Curvature is a way of keeping the books, not the content of the physics ── how far can we get on a flat stage?
"Space curves" is relativity's most famous claim and its hardest to swallow. Shown a picture of a weight on a rubber sheet, one nods vaguely and moves on ── plenty of people do exactly that.
Here let us push the straightforward doubt all the way. Does it really have to be curved? Could the stage stay flat, with something happening on top of it?
The conclusion up front ── it could. It holds exactly. But getting there requires diagnosing precisely where the naive approach gets stuck, and supplying exactly one missing thing. And beyond the supplying, a stronger position is waiting.
"Gravity is the speed of light varying from place to place on flat spacetime" ── let us start from that picture (the piece that follows this rereading all the way is Bonus ②). It is the approach of positing a refractive index \(n(x)\approx1-2\Phi/c^2\). Light bending, gravitational redshift and the Shapiro delay all come out correctly this way.
Written as a formula, this picture amounts to assuming the following metric.
$$ds^2=\Omega^2(x)\,\eta_{\mu\nu}dx^\mu dx^\nu$$A flat metric with nothing but a position-dependent scalar factor. This is called conformally flat. And there is an exact criterion.
the metric is conformally flat \(\iff\) the Weyl tensor vanishes
That draws the boundary. And the result splits cleanly.
| spacetime | Weyl tensor | writable with a scalar alone? |
|---|---|---|
| FRW (expanding universe) | \(=0\) | yes. and exactly |
| Schwarzschild (a star) | \(\ne0\) | impossible in principle |
The expanding universe really is conformally flat. Move to conformal time and light runs on exact 45° straight lines. That is no accident ── the expanding universe is literally "flat spacetime × a scale factor," and in that sense it is not curved. Your intuition, as far as cosmology goes, is correct as a theorem.
Schwarzschild is different. It is a vacuum solution, so \(R_{\mu\nu}=0\), which means all of the curvature is Weyl curvature.
this is the tidal force itself
Ricci is zero and yet the curvature invariant is not ── what remains is all Weyl. So it cannot be made conformally flat. However you choose the scalar \(\Omega(x)\), you cannot write down a single star.
This is the interesting part. There is a mechanical explanation for why the refractive-index picture only works for "light in a static weak field."
Null geodesics are conformally invariant. The path of light does not change when you multiply by \(\Omega^2\). So as long as the conformal class of the metric is right, light bending is structurally bound to be reproduced.
Which is to say the refractive-index picture is built so as to hit for light. But the parts that a conformal factor cannot squash ── Weyl curvature, the rate of clocks, the orbits of matter ── are untouchable with one scalar.
So what is missing? A concrete example makes it obvious in an instant.
Consider frame dragging. Around a rotating body, the time for light to circle clockwise and the time to circle anticlockwise differ (this has been measured).
A scalar \(c(x)\) cannot write this. It has only one value at each point, so it cannot return different answers for different directions.
To write it you need \(c(x,\hat n)\) ── a direction-dependent speed of light. Gravitational waves are the same. A TT polarization stretches one direction and shrinks the orthogonal one. A scalar cannot write anisotropic stretching.
Something that gives a speed per direction at each point.
Mathematics calls it a tensor field; physics calls it the metric \(g_{\mu\nu}\).
Here you might think "so it was the metric after all. We are just back to curvature." That is exactly what is not so. Here is where the real subject begins.
General relativity can be constructed as the theory of a self-interacting massless spin-2 field \(h_{\mu\nu}\) living on flat Minkowski spacetime. Gupta (1954), Feynman, Deser (1970).
Deser's argument is the cleanest.
Step 1. On flat spacetime \(\eta_{\mu\nu}\), write a massless spin-2 field \(h_{\mu\nu}\) (a free field).
Step 2. This field couples to the energy-momentum of matter. But the field itself carries energy. So it must couple to itself as well.
Step 3. Add the correction term. That correction term also carries energy, so add again. Iterate.
result
The series closes, and becomes the Einstein equations themselves. And
$$g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$$appears as a consequence, as the "effective metric." It was not assumed at the outset; it came out.
Weinberg showed something stronger still. Any Lorentz-invariant theory of a massless spin-2 particle coupled to a conserved source must, at low energy, become general relativity. Which is to say ──
The equivalence principle is a conclusion, not an assumption.
You do not need to put "gravitational mass = inertial mass" in by hand. It is derived from the setup of a spin-2 field on flat spacetime.
So "space is not warped; a field on a flat stage merely behaves that way" is not a poetic rephrasing but an exact formulation. Curvature becomes not a property of the stage but a field sitting on the stage.
The red path does not differ by a single pixel between left and right. The same array is drawn twice. The only difference is how the background is drawn ── on the left the lattice is distorted; on the right the lattice is left square and the field is shown by shading.
What is observed is only the red line. The background lattice is, in both cases, a line you drew on paper; it is not out there in nature.
Let us draw the honest line. This formulation has a strange price.
Nothing couples to \(\eta_{\mu\nu}\) alone. Rulers, clocks and light all measure with \(g=\eta+h\). So you can say "it is really flat," but no experiment detecting that flatness exists even in principle.
A delicate position. Like a preferred frame, but a completely invisible one.
One more, and this one is a substantive limitation.
An expansion around a flat background does not cover the interior of a horizon, non-trivial topology, or a closed universe well. The reach of the perturbation series is limited.
The shape of it is "completely right locally, insufficient globally." Try to write the interior of a black hole as \(\eta+h\) and the premise that \(h\) is small collapses.
So far this has been "you may book it that way." But this position has a practical payoff. And it is not a thought experiment ── it is running for real right now.
If the lattice does not distort, you can lay square cells out in space and replace derivatives by differences. For someone who writes programs, this is decisively welcome.
$$\frac{\partial f}{\partial x}\;\longrightarrow\;\frac{f_{i+1}-f_{i-1}}{2\Delta x}$$And ── numerical relativity does exactly this.
Lay a flat orthogonal lattice out in memory. At each point put \(g_{\mu\nu}\) (in practice variables called BSSN or Z4c) as field values. Evolve in time by finite differences.
The lattice never distorts. The curvature is not in the lattice but inside the numbers sitting on the lattice points.
When LIGO detected a black-hole merger, the waveform templates matched against the data were built this way. Curved spacetime, computed on a lattice that does not distort.
And structurally this is Deser's formulation of §03 itself. The stage is flat; the curvature is a field \(h_{\mu\nu}\) on the stage. The formulation and the implementation have the same shape.
Honestly, here is what is hard. The point is that none of it is caused by "the lattice not distorting."
| hard part | content |
|---|---|
| ① foliation (gauge) | To take differences you must slice spacetime into "space × time." How you slice is free, but slice it wrong and the lattice points fall into the black hole and the computation blows up. Techniques like 1+log slicing, gamma-driver shift and moving punctures are needed. |
| ② constraint violation | The Einstein equations split into "evolution equations" and "constraints." Finite differences always violate the constraints slightly, and that error grows exponentially. It was only solved by constraint-damping formulations like BSSN, Z4c and generalized harmonic. |
| ③ singularities | The numbers diverge at the centre of a black hole. Either cut it out (excision) or dodge it with clever coordinates (punctures). |
Recall the table from Episode 2. A lattice had two distinct roles: (a) computational tool and (b) basic structure of spacetime.
The lattice of numerical relativity is entirely (a). The spacing \(\Delta x\) is artificial; you refine it and check convergence. Not one word claims "spacetime is made of a lattice."
So Episode 2's eighteen-order fine-tuning problem does not arise at all. Because no preferred frame has been introduced as physics.
"Space does not distort," therefore it can be solved by finite differences ── this is correct, and it is what people actually do.
