This tool is a physics calculator that computes the Lorentz factor, commonly known as the Gamma factor (γ). You use this calculator to understand the consequences of Albert Einstein’s special theory of relativity. The Gamma factor is a fundamental quantity that determines the magnitude of relativistic effects, such as time dilation and length contraction, for an object moving at a significant fraction of the speed of light. Essentially, it quantifies how much the measurements of time, length, and mass change for a moving object relative to a stationary observer. The calculator provides a precise value that is crucial for calculations in high-energy physics, astrophysics, and any field studying objects at very high velocities.
formula
1. Main Formula
The primary formula calculates the Gamma factor directly from the object’s velocity and the speed of light.
Gamma Factor = 1 / (sqrt(1 – (Velocity^2 / Speed of Light^2)))
Simplified Symbolic Formula
γ = 1 / sqrt(1 – (v^2 / c^2))
Explanation of Variables
v (Velocity): The speed of the object. This value must be in the same units as the speed of light (e.g., meters per second).
c (Speed of Light): A universal constant. Its value is approximately 299,792,458 meters per second.
γ (Gamma Factor): The resulting factor. It is a dimensionless number that is always greater than or equal to 1.
2. Alternative Formula (using Beta)
Sometimes, the calculation is simplified by first calculating “Beta” (β), which is the ratio of the object’s velocity to the speed of light.
a) Beta Factor Formula
Beta = Velocity / Speed of Light
Symbolic Formula
β = v / c
b) Gamma Formula Using Beta
This formula is mathematically identical to the primary formula but can be simpler to compute in steps.
Formula
Gamma Factor = 1 / sqrt(1 – Beta^2)
Symbolic Formula
γ = 1 / sqrt(1 – β^2)
Key Interpretations of the Gamma Factor
If Velocity = 0: The Gamma Factor is exactly 1. There are no relativistic effects.
As Velocity approaches the Speed of Light: The Gamma Factor approaches infinity. This indicates that an object with mass cannot reach the speed of light.
The calculated Gamma Factor is the multiplier for effects like time dilation (how much slower time passes for the moving object) and length contraction (how much shorter the object becomes in its direction of motion).
Gamma Factor for Various Speeds
This table provides the calculated Gamma factor for objects traveling at different percentages of the speed of light. You can use this for a quick reference to see how dramatically the relativistic effects increase as an object gets faster.
| Velocity (as % of speed of light) | Beta (v/c) | Gamma Factor (γ) |
| 1% | 0.01 | 1.00005 |
| 10% | 0.10 | 1.005 |
| 50% | 0.50 | 1.155 |
| 80% | 0.80 | 1.667 |
| 90% | 0.90 | 2.294 |
| 99% | 0.99 | 7.089 |
| 99.9% | 0.999 | 22.366 |
| 99.99% | 0.9999 | 70.712 |
Example
Let’s calculate the Gamma factor for a hypothetical spacecraft traveling at 95% of the speed of light. We will use the alternative formula with Beta for this calculation.
Scenario Details:
- Velocity of spacecraft (v) = 95% of the speed of light, or 0.95c.
- Speed of Light (c) = 299,792,458 m/s.
Calculation Steps:
- First, we calculate the Beta (β) factor.
- β = v / c
- β = 0.95c / c = 0.95
- Next, we square the Beta factor.
- β^2 = (0.95)^2 = 0.9025
- Then, we subtract the result from 1.
- 1 – β^2 = 1 – 0.9025 = 0.0975
- Now, we take the square root of that value.
- sqrt(0.0975) = 0.3122
- Finally, we calculate the Gamma factor by taking the reciprocal.
- γ = 1 / 0.3122 = 3.203
The Gamma factor for a spacecraft traveling at 95% of the speed of light is approximately 3.203. This means that for an observer on Earth, time on the spacecraft would appear to pass 3.203 times slower.
Most Common FAQs
The Gamma factor can never be less than 1 because of its mathematical definition. The velocity of an object (v) can never exceed the speed of light (c), so the ratio v/c is always between 0 and 1. Squaring this ratio keeps it in the same range. When you subtract this value from 1, the result in the square root is also always between 0 and 1. The square root of a number between 0 and 1 is still a number between 0 and 1. Taking the reciprocal (1 divided by that number) will always result in a value that is 1 or greater. A Gamma of 1 occurs only when the object is at rest (v=0).
As an object’s velocity approaches the speed of light, its Gamma factor approaches infinity. In the formula, as ‘v’ gets closer to ‘c’, the term (v^2 / c^2) gets closer to 1. This causes the denominator, sqrt(1 – (v^2 / c^2)), to get closer to 0. Dividing 1 by a number that is approaching zero results in an infinitely large number. This mathematical infinity shows why it is impossible for an object with mass to ever reach the speed of light; it would require an infinite amount of energy, and its length in the direction of motion would contract to zero.
No, you do not need to consider the Gamma factor for everyday objects. For things like cars, airplanes, or even space shuttles, their velocities are an extremely tiny fraction of the speed of light. At these low speeds, the Beta (v/c) value is so close to zero that the Gamma factor is practically equal to 1. For example, for a jet traveling at 1,000 km/h, the Gamma factor is about 1.0000000000004, meaning relativistic effects are completely undetectable and irrelevant. The Gamma factor only becomes significant in settings like particle accelerators or when observing high-speed cosmic phenomena.