gives

Positive $k$ describes positive constant spatial curvature, negative $k$ negative curvature, and zero $k$ flat spatial slices. Local curvature does not by itself settle every global topology question.

Some books instead make $a$ dimensionless and place length units in the spatial coordinates or curvature parameter. Either convention works. Combining the formulas without converting conventions does not. Here, when we want a dimensionless normalized scale factor, we will explicitly write $A(t)=a(t)/a(t_0)$.

Observers at fixed comoving coordinates have $d\tau=dt$. The cosmic time is their proper time; it is not a new absolute Newtonian time available in every spacetime. It is singled out by this geometry and matter congruence.

The Hubble parameter is

with units of inverse time. It measures the fractional expansion rate. An expansion factor is a ratio of scale factors, while $H$ is a rate of change. For example, doubling all distances is an expansion factor of two; taking a billion years to do so and taking two billion years imply different expansion rates.

SPACETIME LAB / 08

More distance. The same cosmic grid.

Choose a scale factor. Every comoving separation grows together; no point in the grid becomes a preferred center of the expansion.

ExploreDiagram

Interactive geometry is loading.

Drag to orbit · pinch to zoom

Scale factor

0.61.5

$$\begin{gathered}\ell(t)=a(t)\Delta\chi\\d\ell^2=a^2(t)(dx^2+dy^2+dz^2)\end{gathered}$$

Read the scene. The highlighted segment measures the changing separation of two fixed comoving markers. This is a finite window into a spatially flat FLRW slice, not a universe with an outer edge. The control selects a scale factor; it does not specify a cosmological expansion history.

19.2 Calculating the spacetime curvature#

Write the spatial bracket as $\gamma_{ij}dx^i dx^j$, so $g_{ij}=a^2\gamma_{ij}$ and $\gamma$ has unit constant curvature $k$. The useful connection coefficients are

The first says spatial motion participates in time evolution because spatial distances depend on time. The second says a spatial basis carried through cosmic time changes its scale. The third contains the ordinary intrinsic connection of the constant-curvature spatial geometry.

For the time-time Ricci component, $\Gamma^i{}_{ti}=3H$. The relevant contraction gives

The cancellation uses $\dot H=\ddot a/a-H^2$. There are three equal spatial contributions because there are three equivalent spatial directions.

For the spatial Ricci components, the intrinsic curvature contributes $2k\gamma_{ij}$. The time-dependent pieces are $\partial_t\Gamma^t{}_{ij}=(\dot a^2+a\ddot a)\gamma_{ij}/c^2$, the trace product $3H\Gamma^t{}_{ij}=3\dot a^2\gamma_{ij}/c^2$, and the two remaining mixed products totaling $-2\dot a^2\gamma_{ij}/c^2$. Adding them gives

Contract with $g^{tt}=-1/c^2$ and $g^{ij}=\gamma^{ij}/a^2$:

Notice that $k=0$ does not generally make this vanish. Spatial flatness is not spacetime flatness. Evolving distances generate spacetime curvature even when every spatial slice is intrinsically Euclidean.

Combining Ricci and its trace gives

and

The $tt$ component has inverse-time-squared units because this chart uses $t$, not $ct$. The mixed spatial components have inverse-length-squared units. The coordinate units explain the difference, just as the metric’s $tt$ component carries $c^2$ when $t$ is measured in seconds.

19.3 The Friedmann equations: what controls expansion and acceleration#

Homogeneity and isotropy select a perfect-fluid background stress-energy tensor. Let $\epsilon$ be physical rest-frame energy density and $p$ pressure. In the comoving chart,

Substitute the time-time components into $G_{\mu\nu}+\Lambda g_{\mu\nu}=8\pi G_NT_{\mu\nu}/c^4$:

Divide by three and rearrange:

This is the first Friedmann equation. In mass-equivalent density $\rho=\epsilon/c^2$, the matter term is $8\pi G_N\rho/3$. That conversion explains many apparently different textbook versions.

The spatial equation gives

Eliminate $H^2+kc^2/a^2$ using the first equation:

This is the acceleration equation. Energy density and isotropic pressure both gravitate. The factor three counts the three equal spatial pressures in the rest frame.

