The Ledger of the Way: Cumulative Redshift as a Potential Difference:

Scholl, Martin

The Ledger of the Way: Cumulative Redshift as a Potential Difference

Light climbing out of a gravitational well loses energy and arrives redder — measured on Earth to exquisite precision, and enormous on a neutron star’s surface. Light crossing the cosmos arrives redder too. When one journey involves both — wells within wells, then the long way home — how do the redshifts combine? Multiply or add? Does the path matter? Is there one equation for the whole way?

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

Take a photon born on the surface of a neutron star, deep in its own gravitational well, inside a galaxy’s well, inside a cluster’s well, riding the background curvature a gigaparsec to our telescope — itself sitting inside the Milky Way’s well. What total redshift arrives? This note derives the answer in one equation and shows that in a static cosmology the whole journey collapses into a ledger: a sum of entries, one per well, plus one for the background — and only the endpoints matter. The worked example: twelve kilometres of neutron-star surface are worth 905 megaparsecs of universe. Out of the same equation falls the cleanest falsifier this series owns: the redshift of every source must never drift with time — while the standard model demands it must. Every symbol is introduced before use.

1The Question

2The Master Equation

Recall two definitions, walked slowly. The redshift z measures stretching: 1 + z is the factor by which each received wave is longer than when it was sent. The lapse N(x) is the local rate of clocks: at a location x deep in a well, or deep in the cosmic curvature, clocks run slow compared to a reference clock, and N(x) — a pure number, close to 1 in shallow places, smaller in deep ones — is that rate.

theoremBecause the geometry of this series is static (Postulate 3: the stage never changes), a photon carries one conserved quantity along its entire way: the product of its locally measured energy E(x) and the local clock rate,

Ẽ = N(x) · E(x) = the same number all along the ray.

theoremNothing en route can change Ẽ; only the exchange rate N between local clocks and universal time varies from place to place. Divide two readings of this conservation law — one at the emitter, one at the observer — and the whole journey collapses:

1 + z = N(observer) / N(emitter)

The total redshift is a ratio of two local clock rates. Nothing else. To make it a ledger, take the logarithm (the natural bookkeeping of multiplied factors) and define the potential of the way, Φ(x) = −ln N(x) — a pure number that grows with depth. Then:

ln(1 + z) = Φ(emitter) − Φ(observer)

And the lapse of a many-welled universe factorizes — the background times each well:

N(x) = e^(−k·r) × ∏ over wells i of √(1 − r_s,i / ρ_i(x))

where k = H/c is the background curvature per metre (H the Hubble rate, c the speed of light); r is proper distance from the observer; r_s,i = 2GM_i/c² is the Schwarzschild radius of well number i (G Newton’s constant, M_i the well’s mass — r_s is the size the well’s mass would have as a black hole); and ρ_i(x) is the distance from the point x to that well’s center. The logarithm turns the product into a sum: wells within wells become entries in a ledger, each shallow well contributing simply its depth, GM/(ρc²). Redshift in this cosmology is not stretching en route; it is climbing out, and the books are kept in ln(1+z).

3The Ledger, Executed: a Neutron Star at One Gigaparsec

Read the first and last lines together: the twelve kilometres of a neutron star’s surface contribute as much to the ledger as 905 megaparsecs of universe. This is also why the correction is nothing at the supernova rung and everything at compact ones. A Type Ia’s light leaves not from the white dwarf but from a photosphere some 10¹⁰ km out, so its entry is 1.3×10⁻¹⁰ — worth about half a parsec of background against the neutron star’s 905 megaparsecs, a factor of 1.6 billion. That is a difference of radius, not of physics, and it needs no model of the explosion to establish. Note also the honest minus sign: the observer’s own well partially refunds the climb. The ledger balances at both ends, as a ledger must.

supernova_well_bound.py — ledger

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.

#!/usr/bin/env python3
# -*- coding: utf-8 -*-
"""supernova_well_bound.py — why the local-well correction cannot matter for Type Ia.

ESTABLISHES: the gravitational redshift of the emitting surface of a Type Ia
supernova is of order 1e-10, and that this bound is KINEMATIC — it follows from
the photospheric radius R = v*t, with v and t read directly off the spectra and
the light curve, and from GM/Rc². Varying v over 5000-15000 km/s and t over
10-25 days moves the answer by less than a factor of eight, while it would have
to grow by a factor of ~1e6 to become detectable.

