The Corona in the Well: A Gravitational Correction to the Quasar Hubble Diagram

1. The Idea, Reborn at the Right Rung

Martin Scholl — Independent Researcher  ·  It Is All One — Notes  ·  July 2026 (readable edition)

An idea this series once buried — that part of a source’s redshift might be gravitational, born in a deep well rather than accumulated on the way — resurrects at the one rung where the physics genuinely supports it. Quasar cosmology is built on X-ray light that is born a few gravitational radii from a supermassive black hole: genuinely deep in a well. This note computes what the well does to that light (two effects, both from this series’ own toolbox), shows that the standard quasar-distance method amplifies the X-ray effect two-and-a-half-fold, and finds that a modest, astrophysically plausible drift of coronal depth across the quasar population produces a bias of the same size, sign and location as the celebrated deviation of quasar cosmology from the standard model at high redshift — the anomaly behind “evolving dark energy” headlines. If so, that anomaly is general relativity in the engine, not new physics in the expansion. Tests are stated; every symbol is introduced before use.

1The Idea, Reborn at the Right Rung

The local-well correction is negligible for supernovae, and the reason is kinematic rather than astrophysical — which is precisely what makes it safe to lean on. A white dwarf's own surface would redden light by GM/Rc² ≈ 3×10⁻⁴, a hundred kilometres per second, the size of the shift already measured on Sirius B. The object is not too small. But a Type Ia does not shine from its surface.

auditAt maximum light the photosphere sits wherever the ejecta have carried it, R = v·t ≈ 10¹⁰ km, and the well there is about 10⁻¹⁰ — three parts in ten billion of a magnitude. Nothing. That bound asks nothing of the explosion: R is a velocity times a time, both read straight off the spectra and the light curve, and running v from 5 000 to 15 000 km/s and t from ten to twenty-five days moves the answer by a factor of seven. For the well to reach even 10⁻⁴ the light would have to come from inside 2×10⁴ km — deeper than the progenitor itself. So the correction stays buried at this rung whatever the explosion simulations eventually settle on, and nothing in this note rests on their being right.
supernova_well_bound.py — bound
runs in your browser
The white dwarf itself is not the problem — its surface well is measurable.
  progenitor surface, R = 1500 km     z = 1.379e-03   =   413.3 km/s   2.99e-03 mag
  progenitor surface, R = 6000 km     z = 3.446e-04   =   103.3 km/s   7.48e-04 mag
  (cf. Sirius B, whose ~80 km/s surface redshift has been measured.)

But a Type Ia shines from the ejecta. R = v*t, both read off the data.
       v [km/s]   t [d]      R [km]        z_grav        dimming [mag]
           5      10   4.320e+09   4.787e-10   1.039e-09
           5      18   7.776e+09   2.659e-10   5.774e-10
           5      25   1.080e+10   1.915e-10   4.158e-10
          10      10   8.640e+09   2.393e-10   5.197e-10
          10      18   1.555e+10   1.330e-10   2.887e-10
          10      25   2.160e+10   9.573e-11   2.079e-10
          15      10   1.296e+10   1.596e-10   3.465e-10
          15      18   2.333e+10   8.864e-11   1.925e-10
          15      25   3.240e+10   6.382e-11   1.386e-10

  spread across that whole grid: 6.382e-11 .. 4.787e-10  — a factor of 7.5

How deep would the light have to come from to matter?
  for z = 1e-04:  R = 2.068e+04 km   (7.52e+05x smaller than a typical photosphere)
  for z = 1e-06:  R = 2.068e+06 km   (7.52e+03x smaller than a typical photosphere)

  20678 km is inside the progenitor's own radius. No explosion
  model puts the emitting surface there, so no revision to explosion
  physics can rescue the correction at this rung.

VERDICT: negligible, by a kinematic argument that does not depend on
         simulating the explosion. The idea was right; the object was wrong.

================================================================
The same two objects, priced in the units of The Ledger of the Way
================================================================
  neutron-star surface, R = 12 km      ln(1+z) = 2.1128e-01   worth       905 Mpc of background
  Type Ia photosphere, R = 1.6e10 km   ln(1+z) = 1.3296e-10   worth      0.57 pc  of background

  ratio = 1.59e+09
  The neutron star's twelve kilometres are worth 905 megaparsecs of
  universe; the supernova's photosphere is worth about half a parsec.
  A factor of 1.6 billion, and it is a difference of RADIUS, not of physics.

