Minkowski Spacetime
In 1905 special relativity arrived as a set of rules — recipes for translating one inertial observer's clock-and-ruler readings into another's. Three years later Hermann Minkowski saw that those rules were not a patchwork of corrections to Newtonian kinematics but the geometry of a single four-dimensional continuum, and he announced it with a flourish that has never been bettered: "Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality." That union is Minkowski spacetime — the manifold ℝ⁴ whose points are events (t, x, y, z) and whose entire physical content is fixed by one object, the metric tensor. What distinguishes this geometry from ordinary four-dimensional Euclidean space is a single sign. The Euclidean rule adds squared coordinate differences with a uniform plus; the Minkowski rule subtracts the time contribution. That one minus sign is the whole of relativity. It makes the interval Δs² an invariant every inertial observer computes to the same value, even while they disagree — legitimately and permanently — about elapsed time and spatial distance taken separately. From that invariance flow, with no further postulates, the relativity of simultaneity, time dilation, length contraction, and a finite invariant speed c that no material body can reach. Minkowski spacetime is not a historical curiosity general relativity superseded; it is the local model general relativity is built on. The equivalence principle says spacetime is locally Minkowskian — in any freely falling laboratory the metric reduces to η with vanishing first derivatives — so the flat geometry is the tangent-space arena at every event of every curved spacetime. Master it and the curved theory becomes a controlled deformation of something you already understand.
- Minkowski spacetime is the flat four-dimensional manifold ℝ⁴ carrying the Lorentzian metric η_{μν} = diag(−1, 1, 1, 1), the geometric arena of special relativity.
- Its defining invariant is the interval Δs² = −c²Δt² + Δx² + Δy² + Δz², identical in value for every inertial observer and preserved by the ten-parameter Poincaré group.
- The metric is indefinite, not positive-definite: it sorts every separation into timelike, null, or spacelike, and that trichotomy is the light-cone causal structure.
- Frame-dependent effects — time dilation, length contraction, relativity of simultaneity — are all consequences of one geometric fact: Δs² is invariant while t and x individually are not.
- Free particles travel straight timelike worldlines that maximize proper time dτ = √(−η_{μν} dx^μ dx^ν)/c; a boost is a hyperbolic rotation and rapidity is its additive angle.
- Key idea: replacing the Euclidean rule dℓ² = dx² + dy² + dz² with the Lorentzian rule ds² = −c²dt² + dx² + dy² + dz² — a single minus sign — fuses space-plus-universal-time into a four-dimensional geometry whose invariant interval encodes all of special relativity and whose flat tangent space underlies all of general relativity.
Replacing the Euclidean rule dℓ² = dx² + dy² + dz² with the Lorentzian rule ds² = −c²dt² + dx² + dy² + dz² — a single minus sign in front of the time term — turns three-dimensional space plus universal time into a four-dimensional geometry whose invariant interval encodes all of special relativity.
Where this lives today
Minkowski geometry is a working tool and a live testing ground. At Fermilab, the Muon g−2 experiment stores muons at the "magic" momentum of 3.094 GeV/c, where γ ≈ 29.3; time dilation stretches their 2.2 μs rest lifetime to about 64 μs, long enough for tens of thousands of storage-ring orbits and a part-per-billion measurement of the muon's magnetic moment — impossible without the flat-spacetime dilation formula, and every collider from the LHC down budgets decay lengths with γ. The exactness of the −+++ structure is itself under experimental assault: the Standard-Model Extension parametrizes hypothetical Lorentz-invariance violations, and modern optical-cavity and atomic-clock comparisons — Michelson–Morley experiments with 21st-century sensitivity — now bound anisotropies of the speed of light at the 10⁻¹⁸ level. And every time LIGO or Virgo registers a gravitational wave, the analysis begins by treating the detector's neighbourhood as a patch of Minkowski spacetime perturbed by a tiny strain h_{μν} on top of η_{μν} — the flat metric is the backdrop against which curvature is measured.