A positive expansion rate $H>0$ does not imply accelerating expansion $\ddot a>0$. A ball thrown upward moves upward while slowing. Likewise, a matter-filled model can grow in size while its growth rate decreases.

Conversely, a positive cosmological constant contributes positively to $\ddot a/a$. If it dominates, expansion accelerates. The equations make the condition quantitative instead of relying on the ambiguous phrase “repulsive gravity.”

19.4 Energy conservation during expansion#

Use the mixed components $T^t{}_t=-\epsilon$ and $T^i{}_j=p\delta^i{}_j$. The time component of their covariant divergence is

Setting it to zero gives

For a fixed comoving volume, its physical volume is proportional to $a^3$. Multiply the conservation equation by $a^3$:

This has the familiar form $dE=-p\,dV$. As the volume expands, positive pressure reduces the energy within that comoving volume. The fluid does expansion work in this local continuum sense.

These equations are not three independent pieces of information. Differentiate the first Friedmann equation and use the continuity equation; away from a turning point, dividing by $H$ recovers the acceleration equation. At $H=0$, use the original Einstein and conservation equations rather than dividing by zero. The undivided equations remain valid at a turning point.

For a separately conserved component with constant equation-of-state parameter

the continuity equation becomes

Integrate:

The integration is a statement about dilution and work, not an additional gravitational law. If components exchange energy, each gets an exchange term and need not obey this separate scaling; the total still obeys conservation.

The dust model does not mean microscopic dust grains specifically. It means negligible pressure relative to energy density at the scale and accuracy being modeled.

If $\Lambda$ is moved to the matter side, its effective density and pressure are

Either keep $\Lambda$ explicit in the Friedmann equations or include this component in total $\epsilon,p$ and remove the explicit term. Doing both counts the same effect twice.

A comoving volume filled with this effective component gains total energy as it grows, since its energy density remains constant. This does not violate the continuity equation; its negative pressure makes the right-hand side $-p\,dV$ positive. In a general expanding spacetime there is no global timelike translation symmetry supplying a universally conserved total energy of the elementary mechanics kind.

19.5 Solving for the scale factor#

Consider a spatially flat universe dominated by a single separately conserved constant-$w$ component, with no additional explicit $\Lambda$. For $w>-1$, choose the expanding branch. Combining the first Friedmann equation with the density scaling gives

where $C>0$ is a constant with inverse-time units. Move the power of $A$ to the left and integrate:

The time $t_B$ is the integration constant locating $a=0$ in this classical idealized solution. The approximation is not a license to extrapolate an arbitrarily chosen matter model into the quantum-gravity regime.

For dust, $a\propto t^{2/3}$; for radiation, $a\propto t^{1/2}$ after shifting $t_B$ to zero. Both expand while decelerating. Radiation decelerates more strongly because its pressure also contributes to the acceleration equation.

For a positive cosmological constant alone in a spatially flat expanding slicing,

For $w=-1$, the density is constant, so the first Friedmann equation makes $H$ constant. Integrating $\dot a/a=H$ gives the exponential directly. This case was excluded when we divided by $1+w$ in the power-law integration.

More generally, a positive-density single component with $w<-1/3$ produces acceleration in the flat model. That criterion depends on the total effective $\epsilon+3p$ when multiple components are present.

A realistic background calculation combines components with different scalings. Radiation fades fastest, matter more slowly, and a cosmological-constant density stays fixed. Different terms can therefore dominate at different epochs without any of them abruptly changing its fundamental identity.

19.6 Cosmological redshift, derived from neighboring wave crests#

Define a radial comoving distance coordinate along a ray by

Radial null propagation gives $c\,dt=\pm a(t)d\psi$. For a fixed comoving source and receiver, the comoving distance traversed by a light signal is

A neighboring wave crest leaves at $t_{\rm em}+\delta t_{\rm em}$ and arrives at $t_{\rm rec}+\delta t_{\rm rec}$. It crosses the same comoving separation. Subtract the two integrals, treating the periods as short compared with the expansion time:

The comoving clocks measure these coordinate intervals as proper periods. Frequency is inverse period, so

A photon observed at redshift $z=2$ was emitted when the scale factor was one third its value at observation. Its observed wavelength is three times its emitted wavelength, assuming no additional peculiar-motion or local gravitational shifts.

Photon energy is proportional to frequency, so it scales as $a^{-1}$. Combined with number-density dilution $a^{-3}$, this independently explains radiation’s $a^{-4}$ energy-density law. The light-propagation and fluid-conservation calculations give the same density scaling.

This redshift differs from comparing stationary observers in a static potential. Generic FLRW spacetime has no corresponding global timelike Killing symmetry. A useful alternative interpretation builds the redshift from many small local Doppler shifts between neighboring comoving observers. What one should not do is pretend all widely separated cosmological observers share one global special-relativistic inertial frame.

19.7 Conformal time and radial light rays#

Define dimensionless conformal time by

Then

The radial functions follow by integrating $d\chi/d\psi=\sqrt{1-k\chi^2}$ near the origin: $\chi=\sin\psi$ for $k=+1$, $\chi=\psi$ for $k=0$, and $\chi=\sinh\psi$ for $k=-1$. Thus $S_{+1}(\psi)=\sin\psi$, $S_0(\psi)=\psi$, and $S_{-1}(\psi)=\sinh\psi$.

In the closed case, $0<\psi<\pi$ continues smoothly through $\psi=\pi/2$, where $\chi$ ceased to be a usable radial coordinate. The endpoints are the two poles of this spherical spatial chart. Radial light rays now satisfy $d\psi=\pm d\eta$.

A plot of $\psi$ against $\eta$ now draws radial light at slopes $+1$ and $-1$. Multiplying a metric by a positive conformal factor preserves its null cones. It does not preserve proper times, physical lengths, or affine parameters of null geodesics. The prefactor $a^2$ still determines the physical lengths and times represented by that plot.

This distinction becomes especially useful for horizons: what matters is how much conformal time has elapsed or remains, not merely whether today’s expansion rate sounds large.

19.8 The Hubble radius and the two cosmological horizons#

At fixed cosmic time, radial proper distance from the origin along a spatial slice is $D=a(t)\psi$. A comoving object has fixed $\psi$, so

This rate can exceed $c$ at sufficiently large $D$. It is a rate of change of a nonlocal separation defined using cosmic simultaneity, not the velocity measured when one object passes another in the same local inertial frame. Special relativity’s local causal limit remains intact.

For a radial light ray,

An inward-directed ray can initially have increasing proper distance when $HD>c$. Whether it later approaches us depends on the subsequent expansion history. This makes the Hubble radius a useful instantaneous scale but not generally an event horizon.

The relevant horizon exists with finite distance only when the corresponding integral converges, with global topology and the spacetime’s actual domain also taken into account. The event-horizon upper limit is the future endpoint, often infinity; its existence therefore depends on future evolution.

For an ideal flat dust universe $a\propto t^{2/3}$ beginning at $t=0$,

The future event-horizon integral diverges if that evolution continues forever: there is no cosmological event horizon in this model. Two distinct present-day distances and one absent horizon have emerged from a single simple scale factor.

For exponentially expanding flat de Sitter slicing, the future event horizon is $D_e=c/H$. Its equality with the Hubble radius is a property of that special evolution, not a universal identity.

19.9 Does expansion stretch your atoms? And what exactly is dark energy?#

FLRW describes a smoothed cosmological background. A bound atom, planetary system, or galaxy is a local solution with its own stresses and gravitational field. You cannot obtain its size evolution by multiplying every internal distance by the background scale factor while ignoring the forces that bind it.

Cosmological effects can appear as tiny tidal terms in suitable local approximations. Whether they matter is determined by comparing them with binding dynamics. “Everything stretches” is not the field equation; neither is “cosmology can never affect a bound system.” Specify the system and compare scales.

Dark matter and dark energy also play different dynamical roles. In the usual cosmological modeling, cold dark matter behaves approximately as pressureless matter, contributes to gravitational clustering, and has background density scaling approximately as $a^{-3}$. Dark energy labels the component or effective physics invoked for accelerated expansion; a cosmological constant is the simplest constant-$w=-1$ realization. The equations alone do not identify dark matter’s particle properties or prove that dark energy is exactly a cosmological constant.

