Consider motion in one spatial dimension. For any velocity...

Consider motion in one spatial dimension. For any velocity $v_{n}$ define the parameter $\theta$ via the relation $v=c \tanh \theta,$ where $c$ is the speed of light in vacuum. This quantity is called the velocity parameter or the rapidity corresponding to velocity $v$. a) Prove that for two velocities, which add via a Lorentz transformation, the corresponding velocity parameters simply add algebraically, that is, like Galilean velocities. b) Consider two reference frames in motion at speed $v$ in the $x$ -direction relative to one another, with axes parallel and origins coinciding when clocks at the origin in both frames read zero. Write the Lorentz transformation between the two coordinate systems entirely in terms of the velocity parameter corresponding to $v$ and the coordinates.

Consider motion in one spatial dimension. For any velocity $v_{n}$ define the parameter $\theta$ via the relation $v=c \tanh \theta,$ where $c$ is the speed of light in vacuum. This quantity is called the velocity parameter or the rapidity corresponding to velocity $v$.
a) Prove that for two velocities, which add via a Lorentz transformation, the corresponding velocity parameters simply add algebraically, that is, like Galilean velocities.
b) Consider two reference frames in motion at speed $v$ in the $x$ -direction relative to one another, with axes parallel and origins coinciding when clocks at the origin in both frames read zero. Write the Lorentz transformation between the two coordinate systems entirely in terms of the velocity parameter corresponding to $v$ and the coordinates.

Key Concepts

Rapidity

Rapidity is a parameter defined via the hyperbolic tangent function (v = c tanh?) that provides an alternative way to represent velocities in special relativity. Unlike velocities, rapidities add linearly when combining boosts along the same direction, which simplifies the analysis of successive Lorentz transformations and offers a natural parameterization of velocity space.

Hyperbolic Functions in Special Relativity

Hyperbolic functions, such as tanh, cosh, and sinh, play a critical role in special relativity by relating the velocity of an object to its corresponding Lorentz factor. The use of hyperbolic functions allows for a compact representation of Lorentz transformations and reveals the analogy between Lorentz boosts and rotations in a hyperbolic space, highlighting the geometric interpretation of relativistic kinematics.

Lorentz Transformation

The Lorentz transformation governs how time and space coordinates change between inertial reference frames moving at constant relative velocity. It incorporates the finite speed of light and thus ensures that the laws of physics, including the constancy of light speed, remain invariant across different frames. Expressing these transformations in terms of rapidity not only simplifies calculations but also brings out the underlying hyperbolic geometry.

Boost Composition

Boost composition refers to the process of combining two Lorentz boosts (i.e., transformations corresponding to different velocities) along the same direction. When expressed in rapidity, the composition becomes a simple algebraic addition of rapidities. This property significantly simplifies the derivation of the velocity addition formula and provides a clear geometric interpretation of the successive application of relativistic boosts.

Transcript

00:01 In this problem, we have a frame s, and frame s prime.

00:05 These are the velocities.

00:08 This v is relative to s.

00:10 U prime is relative to s prime.

00:12 You, we do not know.

00:14 I'm going to make it easy and have everything in the same direction.

00:16 We could get fancier.

00:18 But this is how you saw were introduced the velocity, addition to velocity formula.

00:25 Note, this is the formula when you're looking for the velocity relative to s.

00:31 Not s prime.

00:33 You may have seen the u prime equals form.

00:37 This is just the inverse of that.

00:40 Now, the purpose really of this problem is to understand, to investigate this, a new quantity that we're going to define and use.

00:48 And that's this quantity here, theta, the rapidity parameter.

00:53 This is how we define it.

00:55 And this is the hyperbolic tangent, not the trigonometric tangent, hyperbolic tangent.

01:03 So let's introduce it and see what it does for us.

01:06 So c, tan, h, theta, prime, plus c, and this is for any velocity you introduce, it's not just for the 1v, for the u's also, as you can see what i'm just did, tan h theta sub v over 1 plus c, tan h, theta sub v.

01:33 C, tan, h theta prime over c squared.

01:40 C squared goes away.

01:43 And so we can write this now, c, tan h, theta prime, plus tan, tan, h, theta, v over 1 plus tan h, theta v, times tan h, theta, now, just like with the tritometric functions, there are relationships, some of angles.

02:18 You know, cosine of a plus b, you may remember that one.

02:22 Same thing goes on here with this one, though, is for hyperbolic tangent.

02:29 And this becomes c hyperbolic tangent of theta prime plus theta v, nice and simple.

02:42 Now let's put in the form for you, because we want everything in terms of a rapidity parameter.

02:48 So that's going to be c, tan, h, theta, no prime on it, because u does not have a prime, is equal to c, tan h, theta, prime, plus theta v.

03:05 Well, that's interesting.

03:07 The c's go away.

03:08 This gives me a theta is equal to theta prime plus theta v.

03:14 Very interesting.

03:15 We have something that adds, additive quantity, and why did i write this over here, this galilean? because if this was a galilean, we were in the classical world, and we were looking for what you would be, that's relative to s, we would just take u prime and add v, what the s prime is doing.

03:41 We add.

03:43 Technically, in a more formal work, we call this type of quantity a boost.

03:51 Taking something you prime and then we're adding something to it, we're boosting it.

03:56 That is seen in relativity and more formal work.

03:59 This is a boost.

04:02 It's called a boost and stuff.

04:04 That's what happens when you transform between one frame to another.

04:10 In our case, we transform from s prime to s.

04:12 This is the boost.

04:14 But it's seen in the rapidity parameter.

04:19 It's this additive nature.

04:21 It's seen in the rapidity parameter, not.