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Signed Relativistic Velocity and Frame Transformation

Transform an object’s velocity between aligned inertial frames, with signed direction, editable inverse inputs and meters, kilometers, miles or fractions of light speed.

Formula & worked example
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Calculator inputs

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How to use this calculator

Enter the known values in the units shown. Results update as you type.
Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
Use the worked example to check the method. Reset restores the starting fields.

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Formula and method

For a frame moving at 0.6c and an object moving at 0.6c within that frame, the stationary-frame object velocity is 15/17c, approximately 0.882353c.

u = (u′ + v) / (1 + u′v/c²); v = (u − u′)/(1 − uu′/c²); u′ = (u − v)/(1 − uv/c²).

Worked example

Enter these known values and leave the other values blank.

Moving-frame velocity in stationary frame: 0.6 c
Object velocity in moving frame: 0.6 c

Object velocity in stationary frame: 0.8824 c

Assumptions and limitations

The two reference frames are inertial and their axes are parallel. All velocities are signed components along the common axis. Positive and negative numbers identify opposite directions.
The moving-frame velocity v is measured in stationary frame S. The object velocity u′ is measured in moving frame S′, while u is measured in stationary frame S. Choose two known values and leave the remaining quantity blank.
A physical inertial frame must satisfy |v|<c. Object velocities may lie between −c and +c, including light rays at either endpoint. The exact defined vacuum light speed is 299,792,458 m/s.
For a light ray measured at +c in both frames, or at −c in both frames, its speed does not identify a unique frame velocity. The frame inverse remains unknown; supply a subluminal frame velocity to determine the other object quantity.
All three quantities remain editable with independent m/s, km/s, miles/s or c unit controls and locks. Changing units preserves the physical signed velocity. Locked inconsistent known values produce a conflict.
This one-dimensional transformation does not calculate transverse velocity components, accelerated frames, gravitational effects or cosmic Hubble recession. It is not ordinary addition of speeds near light speed.

Common questions

Do 0.6c and 0.6c add to 1.2c?

No. Transforming velocity between inertial frames also transforms elapsed time. The resulting velocity is 15/17c, about 0.882353c, which stays below c.

Can the inputs be negative?

Yes. A negative component means motion opposite the chosen positive axis. The frame must stay strictly between −c and +c and each object component must stay within those endpoints.

Does the speed of light change between frames?

For a subluminal frame velocity, a light ray at +c or −c retains that same signed light speed in the other frame.

Why is frame velocity blank when both object velocities equal c?

Many different subluminal frame velocities give the same light-ray speed. The two equal light-speed measurements cannot determine a unique frame velocity.

References

Preserve the exact relativistic velocity formula and its ordinary inverses. When an object is measured at the same signed light speed in both frames, the frame velocity is not uniquely determined and remains unknown instead of evaluating zero divided by zero. The inertial frame velocity must be strictly subluminal. Retain signed object velocities and valid ±c light rays; use exact-sign endpoint checks so nearby values are not rounded onto or beyond the boundary.

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