The motion of a particle in four-dimensional spacetime is described by its worldline—a smooth curve that traces the particle’s history through spacetime. This curve is naturally parametrized by the particle’s proper time , which is the time measured by a clock moving with the particle. In other words, is the particle’s own internal time and is invariant under Lorentz transformations; all observers agree on the value of proper time along the worldline.
At each moment of its journey, the particle’s spacetime position is described by the four-position vector:
Here, is a contravariant four-vector, with indexing the time and spatial coordinates. The zeroth component ensures that all components of the vector have consistent units of length.
Now consider two distinct events in spacetime: event at and event at . The displacement four-vector between these events is defined as:
A key scalar quantity constructed from this displacement vector is the spacetime interval (or line element). It is defined as the Minkowski inner product of the four-vector with itself:
The specific sign convention depends on the metric signature adopted (commonly or ). Regardless of convention, the crucial fact is that is invariant under Lorentz transformations—its value remains the same in all inertial frames. This invariance lies at the heart of relativistic physics and ensures consistent descriptions of causality and simultaneity.
For a particle traveling along its worldline, the infinitesimal spacetime interval between two nearby events reflects the physical separation in both space and time between those events. Since the particle is always located at its own position in its rest frame, the only coordinate that changes in it's rest frame must be time (i.e no space coordinates change), which is it's proper time, denoted . Thus, the proper time differential is directly related to the spacetime interval via the metric:
Alternatively, from time dilation (see Time Dilation, we can relate the proper time to the coordinate time in a given inertial frame :
where is the Lorentz factor, and is the particle’s instantaneous velocity in frame .
Thus, proper time provides a natural clock along the worldline, connecting spacetime geometry to measurable quantities experienced by the particle itself.
An important quantity in relativistic mechanics is the proper velocity .
In classical mechanics, velocity is defined as:
where both displacement and time interval are measured in the same inertial frame .
In contrast, proper velocity is defined as:
This is a hybrid quantity:
Proper velocity is a 4-vector tangent to the particle’s worldline with components
We can compute its length via
Since is constant and independent of the reference frame, the magnitude of proper velocity is Lorentz invariant.
In relativistic mechanics, conservation of momentum holds only when using the relativistic energy-momentum 4-vector, defined by:
Here:
Let us compute the Lorentz-invariant scalar :
Since , the length of the 4-velocity squared, is , this becomes
Alternatively, writing it explicitly gives
which can be rearranging to give the famous energy-momentum relation
This relation holds for all particles, regardless of their velocity.
In relativistic mechanics, the center-of-momentum (COM) frame is the inertial frame in which the total three-momentum of a system of particles is zero:
This generalizes the classical idea of the center-of-mass frame, but it is better suited to relativistic contexts where mass and energy are frame-dependent.
The COM frame plays a central role in:
In the COM frame, since total momentum vanishes, the total energy is minimized for a given system and is equal to the invariant mass times :
For any isolated system of particles, the invariant mass is defined via the total energy-momentum four-vector:
Then the invariant mass is:
This scalar quantity is Lorentz invariant and has a direct physical meaning: in the COM frame, it equals the total energy divided by .
The COM frame is not unique to bound systems—it can be defined for any set of particles, even if they are free. It is often the most natural frame for studying particle interactions, such as in collider physics, where initial particles are arranged to collide head-on in their COM frame to maximize available energy for new particles.
Just as the four-velocity describes how a particle moves through spacetime, the four-acceleration describes how its four-velocity changes along its worldline. It is defined as the derivative of the four-velocity with respect to proper time :
This is the natural generalization of classical acceleration, but now formulated in a covariant way that respects the structure of spacetime.
Four-acceleration is a four-vector, so it transforms properly under Lorentz transformations. Additionally, since the norm of the four-velocity is constant:
its derivative must vanish
Expanding the left hand side
implies
In words: the four-acceleration is always orthogonal to the four-velocity. This is a purely relativistic result with no Newtonian analogue, and it reflects the fact that the "speed through spacetime" is constant.
We now introduce the relativistic generalization of force: the 4-force.
In Newtonian mechanics, force is the time derivative of momentum:
In relativity, the 4-force is defined as the derivative of the 4-momentum with respect to proper time :
Explicitly:
is a 4-vector, so it transforms properly under Lorentz transformations. It is also always orthogonal to the 4-velocity
since , as shown above.
In the limit where a particle’s speed is much smaller than the speed of light (), relativistic dynamics must reduce to classical Newtonian mechanics. This correspondence is not only a useful check—it also offers deep insight into the structure of spacetime.
Recall the spatial part of the 4-momentum:
When , the Lorentz factor expands as:
Thus:
The correction term becomes negligible at low speeds, and we recover the classical definition of momentum:
Similarly, the temporal component of the 4-momentum gives the relativistic energy:
This shows that total energy includes both the particle’s rest energy and its classical kinetic energy. The rest energy dominates unless the particle is moving at relativistic speeds.
Likewise, the spatial part of the 4-force:
reduces to Newton’s second law when :
These limiting behaviors confirm that the relativistic formulation is a true generalization: it agrees with classical mechanics in the regime where classical mechanics is valid, but extends smoothly to the high-speed domain where relativistic effects become essential.
A particle of rest mass moves in the lab frame with velocity along the -axis. It collides head-on and sticks to an identical particle that was initially at rest. After the collision, the composite object moves together as one particle.
(a) Compute the total 4-momentum before and after the collision.
(b) Use this to find the rest mass of the composite system.
(c) Explain physically why the final rest mass is greater than . Where does the extra mass come from?
Step 1: Compute total 4-momentum
Total 4-momentum before collision:
Step 2: Use invariant
Compute:
Factor and simplify:
Recall , so:
Simplify:
So: