II Brazilian School of Neutrinos

Friday · 02 · 10 · 2026

ν_e ν_μ ν_τ

Seminar

Neutrinos in
curved spacetime

What happens to flavour oscillation when the journey follows a geodesic — and not a straight line.

Outline

Three parts, one question

I

Curved spacetime

Einstein, Schwarzschild, compact objects: where only General Relativity describes things well, and what the metric does to a neutrino.

II

The supernova, and neutrinos in the well

How the star collapses, where the shock wave comes from, and what happens to flavour when the emission takes place deep in a gravitational well.

III

GHOST: results

The shock front at the MSW resonance: the criterion, the cost in P_H, the bias at DUNE and where to look.

Premise · Wolfenstein 1978 · Mikheyev & Smirnov 1985

MSW: a two-level problem

H = ½ (V − Δ cos 2θΔ sin 2θΔ sin 2θ−(V − Δ cos 2θ))effective Hamiltonian, two flavours

Δ = Δm²2E

V(r) = √2 G_F n_e(r)

The resonance happens when the diagonal term vanishes:

V = Δ cos 2θ

H_res ~ ½ (0Δ sin 2θΔ sin 2θ0)off-diagonal only: 50/50 eigenstates — maximal mixing

An avoided crossing with adiabatic evolution — and the bridge to supernovae, where the profile n_e(r) sweeps through the resonance.

Adiabaticity · 1

In the basis that follows the matter

Change from the flavour basis to the instantaneous matter basis:

(ν_eν_x) = U(θ_m(x)) (ν₁^mν₂^m)

Since U depends on x, the Schrödinger equation picks up a term:

i ddx (ν₁^mν₂^m) = [ (−Δ_m/200+Δ_m/2) − i U^†dUdx ] (ν₁^mν₂^m)

−i U^†dUdx ~ (0i dθ_m/dx−i dθ_m/dx0)

Bridge to the previous slide: U(θ_m) diagonalizes the H there:

Δ_m cos 2θ_m = Δ cos 2θ − V · Δ_m sin 2θ_m = Δ sin 2θ

The previous slide gives the Hamiltonian at one point; diagonalizing it defines θ_m and Δ_m; here we ask what happens when θ_m changes along the path.

The density gradient reappears as an off-diagonal term in the basis of instantaneous eigenstates.

A head-to-head competition:

Δ_m versus |dθ_m/dx|

Adiabatic condition:

|dθ_m/dx| ≪ Δ_m

More fundamental than “the density varies slowly”: what matters is how fast the basis rotates compared with the gap.

Adiabaticity · 2

Why the resonance is the critical place

At resonance, V = Δ cos 2θ and θ_m = π/4. The level separation is smallest:

Δ_m^res = Δ sin 2θ

An avoided crossing. Far from it the levels are well apart; near it they come as close as they can — and that is where the basis turns fastest (pink area).

That is where the ν₁ᵐ ↔ ν₂ᵐ transition is easiest. So adiabaticity is evaluated at the resonance.

Adiabaticity · 3

The adiabaticity parameter γ and Landau–Zener

γ_res = Δm²2E sin² 2θcos 2θ |d ln n_edx|⁻¹_resadiabaticity parameter = gap ÷ rotation of the basis, at resonance

P_jump ≃ e^{−πγ_res/2}probability of jumping branch (Landau–Zener, nearly linear profile)

if…	γ	P_jump	why
larger energy E	↓	↑	Δ = Δm²/2E drops: smaller gap
larger Δm²	↑	↓	larger gap
larger mixing θ (up to 45°)	↑	↓	larger minimum gap Δ sin 2θ
density varies slowly	↑	↓	scale \|d ln n_e/dx\|⁻¹ large: the basis turns slowly
shock front (step)	→ 0	max.	scale → 0: sudden jump

γ ≫ 1 → adiabatic: follows the branch · γ ≲ 1 → jumps branch

The resonance determines where the eigenstates mix maximally; adiabaticity determines whether the neutrino follows those eigenstates as they change.

