The time dilation calculator above turns a speed into a Lorentz factor and applies it to a time interval. Give it the proper time measured by a moving clock and it returns the longer interval measured in the frame that watches that clock go past. It also works backwards, taking a Lorentz factor you want and returning the speed that produces it, which is the more useful direction when you are designing a thought experiment rather than analysing one.
Arb Digital builds free tools that behave honestly at the edges of their range. Time dilation calculators have a specific failure mode: as the speed approaches light speed, the quantity 1 − β² is a subtraction of two nearly equal numbers, and ordinary floating-point arithmetic loses most of its significant digits doing it. This one factorises the expression to keep the precision, and when an input is genuinely indistinguishable from light speed at double precision it says so rather than returning an infinity or a silently wrong number.
What This Time Dilation Calculator Does
Special relativity says that a clock moving relative to you ticks slower than one at rest beside you. OpenStax University Physics Volume 3, section 5.3 on time dilation, states the relationship as Δt = γΔτ, with the Lorentz factor γ = 1 ÷ √(1 − v²/c²). The proper time Δτ is the interval measured by a clock in the frame where the two events happen at the same place; the dilated interval Δt is what any other inertial frame measures, and it is always longer.
The speed of light used throughout is 299,792,458 metres per second. NIST lists that value as exact, with no uncertainty attached, because since 1983 the metre has been defined in terms of it. That is worth noting: the number is not a measurement that might be refined later, it is a definition, so every conversion between a fraction of c and an absolute speed here is exact by construction.
The grid reports the Lorentz factor itself, the gap that opens between the two clocks, the speed in metres per second whichever way you entered it, and the length a one-metre rod would contract to at that speed. Length contraction is included because it is the same factor working on space rather than time, and seeing both at once makes the geometry clearer than either alone.
How to Use It
- Choose how to specify the speed. A fraction of c is the natural input for relativity problems; an absolute speed is easier when the number came from somewhere else.
- Or work backwards from a Lorentz factor. Enter the factor you want and the tool returns the required β and speed, which is how accelerator specifications are usually quoted.
- Enter the proper time. This is the interval on the moving clock, in its own frame. It is always the smaller of the two intervals.
- Read the difference, not just the total. At everyday speeds the dilated time is indistinguishable from the proper time, and only the difference figure shows anything at all.
- Watch what happens as β approaches 1. The tool keeps its precision much further than a naive implementation, and tells you plainly when it has run out.
The Formula: How Time Dilation Is Calculated
The Lorentz factor is γ = 1 ÷ √(1 − β²) where β = v/c. The dilated interval is Δt = γΔτ. Inverting for speed from a target factor gives β = √(1 − 1/γ²), and length contraction is the reciprocal, L = L0/γ.
Internally the calculator evaluates 1 − β² as (1 − β)(1 + β). Algebraically the two are identical; numerically they are not. At β = 0.999999999, squaring first produces a subtraction where the leading sixteen digits cancel and almost nothing meaningful survives, while the factored form computes 1 − β directly from a number whose small size is fully represented. That single change extends the usable range by several orders of magnitude in γ.
Work the defaults. At β = 0.8, 1 − β² = (0.2)(1.8) = 0.36, so γ = 1 ÷ 0.6 = 1.6667. A proper time of one year becomes 1.6667 years in the observing frame, a difference of about 0.667 years or 243.5 days. A one-metre rod contracts to 0.6 m along its direction of travel, and the speed is 0.8 × 299,792,458 = 239,833,966.4 m/s.
What the Calculator Does Near Light Speed
Every quantity in relativity diverges at β = 1, and how a calculator handles that is a fair test of whether it was written carefully. This one refuses β ≥ 1 with a plain statement that no object with mass can reach light speed, rather than showing an infinity as though it were a result.
The more interesting case is just below 1. Doubles carry roughly fifteen to sixteen significant decimal digits, so a β typed as 0.9999999999999999 — sixteen nines — is stored as a value whose difference from 1 is at the edge of what the format can express, and a β with seventeen nines is stored as exactly 1. There is no way around that in a browser, so the tool detects the condition and says the input is indistinguishable from light speed at double precision instead of pretending to a result it cannot compute. The honest failure is the useful one, because a number like 4.5 × 107 for γ produced from cancelled digits looks entirely plausible and is meaningless.
Within the usable range the factored form performs well. At β = 0.999999999, γ is about 22,360, and the calculation still has ample precision left. Particle accelerators live in exactly this territory: protons in a large collider run at Lorentz factors in the thousands, and quoting their speed as a decimal fraction of c is almost useless because the interesting digits are all at the far end of a string of nines. Quoting γ instead, or the energy, carries the information properly. That is why the third input mode exists.
Which Clock Is the Slow One?
The most common confusion about time dilation is that it appears to be symmetric and therefore self-contradictory. If I see your clock running slow and you see mine running slow, who is right? Both are, and there is no contradiction, because the two of you are comparing different pairs of events.
The asymmetry that resolves it is proper time. The proper interval between two events is the one measured by a single clock present at both of them, and that is a frame-independent fact rather than a matter of viewpoint. Any other frame needs two spatially separated clocks to time the same pair of events, and comparing a single clock against a synchronised pair is where the apparent paradox comes from. The clock that was present at both events always reads the smaller interval, and that is the value to enter in the proper time field.
