Length Contraction Calculator
Compute the contracted length of an object measured by an observer relative to whom it moves at high velocity, using special relativity. Returns the contracted length in the same units as the rest length.
Last updated: May 2026
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About this calculator
The formula is L = L₀ × √(1 − (v/c)²), where L₀ is the proper length (measured in the object's rest frame), v is the relative velocity, and c is the speed of light. The contraction factor 1/γ is symmetric to the time-dilation factor γ — moving rulers shrink in the direction of motion, by the same factor that moving clocks slow. Length contraction occurs only along the direction of motion; transverse dimensions are unaffected. Edge cases: at v = 0, L = L₀ (no effect). As v → c, the contraction factor approaches 0 and the apparent length approaches zero — an ultra-relativistic spacecraft of proper length 100 m at 0.99c is seen by Earth observers as ~14 m long. For v ≥ c the formula gives imaginary or zero values; massive objects cannot reach light speed. The contraction is real in the sense that the leading and trailing edges of the object occupy specific spatial positions at the same instant in the observer's frame, separated by the contracted distance — but it is frame-dependent: in the object's own rest frame the length remains L₀. Length contraction has been confirmed indirectly through muon decay experiments and through the Terrell-Penrose effect (which actually shows photographic 'snapshots' of fast objects look rotated, not contracted, because of light-travel-time differences from different parts of the object).
How to use
Example 1 — moderately relativistic spacecraft. Proper length 100 m, velocity 2 × 10⁸ m/s (≈ 0.667c), c = 2.998 × 10⁸ m/s, lengthUnit 1 (metres). Step 1: v/c = 2e8/2.998e8 ≈ 0.6671. Step 2: (v/c)² ≈ 0.4450. Step 3: 1 − 0.4450 = 0.5550. Step 4: √0.5550 ≈ 0.7450. Step 5: L = 100 × 0.7450 ≈ 74.5 m. Verify: γ at 0.667c is 1/√0.555 ≈ 1.343, and 1/γ ≈ 0.745, matching the contraction factor ✓. The 100-m ship appears 74.5 m long to a stationary observer. Example 2 — ultra-relativistic muon path. Proper length 10 km (atmospheric height a stationary muon would traverse, but we'll compute the path length in the muon's frame for v = 0.998c). Convert: 10 km = 10,000 m, lengthUnit 1, v = 0.998 × c ≈ 2.992 × 10⁸ m/s. Step 1: v/c = 0.998. Step 2: (v/c)² = 0.996004. Step 3: 1 − 0.996004 = 0.003996. Step 4: √0.003996 ≈ 0.06322. Step 5: L = 10,000 × 0.06322 ≈ 632 m. Verify: the same path is only ~632 m in the muon's rest frame, which it can traverse easily within its 2.2 μs proper-frame half-life (632 m / 0.998c ≈ 2.1 μs) — and this is the physical explanation for why cosmic-ray muons created at altitude reach Earth's surface ✓.
Frequently asked questions
Does length contraction mean a moving object is 'physically squeezed'?
Not in the sense of mechanical compression — no forces act on the object. Length contraction is a geometric effect of how space and time are related between inertial frames: the two ends of a moving object, at a single instant in the observer's frame, are separated by less distance than the object's proper length. In the object's own rest frame, nothing is squeezed and it has its full length L₀. The contraction is real (you can measure it; it has observable consequences) but frame-dependent — there is no absolute 'length' that is being shrunk. The Lorentz transformations show how spatial coordinates and times mix between frames, and length contraction is one consequence; time dilation and relativity of simultaneity are others. Together they make c the same in all inertial frames, satisfying Einstein's second postulate. So length contraction is not a physical force on the object but a relationship between how different observers measure the same physical object.
Why is length contraction only along the direction of motion?
Lorentz transformations only mix the time coordinate with the spatial coordinate parallel to the direction of relative motion; transverse spatial coordinates are unchanged. This is a consequence of the symmetry of the boost: a velocity in the x-direction affects x and t, but not y and z. Physically, if transverse dimensions also contracted, we could derive contradictions — for instance, two trains moving in opposite directions on parallel tracks could each 'see' the other narrower and thus argue about whether a train of standard gauge can fit on a track of standard gauge, with no symmetric resolution. By keeping transverse dimensions invariant, Lorentz transformations preserve the consistency of all observers' measurements. So a relativistic spacecraft is contracted along its direction of motion but not in width or height — if it's a cube at rest, it appears as a rectangular slab in motion.
How does length contraction relate to time dilation and the constancy of c?
All three are necessary consequences of Einstein's two postulates (principle of relativity and constancy of c) acting together. Time dilation by factor γ and length contraction by factor 1/γ are reciprocal: they appear in the Lorentz transformation matrix with these exact factors so that c remains the same in all inertial frames. Concretely: a light pulse takes time L/c to traverse a stationary ruler of length L. The same pulse, viewed in a frame where the ruler moves with velocity v, must also take L'/c if c is constant — but in that frame, the ruler is contracted (L' = L/γ) and the time taken is dilated. The numbers work out exactly to keep c invariant. This interlocking explains why you can derive any one of (time dilation, length contraction, relativity of simultaneity, velocity addition) from any of the others given the constancy of c. They are not independent phenomena but different aspects of the same underlying spacetime geometry.
What are the common mistakes when applying length contraction?
The biggest mistake is treating length contraction as a force or compressive stress on the object — it is purely a geometric/kinematic effect with no internal stress. The second is forgetting that the contraction is symmetric: A sees B contracted, B sees A contracted. The third is contracting transverse dimensions — only the direction of motion contracts. People also confuse the observed photographic appearance of a fast object with its measured length: Terrell rotation shows that photographs of relativistic objects look rotated, not simply contracted, because light from different parts of the object reaches the camera at different times. Using v ≥ c gives nonsense (zero or imaginary lengths) — massive objects cannot reach c. Finally, do not extrapolate length contraction to non-inertial frames (accelerating objects) without careful analysis using proper time integration along worldlines; the simple formula applies only between two inertial frames at constant relative velocity.
When should I not use this calculator?
Do not use it for accelerating objects or non-inertial frames — the simple formula assumes constant relative velocity. It is not appropriate for cosmological length-scale problems where curvature of spacetime matters; use general relativity's metric formulation instead. Do not use it to compute the actual photographic appearance of fast objects — what a camera records is affected by Terrell rotation and light-travel-time differences, not just simple length contraction. It is unsuitable for transverse dimensions or curved trajectories; the formula only contracts along the line of relative motion. For v ≥ c the formula gives unphysical results (zero or imaginary lengths) — massive objects cannot reach c; massless particles like photons don't have a rest frame and length contraction is not meaningfully defined for them. For very high precision (parts per million or better), use the full CODATA value of c and double-precision arithmetic to avoid floating-point loss when v is very close to c.