Doing the algebra at resonance gives γ, the ratio of the gap to the rate at which the basis turns. Landau–Zener gives the probability to jump branches, e^{−πγ/2}. Three computed numbers: in the Sun, for ⁸B at 10 MeV, γ ≈ 3000 — fully adiabatic. In a supernova, at the H resonance at 20 MeV: with the shock front as the simulation stores it, wide, γ ≈ 5 and there is almost no jump; with the real front, centimetres wide, γ ≈ 0.01 and the jump is almost certain. This sets up the relativistic question: with respect to which parameter do we measure dθm/dx, and which energy enters Δ? That is where tetrads, local energy and proper length come in. How to read it: γ is the gap divided by the rotation of the basis. Higher energy shrinks the gap, γ drops and the jump becomes more likely; larger Δm² or mixing widen the gap and protect adiabaticity; slowly varying density gives large γ; a shock front, with almost zero scale, gives γ → 0 and the maximal jump.

Adiabaticity · 4 · supernova M15-7b

The shock front crosses the resonance

I

Part one

Curved
spacetime

Gravity is not a force: it is geometry. Far from stars, the difference is a tiny correction. Next to a compact object, it is everything.

Einstein · 1915Schwarzschild · 1916compact objectsgravitational wavescollapsars

Eclipse of 29 May 1919 · ESO / Landessternwarte Heidelberg · CC BY 4.0

25 November 1915

Einstein and his field equations

Albert Einstein, 1921 — **Albert Einstein**1921

G_μν = 8πGc⁴ T_μνcurvature = energy and momentum content

d²x^μdλ² + Γ^μ_αβ dx^αdλ dx^βdλ = 0geodesic

ds² = g_μν dx^μdx^νthe metric is the field

“Spacetime tells matter how to move; matter tells spacetime how to curve.”
John A. Wheeler

The classical tests

Sobral, 1919: light bends

effect	GR predicts
perihelion of Mercury (1915)	43″ / century
light grazing the Sun (1919)	1.75″
redshift, Pound–Rebka (1959)	2.5×10⁻¹⁵ over 22.5 m
Shapiro delay (1964)	~200 μs
GPS clocks	+38 μs / day

Sobral expedition (Ceará, Brazil): 1.98 ± 0.12″. Príncipe: 1.61 ± 0.30″. The sky over north-eastern Brazil confirmed Einstein.

1859 – 1915 · the perihelion of Mercury

Mercury’s orbit does not close

perihelion advance	per century
observed (relative to the stars)	574″
other planets, by Newton	531″
unexplained remainder	43″

Δφ = 6πGM_☉c² a(1 − e²) = 0,104″ per orbitGR · Einstein, November 1915 · 415 orbits per century → 43.0″

Le Verrier found the remainder in 1859 and proposed a hidden planet, Vulcan. It was never found. Einstein got the 43″ with no adjustable parameter.

“For a few days, I was beside myself with joyous excitement.”
Einstein to Ehrenfest, January 1916

1959 · Pound & Rebka · Jefferson tower, Harvard

Light loses energy as it climbs

14.4 keV γ rays from ⁵⁷Fe (Mössbauer effect). Pound & Rebka, PRL 4, 337 (1960): ~10%; Pound & Snider, Phys. Rev. 140, B788 (1965): ~1%. Every 12 s source and absorber swap places, as in the experiment.

Pound and Rebka, 1959, in the 22.5-metre tower of the Jefferson laboratory at Harvard. Gravity shifts the frequency of light by gh/c², 2.5 parts in 10¹⁵. To measure it they used the Mössbauer effect of iron-57: a 14.4 keV line narrow enough to see that shift. Light that falls arrives with slightly more energy and is no longer absorbed; to compensate, they moved the absorber away at 0.74 micrometre per second — about 2.7 millimetres per hour. In the middle diagram, on top is the line of the arriving photon, shifted by gravity to +0.74; below, the line the absorber accepts, which slides by Doppler as it moves. When the two match, absorption is maximal — at v = gh/c = 0.74 μm/s. Every 12 seconds source and absorber swap places and the match moves to −0.74, just as they did to cancel systematics. On the right, the same law at a 1.4 solar-mass neutron star: with no exaggeration, 500 nanometres leave green and arrive orange, at 618 nm. It is the same factor neutrinos suffer on leaving the proto-neutron star.

