The symmetry of flat spacetime was not Poincaré but infinite-dimensional ── and it is measurable
Episode 3 confirmed that the stage may stay flat. Curvature can be a field on the stage rather than a property of it.
So how much symmetry does that flat stage have? Anyone would answer "the Poincaré group" ── four translations and six Lorentz transformations, ten in all.
That was wrong. And it has been known since 1962. The true symmetry at infinity is infinite-dimensional, and that fact does not stay in abstract mathematics: it shows up in the extremely concrete form of a gravitational-wave detector's mirrors failing to return to where they were.
This episode chases that infinite-dimensional symmetry, and ends up at holography for flat spacetime.
"Get far enough from matter and spacetime becomes flat, so the asymptotic symmetry must be the Poincaré group" ── a natural inference.
Bondi–van der Burg–Metzner (1962) and Sachs (1962) computed the asymptotic symmetry at null infinity \(\mathscr{I}^+\) (scri-plus) in order to handle gravitational waves exactly. What came out was ──
The key is that you reach infinity separately in each direction. Each direction on the celestial sphere corresponds to a different point of \(\mathscr{I}^+\).
An ordinary translation shifts the retarded time \(u\) by a constant. But ──
$$u\;\longrightarrow\;u-f(\theta,\phi)$$shifting it by an angle-dependent amount also preserves the asymptotic structure. That is a supertranslation. Expand \(f\) in spherical harmonics and the structure becomes visible.
the breakdown
\(\ell=0\) (one) is time translation, \(\ell=1\) (three) are the space translations. The translation part of the Poincaré group was only the first four rungs of an infinite tower.
Here is something to be wary of. Previously we noted that enlarging spacetime symmetry is forbidden by the Coleman–Mandula theorem. Does BMS not run into that prohibition?
It does not. The reason is one line.
Supertranslations are spontaneously broken.
The vacuum is not BMS invariant ── there are infinitely many degenerate vacua related to each other by supertranslations. And what Coleman–Mandula restricts are the unbroken symmetries of the S-matrix.
What a broken symmetry produces is not a constraint on scattering but a Nambu–Goldstone particle. And in this case, that is ──
The soft graviton is the Goldstone particle of broken supertranslations.
Here is the interesting part. Three independently discovered phenomena turned out to be three faces of the same structure (Strominger and others, 2013–17).
| 1962 | asymptotic symmetry (BMS) | general relativity. the symmetry at infinity is found to be infinite-dimensional |
| 1965 | soft theorem | scattering amplitudes. Weinberg finds the universal pole at \(q\to0\) |
| 1974 / 1991 | memory effect | gravitational-wave astronomy. test masses are permanently displaced |
Add one graviton of momentum \(q\to0\) to any scattering amplitude and ──
$$\mathcal{M}_{n+1}(q\to0)=\left(\frac{\kappa}{2}\sum_i\frac{(p_i\cdot\epsilon)^2}{p_i\cdot q}\right)\mathcal{M}_n+O(q^0)$$The leading term is a pole in \(1/q\), and its coefficient is fixed by the external momenta alone ── it does not depend on the details of the interaction at all.
Weinberg used this to derive the equivalence principle. Unless every particle couples to gravity with the same strength, the soft theorem is inconsistent. Episode 3 wrote that "the equivalence principle is a conclusion, not an assumption"; this is the scattering-amplitude version of it. The same conclusion, reached by an entirely different road.
Freely floating test masses ── think of LIGO's mirrors. A burst of gravitational waves passes through. After it is gone, spacetime returns to being flat.
But the mirrors do not return to where they were. Not an oscillation but a permanent displacement.
| edge | relation |
|---|---|
| symmetry ⟺ soft theorem | Write the BMS invariance of the S-matrix as a Ward identity and you get Weinberg's soft theorem directly. The supertranslation charge splits into a "hard part" (acting on matter) and a "soft part" (creating a zero-momentum graviton), which is why the soft theorem has the shape it does. |
| soft theorem ⟺ memory effect | Each other's Fourier transform. The memory effect is the zero-frequency (= soft) limit seen in position space. |
| memory effect ⟺ symmetry | When the wave passes, the vacuum you are in changes. Both before and after are "flat," but they are different flat vacua, related by a supertranslation. The amount of permanent displacement = the supertranslation parameter. |
The abstract story of infinite vacuum degeneracy has become a measurable quantity: the position of a mirror. Let us see it in a figure.
After the wave has passed, the red ring does not coincide with the grey circle. It stops, still squashed into an ellipse. Look at the waveform on the right and you can see a step that does not return to zero once the oscillation has died away. That is the memory.
Set the inclination to 0° (face-on) and the memory vanishes. This angular dependence comes out as a formula in the next section.
Let us get a number. What matters in a binary merger is the non-linear memory (Christodoulou memory), whose source is the energy flux of the gravitational waves themselves. The memory amplitude from the dominant \((2,0)\) mode is ──
Let us look at the angular factor first.
| inclination \(\iota\) | 0° (face-on) | 30° | 60° | 90° (edge-on) |
|---|---|---|---|---|
| angular factor | 0.0000 | 0.0462 | 0.1348 | 0.1771 |
Seen face-on the memory is zero; edge-on it is maximal. Because the memory is a purely \(+\)-polarization effect.