The physical picture
Draw the history of a clock as a curve in a diagram whose vertical axis is time and whose horizontal axes are space: this is its worldline. Euclidean geometry would measure the curve's length by adding little squared steps, dℓ² = dx² + dt² with every term positive. Minkowski geometry does the opposite for the time step — it subtracts it. The immediate, counter-intuitive consequence is that a clock carried along a straight worldline accumulates the most elapsed time, and any detour (a bent, accelerated worldline) accumulates less. The geometry does not merely permit "moving clocks run slow"; it is that statement, written as a reverse triangle inequality. The minus sign has a second, structural payoff. In Euclidean space the set of points at zero distance from the origin is just the origin. In Minkowski spacetime the set of events at zero interval from a chosen event is not a point but a double cone — the light cone — because −c²t² + x² = 0 is solved by x = ±ct, the paths of light. That cone, and not any plane of "all space at one instant", is the invariant structure every observer agrees on. Boosting to another inertial frame tilts and stretches the coordinate grid, sliding events along hyperbolae, but leaves the light cone rigidly in place. The cone sorts the world into the absolute future and past a given event can influence (inside, timelike) and the "elsewhere" it can never causally touch (outside, spacelike).
The most illuminating way to see a Lorentz boost is not as "adding a velocity" but as a rotation — a hyperbolic one. An ordinary rotation in the x–y plane preserves x² + y², sweeping points along circles, and successive rotations add by angle. A boost in the ct–x plane preserves −(ct)² + x², sweeping points along hyperbolae, and successive boosts add by a hyperbolic "angle" called the rapidity φ, with tanh φ = v/c. Because tanh never reaches 1 no matter how large φ grows, no stack of boosts can ever reach v = c: the speed limit is not a barrier bolted onto the theory but the asymptote of a hyperbola. The same picture explains why velocities do not simply add — rapidities do, and tanh of a sum is not the sum of tanhs.
The core equations
The entire geometry is packed into one line element: the recipe for the squared interval between two neighbouring events. Read it as a Pythagorean theorem with the time term's sign flipped. Everything else in this article is a consequence of, or a computation with, this single expression.
where: ds² — the infinitesimal squared interval between neighbouring events; c — the invariant speed of light, 299 792 458 m/s; t, x, y, z — global inertial coordinates. Physical reading: the interval mixes a duration and a distance with opposite signs, so it can be negative, zero, or positive — and whichever it is, all inertial observers agree. Packaging the coefficients into a symmetric matrix and adopting the four-coordinate x^μ = (ct, x, y, z) with x⁰ = ct lets the line element be written as a single index contraction. This is the form that generalizes: in curved spacetime the constant η is replaced by a position-dependent g_{μν}, but the pattern is identical.
where: η_{μν} — the Minkowski metric components in inertial coordinates; μ, ν — spacetime indices running over 0, 1, 2, 3, summed when repeated (Einstein convention). Physical reading: the metric is the machine that eats two vectors and returns their Lorentz-invariant inner product; its diagonal −+++ pattern is the only structure the theory needs. The sign of the interval is not a bookkeeping detail — it is the causal verdict on a pair of events. With the −+++ convention adopted here, a negative Δs² means the events can be joined by a sub-light signal, a zero means only light can join them, and a positive means nothing can.
where: Δs² — the finite interval between two events. Physical reading: this trichotomy is the light cone written algebraically; it is Lorentz-invariant, so causal order is absolute even though time order is not (for spacelike pairs, different observers disagree on which came first). Along a timelike worldline the interval measures elapsed proper time — the reading of an ideal clock carried on the path. Defining dτ so it is real for timelike motion (ds² < 0) and factoring out the coordinate time immediately yields time dilation.
where: dτ — proper time, the invariant clock reading along the worldline; γ = (1 − v²/c²)^{−1/2} — the Lorentz factor, with v the coordinate speed. Physical reading: because γ ≥ 1, a moving clock's proper time falls short of coordinate time — the geometric statement that motion "uses up" time along the time axis by trading it for motion along space. A Lorentz transformation is precisely a linear change of inertial coordinates that leaves the metric numerically unchanged; that invariance condition is the definition of the Lorentz group, and boosts are its hyperbolic-rotation subgroup parametrized by rapidity.
where: Λ^{μ}{}_{ν} — the transformation matrix; the middle equation is the metric-preservation condition; φ — the rapidity, which adds under successive collinear boosts. Physical reading: cosh φ = γ and sinh φ = γβ, so a boost is a rotation through an imaginary angle — the reason the speed limit is an asymptote, not a wall. Finally, the feature that makes this spacetime flat: its Riemann curvature tensor vanishes identically. This is coordinate-independent — it stays zero in curvilinear coordinates where η looks complicated — and it is exactly what fails when gravity is present.
where: R^{ρ}{}_{σμν} — the Riemann curvature tensor. Physical reading: zero curvature means parallel transport is path-independent and global inertial frames exist; general relativity is the statement that gravity makes this quantity nonzero.