Type Ia supernovae are stellar explosions whose brightness can be calibrated using their observed light curves and spectra. Comparing that calibrated luminosity with the received flux defines a luminosity distance, worked out in Section 19.11. Historically, the relation between these distances and redshifts supplied evidence for accelerated expansion. A primary account is Riess and collaborators, Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant. Such an inference connects calibrated observations to a model for light propagation and cosmic evolution; it is not a direct photograph of negative pressure.

The modern task is to confront expansion, lensing, clustering, and other observables together while checking systematics and assumptions. A successful fit within GR supports that description. It does not establish that every alternative gravitational theory is mathematically incapable of producing the same particular observations.

19.10 Comparing cosmological and black-hole curvature#

FLRW has zero Weyl curvature. To check this rather than infer it from spatial flatness, define

The orthonormal curvature components from the connection calculation are $R_{i0j0}=-\mathcal A\delta_{ij}$ and $R_{ijkl}=\mathcal B(\delta_{ik}\delta_{jl}-\delta_{il}\delta_{jk})$, with the time-space mixed triples zero. They give $R_{00}=-3\mathcal A$, $R_{ij}=(\mathcal A+2\mathcal B)\delta_{ij}$, and $R=6(\mathcal A+\mathcal B)$. Substituting these in Chapter 9’s Weyl decomposition gives, for example,

The spatial components cancel as $\mathcal B-(\mathcal A+2\mathcal B)+(\mathcal A+\mathcal B)=0$. The mixed components are already zero. All curvature is therefore in the Ricci part; the metric is conformally flat, as defined in Chapter 9. The Schwarzschild vacuum exterior has the opposite pattern: its Ricci tensor vanishes while its Weyl tensor carries the tidal field.

There is a useful further check. A radiation-filled FLRW solution with $\Lambda=0$ has $T=-\epsilon+3p=0$, so the traced Einstein equation gives $R=0$. Yet $R_{\mu\nu}$ is nonzero. The zero scalar is a cancellation in the contraction, not the absence of the individual Ricci components.

The Einstein equation controls a particular contraction of curvature. Matter-filled cosmology, vacuum black holes, and vacuum gravitational waves demonstrate why that distinction matters. Geometry has both locally sourced structure and dynamical information carried through the spacetime solution.

Vanishing Weyl curvature does not mean vanishing geodesic deviation. De Sitter spacetime, for example, has isotropic relative acceleration of neighboring comoving geodesics. The Ricci–Weyl split distinguishes parts of the tidal geometry; it does not assign all measurable gravitational effects to Weyl alone.

19.11 Observe an expansion history#

Consider a comoving source seen at redshift $z$, at radial coordinate $\psi$ from us. Let $a_0$ be the scale factor when we observe it. Its present radial distance along the cosmic-time slice is $D_C=a_0\psi$. A sphere at that coordinate has present area $4\pi[a_0S_k(\psi)]^2$, which defines the transverse comoving distance $D_M=a_0S_k(\psi)$. For flat spatial geometry these distances agree.

Suppose a small transverse feature had proper size $\ell$ when its light was emitted. The angular part of the metric gives $\ell\simeq a_{\rm em}S_k(\psi)\theta$ for its small observed angle $\theta$. The angular-diameter distance is the distance a Euclidean observer would infer from that size and angle:

Brightness supplies another measurement. Let the source radiate isotropically with luminosity $L$, its emitted energy per unit proper time summed over all wavelengths. The received flux $F$ is energy per unit detector area per unit observer time, also summed over wavelengths. The photons spread over area $4\pi D_M^2$. Each loses an energy factor $1/(1+z)$, and their arrival intervals grow by $1+z$. Thus

Define the luminosity distance by $F=L/(4\pi D_L^2)$. Comparing the two expressions gives

At $z=1$, the same source has $D_L=4D_A$. The definitions differ because the brightness measurement includes two redshift effects that the angle measurement does not. This calculation assumes the smooth FLRW model, light wavelengths short compared with the curvature scale, and no absorption or conversion of photons along the beam. The worked example now obtains these distances from an expansion history.

Compare those three distances below. Changing the Hubble constant changes the overall distance scale. Changing the matter and vacuum fractions changes how expansion proceeded and therefore changes the shapes of the curves. The vertical axis uses gigaparsecs; one gigaparsec is one thousand megaparsecs, the unit used in the readouts.