Now put in real events.
event E_rad D_L incl h_mem ratio to osc. peak
[Msun c²] [Mpc] [deg]
GW150914 3.00 440 150 1.51e-23 1.5%
GW150914 (if edge-on) 3.00 440 90 5.78e-23 5.8%
GW190521 7.60 5300 90 1.22e-23 2.0%
GW170817 (neutron star) 0.04 40 152 1.95e-24 2.0%
scaling check: h ∝ E_rad / D_L
predicted ratio GW190521/GW150914 (7.6/3.0)/(5300/440) = 0.2103
actual ratio (both edge-on) = 0.2103 ✓
GW150914's memory is small not because of distance but because of orientation. The inclination was 150°, i.e. nearly face-on, so \(\sin^2\iota\) drops it to a quarter.
LIGO's arms are 4 km. Multiply.
GW150914 (actual inclination)
\(1.51\times10^{-23}\times4000\,\mathrm{m}=6.0\times10^{-20}\,\mathrm{m}\) ── about 1/14000 of a proton radius
GW150914 (had it been edge-on)
\(5.78\times10^{-23}\times4000\,\mathrm{m}=2.3\times10^{-19}\,\mathrm{m}\) ── about 1/3600 of a proton radius
And what matters is not the size. It is that it does not go back. An oscillation is over once it passes, but this 6×10⁻²⁰ m stays forever. A black-hole merger somewhere in the universe has permanently rewritten the geometry of your detector.
The memory effect is unobserved. It is only one or two orders below the oscillating waveform, so why is it not caught?
The reason is not the size but the frequency. The memory is an essentially DC step. But LIGO is a band-pass instrument with almost no sensitivity below 10 Hz. It is buried in seismic noise. The signal sits exactly where the instrument is least sensitive.
The strategy is stacking. If the memory SNR of one event is \(\rho\), then \(N\) events give \(\sqrt{N}\rho\).
memory SNR per event N needed for 3σ
ρ = 0.05 3600
ρ = 0.067 2005
ρ = 0.10 900
The literature estimates \(O(10^3)\) events at aLIGO design sensitivity. With LISA it would be far easier (low frequency is its main battleground, so where the memory sits and what the instrument is good at coincide).
Finally, the part of this episode that goes furthest.
Holography's greatest weakness was that the only controlled examples are anti-de Sitter space (\(\Lambda<0\)). Our universe has \(\Lambda>0\).
Celestial holography is the programme of doing holography in flat spacetime. The boundary is not the AdS boundary but the celestial sphere \(S^2\) at null infinity.
The heart of the mechanism is one group-theoretic fact.
A four-dimensional Lorentz transformation is a conformal transformation of the celestial sphere.
So if you rewrite scattering amplitudes in a basis of boost eigenstates rather than momentum eigenstates (using the Mellin transform), a four-dimensional amplitude takes the form of a correlation function of a two-dimensional CFT.
And the infrared triangle translates directly into CFT structure.
| 4d flat spacetime | celestial CFT (on the sphere) |
|---|---|
| scattering amplitude | correlation function |
| supertranslations | current algebra |
| soft graviton | stress tensor |
| superrotations (subleading soft) | Virasoro |
The soft graviton becomes the CFT's stress tensor. The whole infrared physics of flat spacetime maps into the language of a two-dimensional conformal field theory.
The symmetry of flat spacetime is not Poincaré. The asymptotic symmetry at null infinity is the BMS group = Lorentz ⋉ supertranslations, and it is infinite-dimensional. The four ordinary translations were only the first four rungs, \(\ell=0,1\) of the spherical harmonics.
It does not touch Coleman–Mandula. Because supertranslations are spontaneously broken. And the soft graviton is the Goldstone particle of that breaking.
Three things kept apart for fifty years were the same thing. Asymptotic symmetry (1962) = the soft theorem (1965) = the memory effect (1974/91). The soft theorem is BMS's Ward identity, the memory effect is its Fourier transform, and the permanent displacement is the supertranslation parameter itself.
And it is measurable. GW150914's memory is \(h=1.5\times10^{-23}\), which over LIGO's 4 km arms is \(6.0\times10^{-20}\) m ── 1/14000 of a proton radius. More important than the size is that it does not go back. But being a DC signal it sits in the instrument's worst band, and stacking \(O(10^3)\) events is required. Still unobserved.
And holography for flat spacetime. \(SO(3,1)\cong SL(2,\mathbb{C})\) ── the 4d Lorentz group is the conformal group of the celestial sphere. The soft graviton becomes the stress tensor of a celestial CFT. Unfinished, but currently the only route to a holography not tied to \(\Lambda<0\).
This document is Episode 4 of the "Lattice We Build" series, a reading piece for high-school and university students who love physics. Where the sister series "That Clicks" explains known physics, this series shows the work itself.
Established material: the BMS group and supertranslations (Bondi–van der Burg–Metzner 1962, Sachs 1962), Weinberg's soft graviton theorem (1965), the memory effect (Zel'dovich–Polnarev 1974, Christodoulou 1991), the equivalence of these three (Strominger and others 2013–2017), and \(SO(3,1)\cong SL(2,\mathbb{C})\). The memory effect is unobserved ── the \(h_{\rm mem}\) values in the text are expectations computed with the formula above from published event parameters, and the required event counts are forecasts based on estimates in the literature. The memory amplitude formula used keeps only the dominant mode, and coefficients differ between formulations. Celestial holography is an ongoing research programme; there is as yet no completed example of the duality. The resolution of the information paradox by soft hair faces strong objections and is unsettled. The strain in the figure is exaggerated by \(10^{21}\).
Main series: Episode 1｜Episode 2｜Episode 3 ｜ bonus: ① Could physics be written more simply?／② How far does "ct = constant" get you? ｜ sister series: Relativity That Clicks ── to print, use your browser's "Print" → "Save as PDF."
Print / PDF: ⌘+P (Ctrl+P on Windows). Press "play" in the figure to watch the mirror fail to come back.