The twin scenario makes it concrete. A traveller who leaves, turns around and returns is present at both the departure and the arrival, so their clock measures the proper time and comes back reading less. The stay-at-home twin was also present at both events, which is why the situation needs care, but the traveller changed inertial frames at the turnaround and the stay-at-home twin did not. That break in symmetry is real and physical, and it is what makes the outcome definite rather than a matter of perspective.
The Experiments That Settle It
Time dilation is not a theoretical curiosity; it is measured routinely. The OpenStax section linked above describes two of the classic demonstrations. Muons created by cosmic rays high in the atmosphere have a half-life of about 1.52 microseconds at rest, which at any speed below light speed should let almost none of them reach the ground. Large numbers arrive anyway, because at their speeds the Lorentz factor stretches their lifetime in our frame by exactly the predicted amount. From the muon's own frame nothing about its lifetime changed; the atmosphere was simply contracted.
The second is Hafele and Keating's 1971 experiment, which flew caesium atomic clocks around the world on commercial aircraft and compared them against clocks left behind. The measured offsets matched the relativistic prediction within experimental error, at speeds where the effect amounts to fractions of a microsecond. Run the orbital preset here and you can see the scale of it: at 7.66 km/s the Lorentz factor differs from 1 by about 3.3 parts in ten thousand million, which over a year amounts to roughly ten milliseconds.
Satellite navigation is the everyday case. Those systems depend on clock comparisons precise enough that both the special-relativistic slowing from orbital speed and the general-relativistic speeding from weaker gravity have to be corrected for, and the two effects push in opposite directions. This calculator handles only the first. Gravitational time dilation depends on the gravitational potential rather than on speed, and for that side of the problem the Schwarzschild radius calculator is the relevant starting point.
How This Differs From the Adjacent Relativity Tools
The boundary in one sentence: this page converts a velocity into a time-stretching factor, while the mass energy equivalence calculator converts mass into energy through E = mc² and the Schwarzschild radius calculator deals with gravitational rather than kinematic effects. All three involve c, but each answers a different question.
For ordinary motion at ordinary speeds, the classical tools are the right ones and the relativistic correction is invisible. The velocity calculator and the SUVAT calculator handle everyday kinematics, and the speed converter or the time converter simply rescale units. Below about a tenth of light speed the Lorentz factor differs from 1 by less than half a per cent, which is smaller than the uncertainty in most real measurements.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Entering the observer's interval as proper time — proper time is what the moving clock reads, and it is always the shorter of the two.
- Expecting a visible effect at ordinary speeds — at a thousandth of light speed the factor differs from 1 by five parts in ten million, so only the difference figure shows anything.
- Trusting a β with more than about fifteen nines — double-precision arithmetic cannot represent the distance from 1, and any calculator that returns a confident answer there is bluffing.
- Applying this to gravitational time dilation — clocks also run at different rates in different gravitational potentials, and that effect has nothing to do with relative speed.
- Assuming the effect is merely apparent — the clocks genuinely disagree when brought back together, as flown atomic clocks and cosmic-ray muons both demonstrate.
Related Free Tools From Arb Digital
For the rest of relativity, use the mass energy equivalence calculator and the Schwarzschild radius calculator. For classical motion at ordinary speeds, the velocity calculator and the SUVAT calculator are the right tools, and the escape velocity calculator covers orbital speeds. Rescale figures with the speed converter or the time converter, look at light itself with the photon energy calculator, and browse the full free online tools hub for everything else.
Frequently Asked Questions
The interval measured by a single clock that is present at both events, which means a clock travelling with the moving object. It is always the shortest interval any frame measures between those two events, and it is the value to enter in the proper time field.
It is the ratio between the observed interval and the proper interval. A factor of two means a moving clock ticks once for every two ticks of a clock at rest in the observing frame, and it also gives the factor by which lengths contract along the direction of motion.
Because the Lorentz factor diverges there, and no object with mass can reach light speed. The calculator refuses the input and says so rather than displaying an infinity as though it were a physical result.
Beyond about fifteen nines, double-precision arithmetic cannot represent how far the value sits from one, so the tool reports that the input is indistinguishable from light speed instead of returning a number produced by cancelled digits.
It depends entirely on how precisely you can measure. At a tenth of light speed the factor is 1.005, at half it is 1.155 and at nine-tenths it is 2.29. Below a hundredth of light speed the effect is smaller than five parts in a hundred thousand.
Because the two observers are comparing different pairs of events. The clock present at both events measures the proper interval, and that fact is frame-independent. In the twin scenario the traveller changes inertial frames at the turnaround, which breaks the symmetry.
No. This page covers the kinematic effect from relative velocity only. Clocks also run at different rates at different gravitational potentials, which is a separate effect from general relativity and can push in the opposite direction.
Yes, repeatedly. Cosmic-ray muons reach the ground in numbers only explicable by their stretched lifetimes, and atomic clocks flown around the world in 1971 returned offsets matching the prediction within experimental error.
This tool is provided for educational use. It models special-relativistic time dilation between inertial frames only, and does not account for gravitational time dilation, acceleration during a journey or any general-relativistic effect, so treat its output as a physics result rather than a description of any real clock.