The goal here is to extract the moving-clock slowdown from pure geometry — from the single fact that Δs² has the same numerical value in every inertial frame — without ever writing down a Lorentz-transformation matrix. Consider two successive ticks of a clock that moves at constant speed v in the laboratory frame S. In the clock's own rest frame S′ the two ticks happen at the same place, separated only in time by the proper interval Δτ. We compute the one invariant Δs² two ways and equate them; the whole result is that equation.
Step 1. In the rest frame S′ the clock does not move between ticks, so the spatial separation vanishes and only the time term of the interval survives, \Delta s^2 = -c^2\Delta\tau^2. Why: the rest frame is defined by Δx′ = 0, which strips the interval down to its clean proper-time content.
Step 2. In the laboratory frame S the same two events are separated by Δt in time and, because the clock moves, by Δx = v Δt in space, giving \Delta s^2 = -c^2\Delta t^2 + v^2\Delta t^2. Why: this is the identical pair of physical events described in a frame where the clock is in motion — nothing about the events changed, only the coordinates.
Step 3. The interval is frame-independent, so the two expressions are one number; equate them and factor: -c^2\Delta\tau^2 = -c^2\Delta t^2\left(1-\frac{v^2}{c^2}\right). Why: invariance of Δs² is the entire physical input — everything else is algebra.
Step 4. Solve for the coordinate time and read off the Lorentz factor, \Delta t = \frac{\Delta\tau}{\sqrt{1-v^2/c^2}} = \gamma\,\Delta\tau. Why: since γ ≥ 1, the lab records more time between ticks than the clock itself does — the moving clock runs slow, and at v = 0.6c the factor is exactly γ = 1.25.
Go deeper · Derivation B · 1 minBoosts as hyperbolic rotations
The aim is to show that a Lorentz boost is not an ad-hoc velocity substitution but the exact spacetime analogue of a rotation, and that the natural boost parameter is an additive angle. We seek the most general linear map of the (ct, x) plane that leaves the quantity −(ct)² + x² unchanged and is continuously connected to the identity, then read off its physical meaning. The payoff is a one-line explanation of both the γ factor and the relativistic addition of velocities.
Step 1. Demand invariance of the two-dimensional interval, -(ct')^2 + x'^2 = -(ct)^2 + x^2, which is the defining property of a Lorentz transformation restricted to the t–x plane. Why: this is the only condition we impose — it is the two-dimensional shadow of equation (5).
Step 2. The general solution is the hyperbolic rotation ct' = ct\cosh\phi - x\sinh\phi,\qquad x' = -ct\sinh\phi + x\cosh\phi, because the identity cosh²φ − sinh²φ = 1 enforces the invariance exactly as cos² + sin² = 1 does for ordinary rotations. Why: hyperbolic functions are to the invariant −t² + x² what circular functions are to x² + y² — same structure, one flipped sign.
Step 3. Fix the physical meaning of φ by tracking the spatial origin of S′, where x′ = 0 forces x = ct tanh φ, i.e. the new frame moves at \tanh\phi = \frac{v}{c} = \beta. Why: rapidity is not an abstract angle — its hyperbolic tangent is the ordinary velocity in units of c.
Step 4. Substitute back to recover the familiar factors, cosh φ = γ and sinh φ = γβ, and note that composing two boosts adds their rapidities, φ_tot = φ₁ + φ₂. Why: because rapidities add linearly while velocities do not, tanh of the sum gives the velocity-addition law β₁₂ = (β₁ + β₂)/(1 + β₁β₂) for free, and tanh < 1 forbids ever reaching v = c.
Go deeper · Derivation C · 1 minThe twin paradox as a geometric inequality
The purpose here is to dissolve the twin "paradox" by exhibiting it as a theorem about lengths of worldlines: between two timelike-separated events, the straight (inertial) worldline is the one of greatest proper time, and any bent (accelerated) path is shorter. Twin A remains inertial; twin B departs at speed v, coasts out, turns around, and returns, meeting A again at the reunion event. We integrate proper time along each worldline and compare — no accelerated-frame gymnastics required, because proper time is a coordinate-independent integral.
Step 1. Twin A never moves in the chosen frame, so dx = 0 all along the trip and proper time equals coordinate time, \tau_A = \int_0^T \mathrm{d}t = T. Why: for the inertial twin the worldline lies straight along the time axis, the maximal-time direction.