What is not distorting is the computational lattice; nobody is claiming spacetime is discrete. Which is why it is safe.
So far the story has been "rewrite curvature as a field." The flat background remains, merely invisible.
There is a more thoroughgoing position: that neither curvature nor curved spacetime exists as a basic element at all.
In AdS/CFT, the boundary theory has no gravity. And no curvature. It is an ordinary field theory sitting on a fixed flat background. And ──
The curved bulk spacetime emerges from the entanglement structure of the boundary.
Curvature is not a basic existent but one form of the description of a flat field theory.
This is the strongest version of "no warping of space is needed." In §03 the stage remained, flat; here the very concept of a curved stage disappears from the basic description.
And a lot lines up.
| what we wanted | in the boundary theory |
|---|---|
| no warping of space | ○ a flat fixed background |
| no spacing of space (Episode 2) | ○ a continuum field theory |
| exact Lorentz invariance | ○ conformally invariant, even |
| a finite amount of information | ○ the holographic bound |
| computable | ○ an ordinary field theory |
This position never touches Episode 2's \(6\times10^{-20}\), nor the eighteen orders of fine-tuning. Because it creates no preferred frame and introduces no spacing.
Finally, what this position is paying.
The only controlled example is AdS. \(\Lambda<0\). But our universe has \(\Lambda>0\). Holography in de Sitter is unsolved, and that is this route's greatest weakness.
"The boundary is flat" is also, in a sense, a choice. The background the boundary theory sits on is fixed, but the reason we may call it "flat" is conformal invariance; in reality only the conformal class is fixed. Here too we are back to bookkeeping.
And the derivations of \(1/4\) and of the Page curve are for special settings. Three-dimensional gravity, JT gravity, supersymmetric and extremal black holes. It is not that the same thing has been done for a general Schwarzschild black hole. We are at the stage of having got through the wall at one spot.
A scalar gets stuck. The criterion is the Weyl tensor. The expanding universe has Weyl \(=0\), so it can be written exactly as "flat spacetime × a scale factor," while a star has Weyl \(\ne0\) and cannot be written in principle. The reason light alone comes out right is that null geodesics are conformally invariant.
What was missing was the single word "direction-dependent." Something that gives a speed per direction at each point ── that is a tensor field. Frame dragging and gravitational waves can both be written with it.
Put a tensor field on a flat stage and general relativity stands up. Iterate the self-interaction of a massless spin-2 field and the series closes into the Einstein equations (Gupta–Feynman–Deser). \(g=\eta+h\) is a result, not an assumption. And the equivalence principle comes out as a conclusion (Weinberg).
There are two prices. The flat background cannot be observed in principle (= which is exactly why it does not cause Episode 2's eighteen-order problem). And it does not reach global structure ── horizons, topology.
And the limit of this is holography. The boundary theory has neither gravity nor curvature, and curved spacetime emerges from entanglement. With no warping, no spacing and no preferred frame, you get a finite amount of information and computability. But the only controlled example has \(\Lambda<0\), and our universe has \(\Lambda>0\).
This document is Episode 3 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 equivalence of conformal flatness and the vanishing of the Weyl tensor, the conformal flatness of FRW, the Kretschmann invariant of Schwarzschild, the conformal invariance of null geodesics, the reconstruction of general relativity from a spin-2 field on flat spacetime (Gupta 1954, Deser 1970), and that the equivalence principle follows from a massless spin-2 particle (Weinberg). On the other hand, the holographic reading that "curvature is emergent" is a position under research, not an established conclusion. Controlled examples of the duality are limited to asymptotically AdS, and the \(\Lambda>0\) case is unsolved. The figure is schematic: the way the background lattice is distorted is not a quantitative rendering of a metric, and only the light path is a numerical integration using the weak-field refractive index \(n=1+2\mu/r\).
Main series: Episode 1｜Episode 2｜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). With the slider in the figure you can confirm that the same path can be booked two ways.