The idea was right and the object was wrong. A quasar is a supermassive black hole feeding: its ultraviolet light comes from the surrounding disk of hot matter, tens to hundreds of gravitational radii out — but its X-rays come from the corona, a compact region of energized plasma sitting at roughly five to ten Schwarzschild radii (the Schwarzschild radius r_s = 2GM/c² is the size the central mass would have as a bare black hole). Five Schwarzschild radii is genuinely deep: and modern cosmology uses exactly this light, because the relation between quasars’ X-ray and ultraviolet brightness (Risaliti and Lusso) turns them into standardizable candles reaching seven times deeper in redshift than any supernova.

2What the Well Does to the Light

Two effects, both already in this series’ toolbox. First, gravitational redshift dimming: light climbing from radius r loses energy, and arrives at a rate slowed by the same factor — together dimming the flux by (1 − r_s/r). Second, the exit cone (Paper 1, Section 6.1): close to a black hole, not all directions lead out; a fraction of the photons emitted isotropically simply falls in. Combining the two, per emission radius: A corona at five Schwarzschild radii loses 0.31 magnitudes to its own well; the ultraviolet disk at twenty-five loses 0.05. These are not small numbers, and they are not speculative physics — they are the standard picture of the quasar engine, priced with Schwarzschild optics.

3The Amplifier

The Risaliti–Lusso method infers a quasar’s distance by comparing its X-ray flux against its ultraviolet flux through their measured statistical relation. Propagating a dimming through that construction (differentiating the distance formula the method uses), one finds:

bias in inferred distance modulus = 2.5 × (X-ray dimming) − 1.5 × (UV dimming)

The construction amplifies X-ray dimming two-and-a-half-fold. For the typical geometry above, the standing bias is +0.70 magnitudes — but a bias constant across all quasars is silently absorbed when the relation is calibrated, exactly the lesson of the buried supernova correction. What cosmology feels is only the drift of coronal depth across the sample.

4The Scenario That Matches the Anomaly

The quasar Hubble diagram famously deviates from the standard model at redshifts above about 3, by 0.3–0.6 magnitudes in the direction of dimmer and farther — the claim behind “evolving dark energy.” Now price a drift: if the effective coronal radius across the sample slides from ten Schwarzschild radii at low redshift to five at high redshift, the induced bias is +0.44 magnitudes of apparent dimming at high redshift; ten-to-four gives +0.70. Direction and magnitude both match the claimed anomaly. And a selection mechanism lies ready to hand: surveys at high redshift see only the most luminous engines, and the most luminous, fastest-feeding engines are precisely those whose coronae are observed (by X-ray timing and microlensing) to be most compact. No cosmic conspiracy required — the telescope’s own selection walks the sample down the well. The hypothesis, plainly labeled: the high-redshift quasar anomaly may be gravitational redshift plus exit-cone capture of coronal X-rays, amplified by the distance construction and driven by luminosity selection — relativity in the engine, not new physics in the expansion. If so, it is a systematic to be modeled before the quasar record can referee any cosmology: the standard model and metric D alike (whose own signature in that band, −0.15 to +0.07 magnitudes, is smaller than this effect).

5Caveats, Honestly

The corona is not a static emitter — motions and beaming modify the static-limit numbers; the table is the leading-order skeleton. The ultraviolet disk is itself mildly relativistic. The amplification factor depends on the measured slope of the X-ray/UV relation. And whether this specific bias has been proposed and tested in the quasar-cosmology literature must be checked in the literature pass this corpus owes — the idea is simple enough that someone may own it.

6Tests

Correlate the quasar Hubble-diagram residuals with independent measures of coronal compactness (X-ray variability timescales, the X-ray-to-UV spectral index, feeding rate): the hypothesis predicts the residual is an astrophysics axis, not a cosmology axis. (2) Re-fit the quasar record with a coronal-depth term included: the high-redshift anomaly should shrink toward the standard model — or toward metric D; the fit can carry both. (3) Gravitationally microlensed quasars yield direct coronal sizes: the subsample with measured coronae is the calibration set. Each test uses data that already exists.

7Bottom Line

The redshift correction buried at the supernova rung resurrects, correctly, at the quasar rung — with this series’ own exit-cone formula as its engine and a ×2.5 amplifier waiting inside the standard method. At the supernova rung the well was three parts in ten billion of a magnitude; at the quasar rung it is a third of a magnitude before amplification — and it sits exactly where modern cosmology reports its most publicized anomaly.

References

G. Risaliti and E. Lusso, Nature Astronomy 3, 272 (2019); E. Lusso et al., A&A 642, A150 (2020); quasar coronal sizes from X-ray reverberation and microlensing literature; and the papers and notes of this series (Paper 1 §6.1 — the exit cone; the cosmology paper; the Ledger of the Way). (Citations from memory; the literature-verification pass applies.)

Symbols & Terms