Step 2. Twin B moves at speed v on each leg, so the integrand carries the dilation factor, \tau_B = \int_0^T \sqrt{1-\frac{v^2}{c^2}}\;\mathrm{d}t = T\sqrt{1-\frac{v^2}{c^2}} = \frac{T}{\gamma}. Why: every element of B's path tilts away from the pure time direction, and the tilt subtracts from the interval because of the minus sign in the metric.
Step 3. Compare: since √(1 − v²/c²) < 1, we have \tau_B < \tau_A, the reverse of the Euclidean triangle inequality. Why: in Euclidean geometry a detour is longer; in Lorentzian geometry, where the time term is subtracted, a detour is shorter — the bent worldline ages less.
Step 4. There is no symmetry to invoke and no paradox to resolve: only B changes frames (accelerates at the turnaround), so the two histories are physically distinguishable, and at v = 0.6c the traveller returns having aged τ_B = 0.8 T against the stay-at-home's T. Why: the asymmetry is geometric and absolute — one worldline is straight, the other is not, and that is a fact no boost can undo.
Limiting cases
Minkowski spacetime is best understood by the regimes it interpolates between: Newtonian physics at low speed, the light cone at the boundary, and curved spacetime in the tangent-plane limit.
| Case | Result | Meaning |
|---|---|---|
| Newtonian limit (v ≪ c) | Δτ → Δt, and Δs² ≈ −c²Δt² | Proper time approaches coordinate time; the time and space parts decouple into absolute Galilean time plus Euclidean space, recovering pre-relativistic physics. |
| Photon / null limit | Δs² = 0 along the path | Light follows null geodesics; proper time accumulates to zero, so "a photon experiences no time" — a heuristic only, since a photon has no rest frame. |
| Ultra-relativistic (v → c) | γ → ∞, φ → ∞ | The worldline asymptotes to the light cone; rapidity grows without bound while velocity saturates below c — the origin of the cosmic speed limit. |
| Curved-spacetime limit (local) | g_{μν}(p) = η_{μν}, ∂_α g_{μν}(p) = 0 | In local inertial coordinates at any event p, a general metric reduces to Minkowski with vanishing first derivatives — the mathematical form of the equivalence principle. |
Worked examples
1 · Why atmospheric muons reach the ground (fully numeric)
Cosmic-ray muons are created about 15 km up when primaries strike the upper atmosphere. A muon has a rest-frame mean lifetime τ₀ = 2.2 μs and travels at v = 0.995c. How far would it travel without relativity, how far does it actually travel, and does the muon's own frame agree?
Step 1. Compute the Lorentz factor for β = 0.995: γ = 1/√(1 − 0.995²) = 1/√(1 − 0.990025) = 1/√0.009975 ≈ 10.01. This single number carries the whole effect.
Step 2. Naive Newtonian estimate — no dilation. In a lifetime τ₀ the muon covers d₀ = v τ₀ = (0.995)(3.00×10⁸ m/s)(2.2×10⁻⁶ s) ≈ 6.6×10² m, only about 0.66 km. On this reckoning essentially none survive the 15 km descent.
Step 3. Relativistic lab-frame lifetime. Time dilation stretches the mean lifetime to γτ₀ ≈ (10.01)(2.2 μs) ≈ 22.0 μs, so the mean distance travelled is d = v (γτ₀) ≈ (2.985×10⁸ m/s)(22.0×10⁻⁶ s) ≈ 6.6×10³ m, about 6.6 km — a factor of γ ≈ 10 farther, and enough that a large fraction reach the surface.
Step 4. The muon's own frame must agree, and it does by a different mechanism. In the muon's rest frame the muon lives only τ₀ = 2.2 μs, but the atmosphere rushes past length-contracted by γ: the 6.6 km lab distance shrinks to 6.6 km / 10.01 ≈ 0.66 km, which at 0.995c takes 660 m / (2.985×10⁸ m/s) ≈ 2.2 μs — exactly one lifetime. Both frames predict survival; they are the one invariant interval read in two coordinate systems, dilation in the lab, contraction in the muon frame.
2 · Classifying a separation and reading its proper time (fully numeric)
Two events are separated by Δt = 3 s and, along one axis, Δx = 4×10⁸ m (Δy = Δz = 0). Take c = 3×10⁸ m/s. Are they timelike, null, or spacelike separated, and if a clock can be present at both, what does it read?
