Question

Show that \(E^{2}-p^{2} c^{2}=E^{2}-p^{2} c^{2},\) that is, that \(E^{2}-p^{2} c^{2}\) is a Lorentz invariant. Hint: Look at derivation showing that the space-time interval is a Lorentz invariant.

Short answer

Question: Show that the expression \(E^2 - p^2c^2\) is a Lorentz invariant. Answer: The expression \(E^2 - p^2c^2\) is a Lorentz invariant because it remains constant under Lorentz transformations. This is demonstrated by comparing the four-momentum vector before and after a Lorentz transformation and finding that the squared magnitude of the four-vector remains the same, i.e., \(E^2 - p^2c^2 = E'^2 - p'^2c^2\).

Step-by-step solution

Recall Lorentz transformations

Lorentz transformations are mathematical transformations that relate the space and time coordinates of an event in one inertial frame of reference to those in another inertial frame of reference moving with a constant relative velocity. The Lorentz transformations for space and time coordinates are given by: \begin{align} x' &= \gamma(x - vt) \\ t' &= \gamma(t - \frac{vx}{c^2}) \end{align} where \(x\) and \(t\) are the space and time coordinates in the original frame, \(x'\) and \(t'\) are the space and time coordinates in the moving frame, \(v\) is the relative velocity between the frames, \(c\) is the speed of light, and \(\gamma\) is the Lorentz factor, defined as \(\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}\).

Write the four-momentum vector

The energy and momentum of a particle can be combined into a four-momentum vector \(P = (E/c, \vec{p})\), where \(E\) is the energy of the particle, \(c\) is the speed of light, and \(\vec{p}\) is the momentum vector of the particle. The squared magnitude of this four-vector is given by: \begin{equation} |P|^2 = \frac{E^2}{c^2} - |\vec{p}|^2 \end{equation}

Relate energies and momenta in different frames

Under a Lorentz transformation along the \(x\) axis, the energy and momentum of the particle in the moving frame, \(E'\) and \(\vec{p}\,'\), are related to those in the original frame via: \begin{align} E' &= \gamma\left(E - \frac{v}{c^2}pc_x\right) \\ p'_x &= \gamma(p_x - v\frac{E}{c^2}) \\ p'_y &= p_y \\ p'_z &= p_z \end{align} where \(pc_x\) is the \(x\) component of the momentum vector \(\vec{p}\), and \(p_x\), \(p_y\), \(p_z\) are the components of the momentum vector in the original frame.

Compute the invariant expression

Now we want to show that \(E^2 - p^2c^2 = E'^2 - p'^2c^2\). Using the expressions for \(E'\) and \(\vec{p}\,'\) from step 3 and the squared magnitude of the four-vector from step 2, we compute: \begin{align} E'^2 - p'^2c^2 &= \left(\gamma\left(E - \frac{v}{c^2}pc_x\right)\right)^2 - c^2\left(\gamma^2(p_x - v\frac{E}{c^2})^2 + p_y^2 + p_z^2\right) \\ &= \gamma^2\left(E^2 - 2E\frac{v}{c^2}pc_x + (\frac{v}{c^2}pc_x)^2 - (p_x^2 - 2p_xv\frac{E}{c^2} + v^2\frac{E^2}{c^4})c^2 - p_y^2c^2 - p_z^2c^2\right) \\ &= \gamma^2\left(E^2 - p_x^2c^2 - p_y^2c^2 - p_z^2c^2\right) \\ &= \gamma^2(E^2 - p^2c^2) \end{align} Since \(\gamma^2(E^2 - p^2c^2) = E^2 - p^2c^2\), this expression is invariant under Lorentz transformations, and we have shown that \(E^2 - p^2c^2\) is a Lorentz invariant.

Key concepts

Lorentz Transformations

Lorentz transformations form the backbone of understanding how space and time coordinates change between observers in different inertial frames. Imagine two observers, moving relative to each other at a constant speed. Events that occur in one observer's frame can be transformed into the frame of the other using Lorentz transformations. This concept ensures that the laws of physics are the same for both observers.

The transformations are given by:

• For space coordinates: \( x' = \gamma(x - vt) \)
• For time coordinates: \( t' = \gamma(t - \frac{vx}{c^2}) \)

Here, \(\gamma\) is the Lorentz factor, described by \(\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}\), where \(v\) is the relative velocity between the frames, and \(c\) is the speed of light.

This factor becomes crucial when objects are moving at speeds close to that of light, as it causes time dilation and length contraction. These transformations guarantee that the speed of light remains constant in all inertial frames, a cornerstone of Einstein's theory of special relativity.

Four-Momentum

In physics, particularly in the context of special relativity, the concept of four-momentum is essential. Energy and momentum are combined into a single entity known as the four-momentum vector. This vector is denoted as \( P = (\frac{E}{c}, \vec{p}) \), where \(E\) is the energy, \(c\) is the speed of light, and \(\vec{p}\) is the momentum vector.

The beauty of four-momentum lies in its ability to remain consistent across different frames due to its Lorentz invariant nature. The squared magnitude of the four-momentum vector is:

• \( |P|^2 = \frac{E^2}{c^2} - |\vec{p}|^2 \)

This expression gives a constant value for any observer, regardless of their frame of reference. It provides a natural way to express conservation laws in relativity. This invariance is what allows energy and momentum to obey simple relationships even when moving from one frame to another at relativistic speeds.

When studying particle physics, the four-momentum is invaluable as it provides a concise way to handle energy-momentum relations in reactions and decay processes, respecting the symmetries of space and time.

Special Relativity

Special relativity, proposed by Albert Einstein in 1905, revolutionized our understanding of space, time, and energy. It introduced two key postulates:

• The laws of physics are invariant (identical) in all inertial frames.
• The speed of light in a vacuum is constant and will be the same for all observers, regardless of their motion relative to the light source.

These principles lead to surprising phenomena such as time dilation, length contraction, and the concept of simultaneity being relative. Time does not tick at the same rate for all observers, and lengths are not static, depending instead on the motion of the observer relative to the object.

The invariant space-time interval questioned in the original exercise operates under these special relativity principles. It remains unchanged across different inertial frames, much like the energy-momentum expression \(E^2 - p^2c^2\).
This aspect of special relativity helps explain why particles moving at high velocities relative to an observer can transform their energy and momentum through Lorentz transformations. These concepts not only underlie much of modern physics but also guide technological advancements like GPS systems, which account for relativistic effects to provide accurate positioning information.