DOES NOT ESTABLISH: anything about the explosion mechanism, the progenitor
channel, nickel masses, or whether Type Ia supernovae are good standard candles.
It deliberately avoids all of that. The point of the calculation is that the
bound survives whatever the answers to those questions turn out to be: no
explosion model places the emitting surface inside 2e4 km, and nothing weaker
than that would rescue the correction.
"""
import math

G    = 6.67430e-11        # m^3 kg^-1 s^-2   -- "big G", Newton's constant
c    = 2.99792458e8       # m/s              -- "see", the speed of light
Msun = 1.98892e30         # kg               -- one solar mass
M    = 1.4 * Msun         # kg               -- Chandrasekhar-scale ejecta mass

def z_grav(R):
    """Gravitational redshift of a static emitter at areal radius R, weak field."""
    return G * M / (R * c**2)

def dmag(z):
    """Magnitudes of dimming corresponding to redshift z (5*log10(1+z))."""
    return 5.0 * math.log10(1.0 + z)

# ------------------------------------------------------------------ the object
print("The white dwarf itself is not the problem — its surface well is measurable.")
for label, R_km in [("progenitor surface, R = 1500 km", 1.5e3),
                    ("progenitor surface, R = 6000 km", 6.0e3)]:
    z = z_grav(R_km * 1e3)
    print(f"  {label:34s}  z = {z:.3e}   = {z*c/1e3:7.1f} km/s   {dmag(z):.2e} mag")
print("  (cf. Sirius B, whose ~80 km/s surface redshift has been measured.)")

# --------------------------------------------------------- where the light is
print("\nBut a Type Ia shines from the ejecta. R = v*t, both read off the data.")
print("       v [km/s]   t [d]      R [km]        z_grav        dimming [mag]")
rows = []
for v in (5.0e3, 1.0e4, 1.5e4):                 # km/s, photospheric velocity
    for t in (10, 18, 25):                      # days from explosion
        R = v * 1e3 * t * 86400.0               # metres
        z = z_grav(R)
        rows.append(z)
        print(f"     {v/1e3:7.0f}   {t:5d}   {R/1e3:.3e}   {z:.3e}   {dmag(z):.3e}")

lo, hi = min(rows), max(rows)
print(f"\n  spread across that whole grid: {lo:.3e} .. {hi:.3e}  "
      f"— a factor of {hi/lo:.1f}")

# ------------------------------------------------------------------ the bound
print("\nHow deep would the light have to come from to matter?")
for target in (1e-4, 1e-6):
    R_need = G * M / (target * c**2)
    print(f"  for z = {target:.0e}:  R = {R_need/1e3:.3e} km"
          f"   ({(1e4*1e3*86400*18)/R_need:.2e}x smaller than a typical photosphere)")
R_need = G * M / (1e-4 * c**2)
print(f"\n  {R_need/1e3:.0f} km is inside the progenitor's own radius. No explosion")
print("  model puts the emitting surface there, so no revision to explosion")
print("  physics can rescue the correction at this rung.")

print("\nVERDICT: negligible, by a kinematic argument that does not depend on")
print("         simulating the explosion. The idea was right; the object was wrong.")

# ------------------------------------------------ the same fact, in ledger units
# The Ledger of the Way is additive in ln(1+z), so the two rungs can be compared
# directly against a length of background. This reproduces that note's table.
print("\n" + "="*64)
print("The same two objects, priced in the units of The Ledger of the Way")
print("="*64)

R_NS      = 12.0e3        # m      -- canonical neutron-star radius
PER_GPC   = 0.2335        # ln(1+z) contributed by 1 Gpc of background (that note)

z_ns = 1.0/math.sqrt(1.0 - 2*G*M/(R_NS*c**2)) - 1.0   # exact Schwarzschild, not weak field
z_sn = z_grav(1.0e4 * 1e3 * 18 * 86400.0)             # v = 10^4 km/s, t = 18 d

for lab, z in (("neutron-star surface, R = 12 km", z_ns),
               ("Type Ia photosphere, R = 1.6e10 km", z_sn)):
    L = math.log1p(z)
    Mpc = L / PER_GPC * 1000.0
    unit = f"{Mpc:9.0f} Mpc" if Mpc > 1 else f"{Mpc*1e6:9.2f} pc "
    print(f"  {lab:36s} ln(1+z) = {L:.4e}   worth {unit} of background")

print(f"\n  ratio = {math.log1p(z_ns)/math.log1p(z_sn):.2e}")
print("  The neutron star's twelve kilometres are worth 905 megaparsecs of")
print("  universe; the supernova's photosphere is worth about half a parsec.")
print("  A factor of 1.6 billion, and it is a difference of RADIUS, not of physics.")