Step 1. Form the interval with the −+++ convention: Δs² = −c²Δt² + Δx². The time term is c²Δt² = (3×10⁸)²(3)² = (9×10¹⁶)(9) = 8.1×10¹⁷ m², and the space term is Δx² = (4×10⁸)² = 1.6×10¹⁷ m².
Step 2. Subtract: Δs² = −8.1×10¹⁷ + 1.6×10¹⁷ = −6.5×10¹⁷ m², which is negative, so the separation is timelike. A sub-light signal can connect the events and a single clock can be present at both.
Step 3. The proper time that clock reads is τ = √(−Δs²)/c = √(6.5×10¹⁷ m²)/(3×10⁸ m/s) = (8.06×10⁸ m)/(3×10⁸ m/s) ≈ 2.69 s. As a check, this is less than the coordinate time of 3 s — the moving clock (it must move, since Δx ≠ 0) runs slow, consistent with time dilation. Timelike separation; the co-moving clock reads about 2.69 s against 3 s of coordinate time.
3 · Rapidities add where velocities do not (fully numeric)
A frame S′ moves at β₁ = 0.6 relative to S, and a particle moves at β₂ = 0.6 relative to S′, all collinear. Find the particle's speed in S, first by rapidity, then by the velocity-addition law.
Step 1. Convert each velocity to a rapidity via φ = arctanh β. Here φ₁ = φ₂ = arctanh(0.6) ≈ 0.6931. Rapidities are the additive boost angles, so the composite rapidity is simply φ = φ₁ + φ₂ ≈ 1.3863.
Step 2. Convert back: β = tanh(1.3863) ≈ 0.882. The naive Galilean sum 0.6 + 0.6 = 1.2 would exceed c; the correct answer stays safely below 1, as it must.
Step 3. Cross-check with the standard velocity-addition formula β = (β₁ + β₂)/(1 + β₁β₂) = (1.2)/(1 + 0.36) = 1.2/1.36 ≈ 0.882 — identical. Adding two 0.6c boosts gives 0.882c, not 1.2c; rapidity additivity and the velocity-addition law are the same statement, and tanh caps the result below c.
Physics of the portrait
Minkowski spacetime carries one name above all others — the mathematician who took Einstein's kinematics and handed physics its geometry, over the initial objections of Einstein himself, who at first dismissed the four-dimensional reformulation as superfluous learnedness before it became the indispensable language of general relativity.
Origins · 1 minHermann Minkowski and how this was discovered
Hermann Minkowski was a mathematician, not a physicist by trade, and — remarkably — he had been one of Einstein's own instructors at the ETH Zürich, where he is said to have once called the young Einstein a "lazy dog." Trained in number theory and the geometry of numbers, Minkowski turned to electrodynamics after 1905 and saw what Einstein's kinematic postulates had left implicit: that the Lorentz transformations are exactly the isometries of a four-dimensional geometry with an indefinite metric. In his 1908 Cologne lecture "Raum und Zeit" he declared that space and time separately would "fade away into mere shadows," and made worldlines, the light cone, and the invariant interval the primary objects of the theory. The reformulation was initially resisted — Einstein grumbled that physicists now had to relearn their own theory as mathematics — but its power became undeniable once he began the search for a relativistic theory of gravity: curved-spacetime general relativity is unthinkable without Minkowski's flat prototype as its local model. Minkowski did not live to see that vindication; he died suddenly of appendicitis in January 1909, months after the Cologne address, at the age of 44.
Going further
Texts. Misner, Thorne & Wheeler, Gravitation (1973), Ch. 1–2 (the classic geometric treatment of flat spacetime) · Wald, General Relativity (1984), Ch. 1 and §4.1 · Carroll, Spacetime and Geometry (2004), Ch. 1 (special relativity as geometry, boosts and rapidity) · Hartle, Gravity (2003), Ch. 4–5 (worldlines and the twin paradox) · Schutz, A First Course in General Relativity, 2nd ed. (2009), Ch. 1–2 (the invariant interval from first principles) · Minkowski, "Raum und Zeit," Physikalische Zeitschrift 10 (1909) 75, the original source.
