Ordinary Velocity and Proper Velocity
In relativistic physics, velocity can be defined in two distinct ways:
- Ordinary (or coordinate) velocity, denoted by $v$, is the velocity calculated with respect to the laboratory time $t$. $$ v = \frac{dx}{dt} $$ This is the familiar definition used in classical physics. Here, $dx$ represents the spatial displacement measured by an external observer, while $dt$ is the time measured in that observer’s (or laboratory) frame. In special relativity, it corresponds to the speed of an object as seen from an external reference frame relative to which it is moving.
- Proper velocity, denoted by $\eta$, is defined with respect to the object’s own proper time $ \tau $. $$ \eta = \frac{dx}{d\tau} $$ This quantity expresses how much distance the object covers per unit of its own elapsed time.
The two definitions coincide only in the classical (non-relativistic) limit. In special relativity, however, they diverge as the velocity approaches the speed of light.
The distinction between ordinary velocity $v$ and proper velocity $\eta$ becomes essential when proper time is taken into account in describing motion.
So, what’s the relationship between the two?
There’s a simple yet fundamental relationship connecting $\eta$ and $v$:
$$ \eta = \gamma v $$
where $\gamma$ is the Lorentz factor:
$$ \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} $$
or, equivalently,
$$ v=\frac{\eta}{\sqrt{1+\left(\frac{\eta}{c}\right)^2}} $$
Since $\gamma \geq 1$, it follows that the proper velocity is always greater than the ordinary velocity.
$$ \eta \geq v $$
The two velocities are identical only when the ordinary velocity is zero, $v = 0$, meaning the object is at rest.
Note. According to special relativity, no object can exceed the speed of light. This relativistic limit applies only to the ordinary velocity $ v < c $. Therefore, even for extremely large values of the proper velocity $ \eta $, the ordinary velocity $ v $ can never surpass the speed of light $ c $. $$ v=\frac{\eta}{\sqrt{1+(\eta/c)^2}}<c $$ In other words, proper velocity does not violate relativity. In fact, neither ordinary velocity $ v $ nor proper velocity $ \eta $ fully captures the notion of motion in relativity: the correct description is given by the four-velocity vector $ U^\mu = (\gamma c,, \gamma \vec{v}) $, which specifies the direction of motion in spacetime. Its norm, equal to $ -c^2 $, is invariant for all observers and reflects the invariance of proper time.
Practical Example
Consider an object - say, a spaceship - moving in a straight line at a constant velocity $v = 0{,}6c$, where $c$ is the speed of light in vacuum.
The Lorentz factor $\gamma$ is given by:
$$ \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} $$
Substituting $v = 0{,}6c$ gives:
$$ \gamma = \frac{1}{\sqrt{1 - (0{,}6)^2}} = \frac{1}{\sqrt{1 - 0{,}36}} = \frac{1}{\sqrt{0{,}64}} = \frac{1}{0{,}8} = 1{,}25 $$
The result $\gamma = 1{,}25$ tells us that time on board the spaceship runs slower compared to the laboratory frame.
We can now compute the proper velocity $\eta$ using the relation:
$$ \eta = \gamma v $$
Substituting the values:
$$ \eta = 1{,}25 \cdot 0{,}6c = 0{,}75c $$
The spaceship’s proper velocity is $ 0{,}75c $, which is greater than its ordinary velocity $ 0{,}6c $.
This means that for every second of its own proper time $\tau$, the ship covers $0{,}75c$ of distance. In the lab frame, however, it travels at $0{,}6c$ with respect to the observer’s time $t$.
So, in its own frame, the object covers more distance per unit of its own time than it appears to an external observer.
Note. This difference arises because proper time $\tau$ flows more slowly than coordinate time $t$. Therefore, for the same displacement $dx$, the proper velocity $\eta$ turns out to be greater than the ordinary velocity. Naturally, this doesn’t mean the object actually exceeds the speed of light, since the relativistic speed limit applies only to the ordinary velocity $ v < c $.
The Four-Velocity Vector
Proper velocity is part of the four-vector formalism - mathematical objects that unify space and time into a single, consistent framework. It’s far more than just a theoretical curiosity.
Specifically, proper velocity forms the spatial component of the four-velocity vector, expressed as:
$$ U^\mu = \left( \gamma c,\ \gamma \vec{v} \right) $$
where:
- $\mu = 0, 1, 2, 3$ is the four-dimensional index (time plus three spatial components);
- $\gamma \vec{v}$ represents the product of the Lorentz factor and the ordinary velocity;
- $\gamma c$ is the temporal component.
Invariance of the Norm
A key property of the four-velocity is that its norm (its “length” in the relativistic sense) is invariant for all observers:
$$ U^\mu U_\mu = -c^2 $$
In other words, this quantity remains the same in every inertial reference frame and is independent of the object’s velocity.
And so on.
