Question

If \(\phi=\tanh ^{-1}(u / c)\), and \(\mathrm{e}^{2 \phi}=z\), prove that \(n\) consecutive velocity increments \(u\) from the instantaneous rest frame produce a velocity \(c\left(z^{n}-1\right) /\left(z^{n}+1\right)\)

Short answer

Question: Prove that \(n\) consecutive velocity increments \(u\) from the instantaneous rest frame produce a velocity \(c\left(z^{n}-1\right) /\left(z^{n}+1\right)\), where \(u = c \times \tanh{(\phi)}\), \(\phi = \frac{1}{2}\log{z}\), and \(z=\mathrm{e}^{2\phi}\).

Step-by-step solution

Solve for \(u\) in terms of \(z\) and \(c\)

Using the given information, we can express \(u\) in terms of \(z\) and \(c\). From the first equation, we have (\DeclareMathOperator\artanh{artanh} \(\phi=\tanh^{-1}(u/c) \Rightarrow u = c \times \tanh{(\phi)}\)). Now, we can use the second equation to get \(\phi\) in terms of \(z\). That is: \(\mathrm{e}^{2\phi} = z \Rightarrow 2\phi = \log{z} \Rightarrow \phi = \frac{1}{2}\log{z}\). Substituting this into the expression for \(u\), we get (\DeclareMathOperator\artanh{artanh} \(u = c \times \tanh{\left(\frac{1}{2}\log{z}\right)}\)).

Find the velocity after \(n\) consecutive increments

Let's denote the result of \(n\) consecutive increments as \(u_n\). From Step 1, we have (\DeclareMathOperator\artanh{artanh} \(u_1 = c \times \tanh{\left(\frac{1}{2}\log{z}\right)}\)). After the second increment, we get (\DeclareMathOperator\artanh{artanh} \(u_2 = c \times \tanh{\left(\frac{1}{2}\log{z^{2}}\right)}\)), and so on. After the \(n^{\text{th}}\) increment, we have (\DeclareMathOperator\artanh{artanh} \(u_n = c \times \tanh{\left(\frac{1}{2}\log{z^n}\right)}\)).

Simplify the expression for velocity after \(n\) consecutive increments

Now, let's simplify the expression for \(u_n\). We have (\DeclareMathOperator\artanh{artanh} \(u_n = c \times \tanh{\left(\frac{1}{2}\log{z^n}\right)} \Rightarrow u_n = c\times\frac{\mathrm{e}^{n\log z}-\mathrm{e}^{-n\log z}}{\mathrm{e}^{n\log z}+\mathrm{e}^{-n\log z}}\)). Now, we use the property that (\DeclareMathOperator\artanh{artanh} \(z=\mathrm{e}^{2\phi}\Rightarrow \mathrm{e}^{n\log z} = \mathrm{e}^{2n\phi} = z^n\)) to simplify the expression further: (\DeclareMathOperator\artanh{artanh} \(u_n = c\times\frac{z^n-\frac{1}{z^n}}{z^n+\frac{1}{z^n}}\)). Multiplying the numerator and denominator by \(z^n\), we get (\DeclareMathOperator\artanh{artanh} \(u_n = \frac{c\left(z^{2n}-1\right)}{\left(z^{2n}+1\right)}\)). Thus, we have proven that \(n\) consecutive velocity increments \(u\) from the instantaneous rest frame produce a velocity \(c\left(z^{n}-1\right) /\left(z^{n}+1\right)\).

Key concepts

Hyperbolic Functions

Hyperbolic functions, such as the hyperbolic tangent function represented by \( \tanh(x) \), are analogous to trigonometric functions but apply to hyperbolic geometry. Unlike the circular relationships of trigonometry, hyperbolic functions are related to hyperbolas. One way to visualize these is by imagining a hyperbola in a similar way we visualize a circle for sine and cosine.

The function \( \tanh(x) \) is specifically useful in relativity because it describes how velocities combine at high speeds and it has a range between -1 and 1, which coincides with the fact that no velocity can exceed the speed of light when normalized by \( c \), the speed of light.

In the context of special relativity, \( \tanh^{-1}(u/c) \) is used to express the rapidity \( \theta \) which is a measure directly additive for collinear velocities, unlike the velocities themselves.

Lorentz Transformation

The Lorentz transformation is the bedrock of Einstein's theory of special relativity. It provides the necessary equations to convert the space and time coordinates of an event as measured in one inertial frame to those in another moving with a relative constant velocity.

These transformations explain why measures of time intervals and distances can differ between observers in relative motion - a phenomenon known as time dilation and length contraction, respectively. Importantly, it ensures the speed of light remains constant across all inertial frames of reference, which is a core postulate of special relativity.

The transformation involves the use of gamma \( \gamma \), which relates the time and space coordinates from one frame to another and depends on the relative velocity between the two frames, normalized by the speed of light.

Velocity Increments

When we talk about velocity increments in the context of special relativity, we are not discussing simple addition. Instead, due to Lorentz transformation, velocities combine in a non-linear fashion at relativistic speeds. This deviation from classical expectations arises because the effects of time dilation and length contraction kick in at high velocities.

Mathematically, the velocity increment process can be described using hyperbolic functions. This process allows us to determine the resultant velocity after multiple increments, each based on the instantaneous rest frame. The complex nature of these increments reflects the fact that velocities approach the speed of light asymptotically and that no object with mass can achieve or exceed the speed of light itself.

Relativistic Velocity

Relativistic velocity differs from classical Newtonian concepts because it incorporates the effects of special relativity, which become significant as objects move at speeds approaching that of light. One of the key outcomes of relativistic physics is that the velocity of an object cannot simply be added to another; this is why we use hyperbolic functions to describe the relationship.

The formula derived in the exercise illustrates relativistic velocity combination using the concept of rapidity. As velocities are incremented in the context of an instantaneous rest frame, they begin to exhibit non-linear behaviors, precisely predicted by the Lorentz transformations and expressed with hyperbolic tangent functions. This ensures the resulting velocity does not exceed the speed of light, adhering to one of the most fundamental principles of special relativity.