4What the Equation Buys: Two Theorems and a Falsifier

Path independence. The ledger depends only on endpoints — mathematically, because a difference of potentials doesn’t care about the route. Physical consequence: gravitational lensing can bend a ray around any detour whatsoever and cannot change its redshift by one part in anything. 4.1 The Fork: Does the Distance Grow? The objection arrives naturally: “surely the object gets more distant over time, so tomorrow’s redshift is greater.” That is the expanding picture, and the model must choose. Branch one, strictly static: redshift is position in the potential, not history; the drift is exactly zero. Branch two, the slow spiral: if proper distances really grow, consistency already caps the rate — today’s redshift is fully paid by the potential, so any true recession stacks its own Doppler shift on top, and the supernova fit (good to ±0.1 magnitudes) tolerates only a few percent of the naive rate. The three contenders then separate cleanly, per decade of watching: (The units: a drift in redshift is expressed as an apparent velocity change, centimetres per second, over ten years of observation.) Note the standard model’s signature: its drift changes sign near z ≈ 2 — positive nearby, −5 cm/s at z = 4 — because in an expanding universe redshift is accumulated history, and history accumulates differently at different depths. This model, in either branch, can never produce that sign flip. 4.2 The Yardstick: Why the Distance Does Not Grow. The deeper form of the objection: “tomorrow’s distance is measured with tomorrow’s yardstick — it has to increase.” A yardstick is atoms; its length is set by the Bohr radius, ħ/(m_e·c·α). For the yardstick to shrink, the electron’s mass or the fine-structure constant must drift with time — local physics evolving from day to day. Postulate 3 forbids exactly this, and not as decoration: the same time symmetry that conserves Ẽ along the ray (Section 2) and makes the temperature law of the cosmology hold is what freezes the yardstick. The dimensionless statement: a redshift measurement is a ratio, λ_photon/λ_lab — fossil light against a laboratory atom. For z to grow overnight, either the climb changes (static potential: it doesn’t) or the lab atom shrinks (eternity: it doesn’t). Both frozen, ratio frozen, drift zero. The shrinking-yardstick alternative is an honest, known theory — Wetterich (2013), a static geometry with growing particle masses — but it is constructed to be observationally identical to the expanding model: adopting it would surrender this falsifier and reopen every settled contest in different units. The distinctness of metric D lives in the frozen yardstick: redshift is where you are in the potential, not what date it is.

openThe falsifier, named. The Extremely Large Telescope’s high-resolution spectrograph programme (ANDES) is being built to measure precisely these few centimetres per second per decade, against quasar absorption forests, over roughly twenty years. No candles, no calibration ladders, no dust corrections — the same object against the same atomic clock, years apart; everything systematic cancels. One confirmed sign-flipping drift and metric D is dead, no appeal. A persistent null while the standard model demands −5 cm/s at z = 4, and the standard model has a crisis. Few predictions in cosmology are this clean, and this series stakes its background metric on it.

5Remarks

Moving emitters add an ordinary Doppler factor multiplying the static ledger — peculiar velocities are entries of order v/c ≈ 10⁻³, kept in the same logarithmic books. And the ledger’s structure is the Bookkeeping paper’s thesis in miniature: between the drains and the horizons, every account balances, and the way home is a sum of honest entries.

References

R. C. Tolman and P. Ehrenfest, Phys. Rev. 36, 1791 (1930); I. M. H. Etherington (1933); A. Sandage (1962) and A. Loeb (1998) on redshift drift; C. Wetterich, Phys. Dark Univ. 2, 184 (2013); the ELT/ANDES instrument programme; and the papers and notes of this series (Paper 2 and the Four Calculations note; the Postulates). (Citations from memory; the literature-verification pass applies.)