Provenance. All equations are standard SR/GR textbook results and were re-verified for this article. A checker should confirm: (1) the line element and η_{μν} = diag(−1, 1, 1, 1) against MTW Ch. 1–2, Wald §4.1, Carroll Ch. 1; (2) the time-dilation result from interval invariance (Schutz Ch. 1); (3) the boost-as-hyperbolic-rotation matrix with tanh φ = β, cosh φ = γ (Carroll Ch. 1); (4) the numerics — γ(0.6c) = 1.25, γ(0.99c) ≈ 7.09, γ(0.995c) ≈ 10.01, muon lab range ≈ 6.6 km versus naive ≈ 0.66 km, interval Δs² = −6.5×10¹⁷ m² giving proper time ≈ 2.69 s, and the two-boost velocity sum tanh(2 arctanh 0.6) ≈ 0.882; (5) the g−2 magic-momentum γ ≈ 29.3 and dilated lifetime ≈ 64 μs (Fermilab). The muon rest lifetime 2.2 μs is the standard PDG value.
Prerequisites. Postulates of Special Relativity · Spacetime Interval · Lorentz Transformations. Leads to. Minkowski Metric and Signature · Causal Structure and Light Cones · Four-Vectors · Proper Time.
Ask the tutor: Ask me to draw the light cone for any pair of events you name and walk through whether one can cause the other.
Why is there a minus sign in front of the time term, and could we put it in front of space instead?
The minus sign encodes the physical asymmetry between time and space: you can stand still in space but not in time, and light travels at the same speed for everyone. Mathematically it makes the metric indefinite, which is what creates light cones and causal order. You can flip the convention to +−−− (common in particle physics), putting three minus signs on space; then timelike intervals come out positive instead of negative. Nothing physical changes — every observable is a ratio or a sign-checked classification that is convention-independent. What you cannot do is make all four signs the same: that would be Euclidean four-space, with no cones, no causality, and no invariant speed.
If Δs² can be negative, how is it a "squared" anything?
The notation Δs² is traditional but slightly misleading — it is a single real number defined by the quadratic form −c²Δt² + Δx² + Δy² + Δz², not the square of some prior quantity Δs. For spacelike separations it is positive and its square root is a genuine invariant length (the "proper distance"). For timelike separations it is negative, and the physically meaningful root is the proper time, τ = √(−Δs²)/c. For null separations it is exactly zero. So "Δs²" is best read as one symbol naming the interval, whose sign is the whole point; do not expect it to behave like an ordinary square, which is always non-negative.
What is rapidity, and why do textbooks bother with it instead of velocity?
Rapidity φ is the "angle" of a Lorentz boost, defined by tanh φ = v/c. Its virtue is additivity: boost by φ₁, then by φ₂ in the same direction, and the total is simply φ₁ + φ₂ — whereas velocities combine through the awkward addition law. This parallels ordinary rotation angles, which add while their tangents do not. Rapidity also makes the speed limit transparent: v = c corresponds to φ → ∞, so no finite stack of boosts can reach it. Particle physicists use rapidity (and the related pseudorapidity) constantly, because rapidity differences are boost-invariant along the beam axis.
Does a photon really "experience no time"?
The slogan captures a real limit but should not be taken literally. Along a light ray the interval is exactly Δs² = 0, so the proper-time integral dτ = √(−Δs²)/c accumulates to zero — formally, no proper time elapses between emission and absorption. But this is a limit, not a frame: there is no inertial frame moving at c, so a photon has no rest frame in which "its experience" could be defined, and you cannot boost to ride alongside light. The honest statement is that light travels on null worldlines of zero proper length; "the photon experiences no time" is a vivid shorthand for that geometric fact, not a description of a photon's point of view.
How can Minkowski spacetime be the foundation of general relativity if GR is about curvature?
Through the equivalence principle. General relativity says that in a small enough freely falling laboratory, gravity is undetectable and physics looks exactly special-relativistic. Geometrically, at any event p one can choose local inertial coordinates in which the curved metric reduces to g_{μν}(p) = η_{μν} with vanishing first derivatives — the tangent space at every point of any spacetime carries the Minkowski inner product. Curvature shows up only in the second derivatives (the Riemann tensor), i.e. in how these local Minkowski frames fail to mesh globally. So flat spacetime is not superseded by GR; it is the pointwise model GR is stitched together from, which is why mastering it is the prerequisite for the curved theory.
Length contraction and time dilation seem symmetric between two observers — isn't that a contradiction?
It is not, because the two claims are about different pairs of events and different lines of simultaneity. Each observer sees the other's clocks run slow and rulers shrink, and both are right, because "at the same time" and "at the same place" are frame-dependent — the relativity of simultaneity is the ingredient that reconciles the apparent symmetry. A genuine asymmetry appears only when the situation itself is asymmetric, as in the twin paradox: there the two worldlines are objectively different curves between the same two events, and the geometry — not any observer's opinion — decides that the bent one has less proper time.