Question

Show that momentum and energy transform from one inertial frame to another as \(p_{x}^{\prime}=\gamma\left(p_{x}-v E / c^{2}\right) ; p_{y}^{\prime}=p_{y} ; p_{z}^{\prime}=p_{z} ; E^{\prime}=\gamma\left(E-v p_{x}\right)\).

Short answer

Answer: By using the Lorentz transformation equations and the definition of four-momentum, we find the transformation equations for the momentum and energy components as follows: \(p_{x}^{\prime}=\gamma\left(p_{x}-v E / c^{2}\right)\) \(p_{y}^{\prime}=p_{y}\) \(p_{z}^{\prime}=p_{z}\) \(E^{\prime}=\gamma\left(E-v p_{x}\right)\) where \(v\) is the relative velocity between the reference frames, \(c\) is the speed of light, and \(\gamma\) is the Lorentz factor given by: \(\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}\).

Step-by-step solution

Review the Lorentz transformation equations for space and time coordinates

In special relativity, we use the Lorentz transformation equations to relate the position and time coordinates (x, y, z, t) in one inertial frame (unprimed) to the position and time coordinates (x', y', z', t') in another inertial frame moving at a constant velocity relative to the first frame. The relevant Lorentz transformation equations are given by: \(x^{\prime}=\gamma(x-vt)\) \(t^{\prime}=\gamma\left(t-\frac{vx}{c^{2}}\right)\) \(y^{\prime}=y\) \(z^{\prime}=z\) where \(v\) is the relative velocity between the reference frames, \(c\) is the speed of light, and \(\gamma\) is the Lorentz factor given by: \(\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}\).

Apply the definition of energy and momentum in terms of space and time coordinates

By definition, the four-momentum is given by: \((E/c, p_{x}, p_{y}, p_{z})\) where E is the energy and \(p_{x}\), \(p_{y}\), \(p_{z}\) are the components of momentum in the unprimed frame. In the primed frame, these four-momentum components can be denoted as: \((E^{\prime}/c, p_{x}^{\prime}, p_{y}^{\prime}, p_{z}^{\prime})\) Both sets of these four-momentum components transform under the same Lorentz transformations as the space and time coordinates \((x, y, z, t)\) and \((x', y', z', t')\).

Combine the Lorentz transformation equations and the definition of four-momentum

We now use the Lorentz transformation equations for the space and time coordinates and the definition of four-momentum to write the transformation equations for the momentum and energy components. \(p_{x}^{\prime}=\gamma\left(p_{x}-v E / c^{2}\right)\) \(p_{y}^{\prime}=p_{y}\) \(p_{z}^{\prime}=p_{z}\) \(E^{\prime}=\gamma\left(E-v p_{x}\right)\) This completes the proof of the transformation equations between the momentum and energy components of two inertial frames in special relativity.

Key concepts

Lorentz Transformation

The Lorentz transformation is a fundamental concept in special relativity that describes how measurements of space and time by two observers moving at a constant velocity relative to each other are related. These equations take into account the constant speed of light, denoted by \( c \), and the relative velocity between observers, denoted by \( v \).

• Space and time coordinates are not absolute but relative to the motion of the observer.
• Time dilation and length contraction are natural consequences of these transformations.
• They ensure that the speed of light remains constant in all inertial frames of reference.

The Lorentz factor, \( \gamma \), is pivotal in these transformations, accounting for the effects of relative motion on measured time intervals and spatial distances. This effect becomes significant as the relative velocity approaches the speed of light.

Special Relativity

Special relativity revolutionized physics by introducing new concepts about space and time. Its fundamental postulates—constancy of the speed of light in a vacuum for all observers and the equivalence of physical laws in all inertial frames—led to a deeper understanding of how motion is perceived.

• Mass increases with velocity and approaches infinity as one approaches the speed of light.
• Time measured by an observer in motion relative to a stationary observer will appear to pass slower, a phenomenon known as time dilation.
• Simultaneity of events becomes relative, depending on the observer's state of motion.

Special relativity has far-reaching implications not just on cosmological scales but also in technologies such as GPS, which needs to correct for relativistic effects to maintain accuracy.

Four-Momentum

In the realm of special relativity, four-momentum extends the classical concept of three-dimensional momentum to include energy as a fourth component. This unites energy and momentum into a single four-vector, which is conserved in isolated systems.

• The four components are \( E/c \), \( p_x \), \( p_y \), and \( p_z \), where \( E \) represents energy and the others represent spatial momentum components.
• The conservation of four-momentum is analogous to the conservation of energy and momentum in classical mechanics, but it applies across reference frames moving at relativistic speeds.
• Four-momentum is a covariant vector, meaning it transforms predictably between inertial reference frames using the Lorentz transformation.

Inertial Reference Frames

An inertial reference frame is a frame of reference in which an object either is at rest or moves at a constant velocity—meaning no net force is acting upon it. The laws of physics, including Newton's laws of motion, hold true in all inertial frames.

In special relativity, inertial reference frames are crucial because:

• They are the