The river model · Hamilton & Lisle 2008

Space falls · neutron star

Schwarzschild in Gullstrand–Painlevé · 1.4 M☉ · R = 12 kmdrag the lattice to rotate

Each node is a freely falling local frame, released from rest far away. In Gullstrand–Painlevé coordinates space is flat and flows inward at the escape velocity, v = √(2GM/r): the lattice is carried by the river. Near the mass the nodes run faster — the lines bend, cells stretch radially and squeeze sideways: the tide. At the star’s surface the river already runs at 0.59 c.

The river model · Hamilton & Lisle 2008

Space falls · black hole

Schwarzschild in Gullstrand–Painlevé · 6 M☉ · horizon at 17.7 kmdrag the lattice to rotate

The same lattice, now with no surface to stop the fall. At the horizon the river runs at exactly c: a light signal aimed outward can only stand still there, and inside it is dragged inward. That is why nothing gets out. The dashed line is the rate of a clock held at rest — it drops to zero at the horizon, where staying at rest is no longer possible.

December 1915 · Russian front

Schwarzschild and the first exact solution

ds² = −(1 − r_sr) c²dt² + dr²1 − r_s/r + r²dΩ²r_s = 2GM/c² · valid outside any non-rotating spherical body

1 r_s

event horizon

1.5 r_s

photon sphere

3 r_s

ISCO: last stable orbit

2.95 km

r_s of the Sun (Earth: 8.9 mm)

He wrote to Einstein on 22 December 1915 while serving as an artillery officer. He died in May 1916.

What we need from the metric

Curvature does two things to a neutrino

1 · It changes the energy

A neutrino emitted at r with local energy E arrives at infinity with

E_∞ = E √(1 − r_s/r)gravitational redshift, r_s = 2GM/c²

The whole spectrum slides down, and the source's clock runs slower as seen from outside.

2 · It bends the path

The trajectory is a geodesic, not a straight line. Next to a compact object the deflection is large, and different paths can connect the same source to the same detector.

Photons and neutrinos follow the same geometry, which SN 1987A confirmed to ~0.5%.

Both matter because, in a supernova, emission happens deep in the well and detection happens far from it.

And there is a third, subtler one

The oscillation phase is accumulated along the geodesic, with the particle's proper time. In flat spacetime it reduces to the familiar Δm²L/4E; in a curved metric it does not.

We come back to this in three slides.

None of this is optional when the density that sets the MSW resonance is measured in the local frame and the detector sits 10 kpc away.

How much, in numbers

What arrives is not what left

where	R	u = r_s/R	z
neutrinosphere, 1.4 M☉	30 km	0.138	0.077
cooler neutrinosphere	50 km	0.083	0.044
cold neutron star	12 km	0.345	0.235
collapsar disk, 3 M☉	3 r_s	0.333	0.225

During accretion, the spectrum DUNE sees left the star ~8% more energetic than it arrives.

And so does the time interval: what the star lives in 1 s, we measure in 1.08 s. The neutrino light curve is read with the wrong clock if the metric is ignored.

Why this matters for the resonance

The MSW condition involves the local energy. Using the detected energy instead shifts the resonance density by ~8%, and therefore shifts the radius where the conversion happens.

u and z computed with r_s = 2GM/c²; z = (1 − u)^−1/2 − 1.

Dashboard · Schwarzschild null geodesics

Light and neutrinos near a black hole

interactive · RK4 in the browseru″ + u = (3/2) r_s u², u = 1/r

impact parameter b

Every ray arriving with b < 3√3 M falls in. Just above that, it loops around the photon sphere before escaping.

An ultra-relativistic neutrino follows, to first order, the same null geodesic. Mass enters as a correction ∝ m²/E² — and that correction is exactly what carries the oscillation phase.

Compactness · 1/8