{
  "id": "time-dilation",
  "version": "05a974b0d65d",
  "status": "published",
  "name": "Time Dilation Calculator",
  "question": "What is the time dilation?",
  "summary": "Works out special-relativity time dilation: how long a time on a moving clock lasts for an observer at rest, Δt = γΔτ, with the Lorentz factor γ, from a speed in % of light speed or in m/s, km/h or mph.",
  "category": "physics",
  "subcategory": "mechanics",
  "url": "https://www.acalculator.org/physics/time-dilation-calculator",
  "markdown": "https://www.acalculator.org/physics/time-dilation-calculator.md",
  "kind": "function",
  "method": "Δt = γ × Δτ, γ = 1 ÷ √(1 − v²/c²), c = 299,792,458 m/s.",
  "assumptions": [
    "Special relativity: the clock moves at a constant speed relative to the observer; gravity (general relativity) is left out.",
    "The speed of light is exactly 299,792,458 m/s; 1 mph = 0.44704 m/s; 1 km/h = 1/3.6 m/s; a year is 365 days."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "by": {
        "title": "Speed as",
        "description": "Type the speed as a percent of the speed of light, or as a speed in m/s, km/h or mph.",
        "type": "string",
        "enum": [
          "percent",
          "speed"
        ]
      },
      "b": {
        "title": "Speed (% of c)",
        "description": "The speed of the moving clock as a percent of the speed of light, from 0 up to (not including) 100.",
        "type": "number",
        "x-unit": "percent",
        "minimum": 0,
        "exclusiveMaximum": 100
      },
      "v": {
        "title": "Speed",
        "description": "The speed of the moving clock, below the speed of light (299,792,458 m/s).",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "speed",
        "minimum": 0,
        "exclusiveMaximum": 299792458
      },
      "t": {
        "title": "Time on the moving clock",
        "description": "The proper time Δτ: how long the event lasts on a clock that moves with it.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "time",
        "exclusiveMinimum": 0,
        "maximum": 1000000000000000
      }
    }
  },
  "outputs": {
    "dilated": {
      "label": "Time for an observer at rest",
      "description": "The dilated time Δt = γ × Δτ, in the unit of the time you typed.",
      "format": "quantity",
      "unit": "s"
    },
    "gamma": {
      "label": "Lorentz factor (γ)",
      "description": "γ = 1 ÷ √(1 − v²/c²), always 1 or more.",
      "format": "number"
    },
    "extra": {
      "label": "Extra time",
      "description": "How much longer the event lasts for the observer at rest: Δt − Δτ, in the unit of the time you typed.",
      "format": "quantity",
      "unit": "s"
    },
    "pct": {
      "label": "Speed (% of c)",
      "description": "The speed as a percent of the speed of light.",
      "format": "number"
    },
    "mps": {
      "label": "Speed (m/s)",
      "description": "The speed in metres per second (c = 299,792,458 m/s).",
      "format": "number"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "by": "percent",
      "b": 80,
      "v": 200000000,
      "t": "1 yr"
    },
    "outputs": {
      "dilated": 52560000,
      "gamma": 1.6666666666666667,
      "extra": 21024000,
      "pct": 80,
      "mps": 239833966.4
    },
    "text": "At 80% of the speed of light, 1 yr on the moving clock lasts 1.666666667 yr for an observer at rest."
  },
  "examples": [
    {
      "given": {
        "by": "percent",
        "b": 95,
        "t": 0.0000022
      },
      "expect": {
        "dilated": 0.000007045638767423835,
        "gamma": 3.2025630761017427
      },
      "source": "OpenStax, University Physics Volume 3, §5.3 Time Dilation (Δt = Δτ ÷ √(1 − v²/c²)), https://openstax.org/books/university-physics-volume-3/pages/5-3-time-dilation (Example 5.3: Δt = 7.05 μs); hand calculation in content.mdx: γ = 1 ÷ √(1 − 0.95²) = 3.20256"
    },
    {
      "given": {
        "by": "percent",
        "b": 80,
        "t": 31536000
      },
      "expect": {
        "dilated": 52560000,
        "gamma": 1.6666666666666667,
        "extra": 21024000
      },
      "source": "OpenStax, University Physics Volume 3, §5.3 Time Dilation (Δt = Δτ ÷ √(1 − v²/c²)), https://openstax.org/books/university-physics-volume-3/pages/5-3-time-dilation; hand calculation in content.mdx: γ = 1 ÷ √(1 − 0.64) = 1 ÷ 0.6 = 5/3, so 1 yr becomes 5/3 yr"
    },
    {
      "given": {
        "by": "speed",
        "v": 5830,
        "t": 1
      },
      "expect": {
        "dilated": 1.0000000001890887,
        "extra": 1.89088757504636e-10
      },
      "source": "OpenStax, University Physics Volume 3, §5.3 Time Dilation (Δt = Δτ ÷ √(1 − v²/c²)), https://openstax.org/books/university-physics-volume-3/pages/5-3-time-dilation (Example 5.1: Δt = 1 s + 1.89 × 10⁻¹⁰ s); hand calculation in content.mdx"
    },
    {
      "given": {
        "by": "percent",
        "b": 60,
        "t": 3600
      },
      "expect": {
        "dilated": 4500,
        "gamma": 1.25,
        "mps": 179875474.8
      },
      "source": "OpenStax, University Physics Volume 3, §5.3 Time Dilation (Δt = Δτ ÷ √(1 − v²/c²)), https://openstax.org/books/university-physics-volume-3/pages/5-3-time-dilation; hand calculation in content.mdx: γ = 1 ÷ √(1 − 0.36) = 1 ÷ 0.8 = 1.25; 1 h becomes 1.25 h"
    }
  ],
  "sources": [
    "OpenStax, University Physics Volume 3, §5.3 Time Dilation (Δt = Δτ ÷ √(1 − v²/c²); Examples 5.1 and 5.3). https://openstax.org/books/university-physics-volume-3/pages/5-3-time-dilation (retrieved 2026-10-03)",
    "NIST, CODATA value: speed of light in vacuum, c = 299,792,458 m/s (exact). https://physics.nist.gov/cgi-bin/cuu/Value?c (retrieved 2026-10-03)"
  ],
  "related": [
    "speed",
    "kinetic-energy",
    "wavelength"
  ],
  "changelog": []
}