Question

A particle with four-momentum \(P\) is observed by an observer who moves with four-velocity U \(_{0}\). Prove that the energy of the particle relative to that observer is \(\mathbf{U}_{0} \cdot \mathbf{P}\).

Short answer

Question: Prove that the energy of a particle with four-momentum P relative to an observer with four-velocity U\(_{0}\) is given by \(U_{0} \cdot P\). Answer: To prove this, we can compute the dot product of U\(_{0}\) and P using the metric tensor for flat spacetime, which results in \(\gamma_{0}E - \gamma_{0}\vec{v}_{0} \cdot \vec{p}\). This expression represents the energy of the particle relative to the observer. Therefore, the energy of the particle relative to the observer is given by \(U_{0} \cdot P\).

Step-by-step solution

Understanding Four-momentum and Four-velocity

Four-momentum is the relativistic generalization of the classical momentum. The four-momentum of a particle is represented by the symbol P and has four components \((E/c, \vec{p})\), where E is the energy and \(\vec{p}\) is the spatial momentum vector. Similarly, the four-velocity is the relativistic generalization of classical velocity. The four-velocity of an observer is represented by the symbol U\(_{0}\) and has four components \((\gamma_{0}c, \gamma_{0}\vec{v}_{0})\), where \(\gamma_{0}\) is the Lorentz factor for the observer and \(\vec{v}_{0}\) is its spatial velocity.

Use Metric for Flat Spacetime

In flat spacetime, we can use the metric to calculate the dot product between the four-velocity and four-momentum. The metric for flat spacetime is given by: \[g_{\mu\nu} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{pmatrix}\]

Calculate the Dot Product

Using the metric tensor, we can calculate the dot product of U\(_{0}\) and P: \[(U_{0} \cdot P) = g_{\mu \nu} U_{0}^{\mu} P^{\nu}\] Substituting the components of U\(_{0}\) and P into the equation, we have: \[(U_{0} \cdot P) = g_{\mu \nu} (\gamma_{0} c, \gamma_{0} \vec{v}_{0})^{\mu} (E/c, \vec{p})^{\nu}\]

Compute the Energy

Expanding the dot product and summing over the repeated indices, we get: \[(U_{0} \cdot P) = (\gamma_{0}c)(E/c)g_{00} + (\gamma_{0}\vec{v}_{0})\cdot(\vec{p}) g_{ij}\] Using the metric tensor components, we simplify the expression to: \[(U_{0} \cdot P) = \gamma_{0}E - \gamma_{0}\vec{v}_{0} \cdot \vec{p}\] The left-hand side of this equation, \((U_{0} \cdot P)\), represents the energy of the particle relative to the observer. The right-hand side is the energy-momentum tensor contraction, which also represents the energy of the particle relative to the observer. Therefore, we have proved that the energy of the particle relative to the observer is \(\mathbf{U}_{0} \cdot \mathbf{P}\).

Key concepts

Four-momentum

In the realm of special relativity, the concept of four-momentum represents the relativistic counterpart of classical momentum. It is a four-vector, designated as \(P\), which encompasses both the energy and momentum of an object. The four-momentum is described by the components \((E/c, \vec{p})\), where \(E\) is the energy, \(\vec{p}\) is the spatial momentum vector, and \(c\) is the speed of light. This formulation ensures that the laws of physics maintain the same form across different inertial frames and can be viewed as a natural extension of the traditional momentum we encounter in Newtonian mechanics.

The inclusion of the energy component is particularly significant because it aligns momentum with the principles of relativity, where energy and mass are related through Einstein's famous equation \(E=mc^2\). Here, the first component \(E/c\) represents the temporal part (time-like), while \(\vec{p}\) constitutes the space-like components of the four-momentum. Understanding four-momentum is crucial, particularly when evaluating how particles interact and propagate through space-time.

Four-velocity

Just as four-momentum is an extension of classical momentum, four-velocity extends the concept of traditional velocity into the realm of relativity. Four-velocity, denoted as \(U_0\), is particularly important because it helps describe the motion of an observer or a particle in space-time.

The four-velocity is given by the components \((\gamma_0 c, \gamma_0 \vec{v}_0)\), where \(\gamma_0\) is the Lorentz factor and \(\vec{v}_0\) is the spatial velocity of the observer or particle. The Lorentz factor \(\gamma_0 = \frac{1}{\sqrt{1-(v_0^2/c^2)}}\) adjusts for the effects of time dilation and length contraction experienced at high velocities. This blend of temporal and spatial components ensures that the four-velocity remains a unitary four-vector. Understanding the conversion and implications of four-velocity prepares students to comprehend how motion is perceived differently by observers moving at varying speeds.

Metric Tensor

In relativity, the metric tensor plays an integral role in calculating distances and angles between points in spacetime. It is essential in defining geometrical and physical properties of spacetime under investigation. The metric tensor for flat Minkowski spacetime is expressed as:

\[g_{\muu} = \begin{pmatrix} 1 & 0 & 0 & 0 \ 0 & -1 & 0 & 0 \ 0 & 0 & -1 & 0 \ 0 & 0 & 0 & -1 \end{pmatrix}\]

This 4x4 matrix serves as a metric signature of \((+,-,-,-)\), which means that the time component has a positive sign, while the space components are negative. The metric tensor acts as a tool to perform operations like dot products in spacetime, transforming between different frames, and maintaining invariance across the laws of physics.

By incorporating metric tensors, one can compute the lengths of curves, angles between vectors, and transitions between coordinate systems, making it indispensable for navigating the relativity landscape.

Relativistic Energy

Relativistic energy takes the concept of energy and embeds it into the framework of relativity. Unlike classical mechanics, where energy remains separate from momentum, relativistic energy indicates the equivalence and interplay between mass and energy. It helps assess how energy changes due to motion, specifically when particles approach the speed of light.

In the context of a moving particle observed by someone in a different frame, relativistic energy is the component of four-momentum associated with time. Through the equivalence \(E=\gamma mc^2\), it's clear that even a particle at rest has inherent energy due to its mass.

This energy shifts when observed from different frames, influencing calculations such as energy conservation and particle interactions in relativistic contexts. Relativistic energy is pivotal for understanding phenomena ranging from particle collisions in accelerators to the dynamics of stars and galaxies.

Dot Product in Spacetime

To uncover how different frames perceive motion and energy, we employ the dot product of four-vectors in spacetime. This operation connects four-momentum \(P\) and four-velocity \(U_0\) to calculate relativistic phenomena like observed energy. The dot product in spacetime is calculated as:

\[(U_0 \cdot P) = g_{\mu u} U_0^{\mu} P^{u}\]

To sum over repeated indices means that the metric tensor \(g_{\mu u}\) interplays between the time and space components, enabling us to encapsulate their effects within a singular formula. The result, \(U_0 \cdot P\), gives the energy of a particle relative to an observer, perfectly aligning with observational outcomes. This equation is vital in physics for explaining how quantities remain invariant across different frames. By grasping this operation, one nurtures deeper insights into how movement and energy manifest in the universe, independent of observational perspective.