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Lorentz Boosts

Previously, we discussed the flaws of Galilean relativity, which led to the need for a new theory of relativity—special relativity. In Galilean relativity, we had the following transformation between two inertial frames of reference, and :

where is the relative velocity between the two frames of reference. In special relativity, this transformation must be replaced with a new one, known as the Lorentz transformation. Strictly speaking, there are six Lorentz transformations, corresponding to the six generators of the Lorentz group. They comprise of three rotations and three boosts.

We will slowly build up a fundamental understanding of the Lorentz transformation. First, we will discuss a simple Lorentz boost in the direction. This is the simplest case, and we will later generalize it to arbitrary directions. Then, we will discuss the full Lorentz transformation, which is technically defined as a transformation that preserves the Minkowski metric. This is a more general transformation that includes both spatiotemporal boosts (hyperbolic rotations) and spatial rotations. This leads us to the full Lorentz group and its orthochronous subgroup . They are Lie groups with their corresponding Lie algebras and .

Table of Contents​

Introduction​

I will assume that the reader has already seen a derivation of the Lorentz boost. If not, see this video for a visual derivation. The derivation is based on the invariance of the speed of light, which is a postulate of special relativity.

In relativity, we define a few values to make the math easier;

  1. The beta parameter is defined as

    Its range is .

  2. The gamma parameter is defined as

    It is always greater than or equal to .

The boost is given by the following transformation:

where and are the transformed basis vectors, and and are the original basis vectors. The first equation is the transformation of the basis vectors, and the second equation is the transformation of the coordinates. As expected, the transformation matrices for the basis vectors and the coordinates are inverses of each other because basis vectors are covariant, while coordinates are contravariant.

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Time Dilation/Length Contraction​

The Lorentz boost can be used to derive the time dilation and length contraction formulas. These are important to understand the physical implications of the Lorentz boost.

Time dilation is a difference in the time coordinate between two events as measured by two observers. Suppose we have two events, and , that occur at the same location in the frame, but a time difference . Now suppose moves with velocity with respect to .

Since is in both frames, we just need to find the coordinates of in the frame. In , the coordinates of are . Then, by applying the Lorentz boost, we have:

In other words, the time changes as . This means that the time interval between two events is longer in the frame than in the frame (since ).

Note that it is not a contradiction that the time interval is longer in the frame than in the frame. It might seem counterintuitive because both and are true. However, in the two cases we are measuring different things;

  1. In the first case, we are measuring the time interval between two events that occur at the same location in the frame.
  2. In the second case, we are measuring the time interval between two events that occur at the same location in the frame.

The value is measured along a worldline which follows the events (i.e. both events occur on the worldline). The shortest time is always measured along the worldline of the observer. The mnemonic to remember is that "moving clocks run slow".

Next, instead of two events at the same location, we can have two events at the same time (perhaps representing two points on a bar), leading to the concept of length contraction. Suppose we have two events, and , that occur at the same time in the frame, but a spatial difference . Now suppose moves with velocity with respect to .

Note that we cannot simply measure the time component of the events in the frame. We need to trace the lines of simultaneity in both frames and find the coordinates of these events:

Lines of Simultaneity in frame

Lines of Simultaneity in frame

In other words, we need to measure different vectors in the frames. For the frame, we measure the vector , which gives the length by definition.

For the frame, we measure the vector . We can make the following observations from the diagram:

  1. only has a time component in , so we can write it as

  2. While has both time and space components in , it only has a time component in . This is by definition (since it is a line of simultaneity). We can thus write

Substituting these into the first equation, we have:

Using the transformation , we can write the equation as:

The -component of both sides must be equal. Since has no -component in , we can write:

Or equivalently,

This means that the length of the object is shorter in the frame than in the frame (since ). The mnemonic to remember is that "moving objects appear shorter".

Hyperbolic Rotations​

The Lorentz boost can be interpreted as a hyperbolic rotation in spacetime.

Recall that a unit circle in the plane can be represented as an implicit equation:

To traverse the circle, we use a circular angle and parametrize the circle with circular trigonometric functions:

On the other hand, a unit hyperbola in the plane can be represented as

To traverse the hyperbola, we use a hyperbolic angle and parametrize the hyperbola with hyperbolic trigonometric functions:

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The hyperbolic angle is not the same as the circular angle. It is defined as the angle between the -axis and the line connecting the origin to the point on the hyperbola.

Suppose a square is placed on top of the plane, with sides placed on the hyperbola. As the sides moves along the hyperbola, one side of the square will shrink, while the other side will grow. This can be expressed as a hyperbolic rotation matrix:

As one would expect from a rotation matrix, its inverse can be obtained by simply flipping the sign of the hyperbolic angle. Its determinant is , which means that it preserves the area of the square. To combine two hyperbolic rotations, we can simply add the angles:

(This can be proven using the hyperbolic addition formulas, which are similar to the regular circular trigonometric identities.)

Notice that this hyperbolic rotation is exactly what a Lorentz boost looks like. In fact, we can make them equivalent with the following:

If then

Proof. The proof is quite simple. The top-left and bottom-right elements are obviously equal (since ). For the other elements, we can first write in terms of and equate with the hyperbolic functions:

This means that

We can then use the identity to equate the two:

Lastly, multiplying this by gives us the diagonal elements:

This means that the Lorentz boost is equivalent to a hyperbolic rotation in spacetime.

To see why we prefer hyperbolic rotations over regular Lorentz boosts, we will prove an important theorem.

Addition of Velocities​

The addition of velocities is a very important concept in special relativity. Suppose a rocket is moving with beta with respect to the (ground) frame. The rocket fires a projectile with beta with respect to the rocket frame. We want to find the beta of the projectile with respect to the frame.

First, let's define the notation:

  • are the coordinates in the ground frame.
  • are the coordinates in the rocket's frame.
  • are the coordinates in the projectile's frame.

First, the transformation from the ground frame to the rocket's frame is given by:

where . The transformation from the rocket's frame to the projectile's frame is given by:

where . Plugging Equation into Equation gives us:

The factor can be expanded by using the definition of :

Combining this with the transformation matrix gives us:

If we define , we can write the transformation as:

As such, we can see that the transformation is equivalent to a Lorentz boost with beta .

Now, let's try to prove this using hyperbolic rotations. We can rewrite the transformation from the ground frame to the rocket's frame as:

where . We can then rewrite the transformation from the rocket's frame to the projectile's frame as:

We can just plug in the first equation into the second equation and use Equation to get:

That's it.

Additionally, Recall that for hyperbolic rotations is . Thus, the new beta is . If we equate the two expressions for , we have a neat identity for the hyperbolic tangent:

Corollary: The Speed Limit​

For any given and where and , we have:

Proof. Since , we have .

Multiplying both sides of by gives us:

Next, adding to both sides gives us:

Finally, dividing both sides by gives us:

The proof is even easier if we use hyperbolic functions. Since , and is always within the range , we automatically have .

The physical implication of this is that no object can possess a speed greater than or equal to the speed of light.

Summary and Next Steps​

In this section, we discussed the Lorentz boost and its implications.

Here are the key things to remember:

  • The Lorentz transformation is a transformation between two inertial frames of reference. It consists of a boost and a rotation.

  • The Lorentz boost is a transformation between two inertial frames of reference that are moving with respect to each other.

  • The beta parameter is defined as the ratio of the velocity of the frame to the speed of light.

  • The gamma parameter is defined as

  • The Lorentz boost, for coordinates, can be written two ways:

    and

    where .

  • No object can possess a speed greater than or equal to the speed of light.

In the next section, we will introduce Minkowski geometry.