Cosmology lab

Ask the universe three distance questions

Which distance did the observation actually measure?

• Comoving radial
• Angular-diameter
• Luminosity

Comoving radial distance · Mpc5179.9

Angular-diameter distance · Mpc1726.6

Luminosity distance · Mpc15540

Curvature fraction Ωₖ0

Change H₀ first, then the matter fraction. The dots mark your source redshift. Axes keep a common scale unless a curve needs more room.

How this is calculated

H₀ is in km s⁻¹ Mpc⁻¹. Readouts are Mpc and plotted distances are Gpc. Density parameters are relative to the present critical density.

$$E(z)^2=\Omega_m(1+z)^3+\Omega_k(1+z)^2+\Omega_\Lambda$$

$$D_L=(1+z)^2D_A,\qquad D_C=\frac c{H_0}\int_0^z\frac{dz'}{E(z')}$$

• Homogeneous FLRW expansion with dust, spatial curvature and a cosmological constant; radiation is set to zero in this lab.
• Geometric optics, isotropic bolometric source luminosity and photon-number conservation.
• Composite Simpson integration; Ωₖ = 1 − Ωₘ − ΩΛ.

This is an ideal background prediction. It does not fit an observational catalogue or include peculiar velocities, extinction, selection effects or lensing along a particular line of sight.

Hogg · Distance measures in cosmology ↗

Measurements and notebook

H₀ is in km s⁻¹ Mpc⁻¹. Readouts are Mpc and plotted distances are Gpc. Density parameters are relative to the present critical density.

z	H	comoving	angular	luminosity
0	70	0	0	0
0.5	91.604	1888.6	1259.1	2832.9
1	123.25	3303.8	1651.9	6607.7
1.5	162.48	4363.9	1745.5	10910
2	207.65	5179.9	1726.6	15540
2.5	257.79	5827.5	1665	20396
3	312.27	6355.7	1588.9	25423
3.5	370.65	6796.1	1510.3	30583
4	432.64	7170.4	1434.1	35852
4.5	498	7493.2	1362.4	41213
5	566.52	7775.4	1295.9	46652

WORKED EXAMPLE

Calculate three distances to one galaxy

Why do brightness and angular size assign different distances to the same source?

Builds onRedshift Friedmann evolution

See the idea

A cosmological distance is a prescription for inferring separation from measurements. Light travels while the scale factor changes. Its energy and arrival rate change, and a source’s transverse size belongs to its emission time. These facts make several useful distance definitions unequal.

Work it out

1. 01
   Normalize the expansion history

   The subscript 0 refers to the observation time, so $H_0=H(t_0)$. Keep this chapter’s length-valued a and define $A=a/a_0=1/(1+z)$. Let $\rho_c=3H_0^2/(8\pi G_N)$ be the present critical mass-equivalent density. Define $\Omega_m=\rho_{m0}/\rho_c$, $\Omega_r=\rho_{r0}/\rho_c$, $\Omega_\Lambda=\Lambda c^2/(3H_0^2)$, and $\Omega_k=-kc^2/(a_0^2H_0^2)$. The Friedmann equation implies their sum is one. Separately conserved dust, radiation, and vacuum scale as $A^{-3}$, $A^{-4}$, and a constant.

   $$E(z)^2\equiv\frac{H(z)^2}{H_0^2}=\Omega_m(1+z)^3+\Omega_r(1+z)^4+\Omega_k(1+z)^2+\Omega_\Lambda.$$

   Why this step works These scalings follow from the fluid equation with pressure-to-energy ratios 0, 1/3, and −1.

2. 02
   Follow a radial null ray

   Since $dt=-dz/[(1+z)H(z)]$, the present line-of-sight comoving distance is $D_C=c\int_0^z dz'/H(z')$. The transverse comoving distance is $D_M=a_0S_k(D_C/a_0)$, where $S_{+1}(q)=\sin q$, $S_0(q)=q$, and $S_{-1}(q)=\sinh q$. This distinction is the geometry of spatial curvature.

   $$D_C=\frac c{H_0}\int_0^z\frac{dz'}{E(z')},\qquad k=0:\ D_M=D_C.$$

   Why this step works A radial distance and the radius inferred from a spherical area agree only in the flat case.

3. 03
   Define angular and luminosity distances

   For small observed angle θ and transverse proper source size $\ell$ at emission, define $D_A=\ell/\theta=D_M/(1+z)$. For isotropic luminosity L and observed flux F, both summed over all wavelengths, define $D_L$ by $F=L/(4\pi D_L^2)$. Redshift reduces each photon’s energy and stretches arrival times, giving two factors of $1+z$ in flux.

   $$D_L=(1+z)D_M=(1+z)^2D_A.$$

   Why this step works The relation assumes metric null propagation, geometric optics, and photon-number conservation along the beam.

Go deeper

A low-z expansion gives $D_C\simeq cz/H_0$; at larger z, replacing the integral by $cz/H(z)$ is generally wrong. For a flat matter-only model $E(z)=(1+z)^{3/2}$, direct integration gives $D_C=2(c/H_0)[1-(1+z)^{-1/2}]$. The angular-diameter distance can turn over with redshift, so more distant objects need not always look smaller at fixed proper size. Absorption, source evolution, peculiar velocities, calibration uncertainties, and lensing must be handled before interpreting a real data set with this ideal homogeneous model.

Further readingDavid Hogg · Distance measures in cosmology

Test the idea

FIRST, PREDICT

At redshift z = 1, is luminosity distance equal to angular-diameter distance under the stated assumptions?

Luminosity distance is half angular-diameter distance.No. Luminosity distance is four times angular-diameter distance.Yes. A source has only one possible distance.

Compare the reasoning

Luminosity distance is half angular-diameter distance.

Both photon energy loss and slower arrival contribute to the luminosity relation.

No. Luminosity distance is four times angular-diameter distance.

The distance-duality factor is (1+z)² = 4.

Yes. A source has only one possible distance.

These distances encode different measurement protocols.

A hint

Apply distance duality before substituting z.

NOW CHANGE THE EXAMPLE

In a flat matter-only model at z=3, find $D_C/(c/H_0)$.

A hint

Use the analytically integrated matter-only formula.

Work through the solution

$2[1-(1+3)^{-1/2}]=2(1-1/2)=1$.

A distance formula is a measurement protocol with assumptions, not just a label on a diagram.

19.12 A collapsing surface can cross its horizon in finite proper time#

A useful exact collapse model is a homogeneous pressureless ball, matched without a surface layer to a Schwarzschild exterior. It neglects pressure, rotation, inhomogeneity, and radiation. Use a closed FLRW interior,

The surface follows a fixed $0<\chi_0<\pi/2$. Starting at rest at maximum size, the dust Friedmann equation has the parametric solution

To check it, differentiate with respect to $\eta$, use $d\tau/d\eta=a/c$, and substitute into the closed dust Friedmann equation with conserved $\rho a^3$. The initial equation fixes $\rho_{\max}=3c^2/(8\pi G_Na_{\max}^2)$, where $\rho$ is mass-equivalent energy density. The areal surface radius is $R=a\sin\chi_0$, and the mass matching condition is

This spherical gravitational mass is not the integral of rest density over the curved proper volume. The matching conditions require continuity of the induced metric and extrinsic curvature in the absence of a surface stress tensor. Zero pressure at the dust boundary makes this interior/exterior matching possible.

The surface reaches $r_s=2G_NM/c^2$ when $a/a_{\max}=\sin^2\chi_0$, hence at $\eta_h=\pi-2\chi_0$. Its proper time then is finite and smaller than the singular endpoint $\tau_{\rm sing}=\pi a_{\max}/(2c)$. For $\chi_0=\pi/6$, crossing occurs at $\eta_h=2\pi/3$, while the surface has shrunk to one quarter of its initial radius.

The event horizon inside the dust is an outgoing radial null line, $d\chi/d\eta=1$, traced backward from that crossing event. It obeys $\chi=\eta-\pi+3\chi_0$ until it reaches the centre. For the specified $\chi_0=\pi/6$ example, it begins at the centre at $\eta=\pi/2$, before the surface reaches its Schwarzschild radius. The horizon’s global definition and its smooth crossing by infalling matter are visible in the same solution. This exact dust model illustrates collapse; it is not a model of realistic stellar microphysics or an extension through its singular endpoint.