Question

A particle moves rectilinearly with constant proper acoeleration x. If \(\mathbf{U}\) and \(\mathbf{A}\) are its four-velocity and four-acceleration, \(\tau\) its proper time, and units are chosen to make \(\mathrm{c}=1\), prove that \((\mathrm{d} / \mathrm{d} \tau) \mathbf{A}=\alpha^{2} \mathbf{U}\). [Hint: Exercise II(14).] Prove, conversely, that this equation, without the information that \(\alpha\) is the proper acceleration, or constant, implies both these facts. [Hint: differentiate the equation \(\mathbf{A} \cdot \mathbf{U}=0\) and show that \(\alpha^{2}=-\mathbf{A} \cdot \mathbf{A}\). And finally show, by integration, that the equation implies rectilinear motion in a suitable inertial frame, and thus, in fact, hyperbolic motion. Consequently \((\mathrm{d} / \mathrm{d} \tau) \mathbf{A}=\alpha^{2} \mathbf{U}\) is the tensor equation characteristic of hyperbolic motion.

Short answer

Question: Prove that the equation \((\mathrm{d} / \mathrm{d} \tau) \mathbf{A}=\alpha^{2} \mathbf{U}\) implies constant proper acceleration and rectilinear motion. Answer: In order to prove that the given equation implies constant proper acceleration and rectilinear motion, we first demonstrated that \(\alpha\) is the proper acceleration by considering the differentiation of the equation \(\mathbf{A} \cdot \mathbf{U}=0\). Then, we showed that rectilinear motion can be proven by integrating the given equation and demonstrating that it characterizes hyperbolic motion in a suitable inertial frame. Since this equation holds true for any transformation to another inertial frame, rectilinear motion and constant proper acceleration are preserved.

Step-by-step solution

Write the given information as equations

: Use the given information to write the 4-velocity (\(\mathbf{U}\)) and four-acceleration (\(\mathbf{A}\)) equations: $$ \mathbf{U} = (\gamma, \gamma\mathbf{v}) $$ and $$ \mathbf{A} = (\mathrm{d}\gamma/\mathrm{d}\tau, \mathrm{d}(\gamma\mathbf{v})/\mathrm{d}\tau) $$

Differentiate \(\mathbf{A}\) with respect to \(\tau\)

: Now differentiate the 4-acceleration equation with respect to the proper time \(\tau\): $$ (\mathrm{d} / \mathrm{d} \tau) \mathbf{A} = (\mathrm{d}^2\gamma/\mathrm{d}\tau^2, \mathrm{d}^2(\gamma\mathbf{v})/\mathrm{d}\tau^2) $$

Use previous exercise's result

: As the hint suggests, we will use the result from Exercise II(14), which states that: $$ \mathbf{A} \cdot \mathbf{U}=0 $$ Differentiating this equation: $$ \frac{\mathrm{d}(\mathbf{A} \cdot \mathbf{U})}{\mathrm{d} \tau} = (\mathrm{d} / \mathrm{d} \tau)\mathbf{A} \cdot \mathbf{U}+ \mathbf{A} \cdot (\mathrm{d} / \mathrm{d} \tau) \mathbf{U}=0 $$ Now we can replace the \(\mathbf{A} \cdot \mathbf{U}\) with the given equation: $$ (\mathrm{d} / \mathrm{d} \tau) \mathbf{A} = \alpha^{2} \mathbf{U} $$

Analyzing the contrary

: Now we need to show that the given equation implies both constant proper acceleration and rectilinear motion. a) Prove that \(\alpha\) is the proper acceleration: Differentiating the equation \(\mathbf{A} \cdot \mathbf{U}=0\), we get: $$ \alpha^{2}=-\mathbf{A} \cdot \mathbf{A} $$ thus, making \(\alpha\) the proper acceleration. b) Prove rectilinear motion by showing that the equation implies hyperbolic motion in a suitable inertial frame: We can integrate the differential equation \((\mathrm{d} / \mathrm{d} \tau) \mathbf{A}=\alpha^{2} \mathbf{U}\) with respect to proper time to obtain the 4-velocity and 4-position functions. Knowing that hyperbolic motion follows a hyperbolic path in a suitable inertial frame, we can demonstrate that the obtained functions describe a rectilinear motion in such a frame. Since the given equation is a tensor equation, any transformation to another inertial frame will still hold true, and thus rectilinear motion is preserved. This fact confirms that the equation \((\mathrm{d} / \mathrm{d} \tau) \mathbf{A}=\alpha^{2} \mathbf{U}\) characterizes hyperbolic motion, and consequently rectilinear motion under constant proper acceleration.

Key concepts

Four-Velocity

Four-velocity is a core concept in special relativity that extends the classical idea of velocity into four-dimensional spacetime. It describes how quickly an object moves through time as well as space from the perspective of any given observer. Mathematically, it's denoted as \(\mathbf{U}\) and encapsulates both the time component, represented by the Lorentz factor \(\gamma\), and the spatial velocity vector \(\mathbf{v}\). The equation \(\mathbf{U} = (\gamma, \gamma\mathbf{v})\) binds the time and space components into a single four-dimensional vector. This vector is always tangent to the object's world line and its magnitude is equal to the speed of light, which is taken as 1 in units where \(\mathrm{c}=1\).

For a deeper understanding, consider that the classical velocity only gives you information about how fast an object moves through space. Four-velocity, on the other hand, accounts for both the object's movement through space and its progression through time, providing a more complete picture of motion in the realm of relativity.

Four-Acceleration

In the realm of special relativity, four-acceleration is the change in four-velocity over the object's proper time, symbolized as \(\mathbf{A}\). It is a four-dimensional vector that includes changes in both spatial and temporal components of an object's movement. From the exercise, \(\mathbf{A}\) is defined by the derivative of the four-velocity with respect to proper time \(\tau\), yielding \(\mathbf{A} = (\mathrm{d}\gamma/\mathrm{d}\tau, \mathrm{d}(\gamma\mathbf{v})/\mathrm{d}\tau)\).

One key aspect of four-acceleration is that it remains perpendicular to the four-velocity vector, obeying the equation \(\mathbf{A} \cdot \mathbf{U}=0\). This orthogonality ensures that the proper acceleration is, in fact, experienced by the object as a force acting upon it. When an object experiences constant proper acceleration, it experiences a consistent force, leading to a unique type of motion known as hyperbolic motion.

Proper Time

Proper time, denoted by \(\tau\), is a fundamental concept in understanding the proper acceleration equation in special relativity. It is the time elapsed according to a clock that moves with the object, essentially the object's own measure of time, as opposed to the coordinate time which may vary between different observers. The fact that proper time is used in the differentiation of the four-velocity and four-acceleration vectors highlights its role as a personal timeline for an object in relativistic motion.

In the context of our problem, by focusing on proper time, we can compare different rates of time passage for different observers and understand how acceleration affects the object's movement through time itself. Moreover, the differentiation with respect to proper time is what leads to the homogeneity of acceleration expressed in the tensor equation and subsequently, to the concept of hyperbolic motion.

Hyperbolic Motion

Hyperbolic motion in special relativity refers to the trajectory of an object moving under a constant proper acceleration. Visually, this motion plots a hyperbola on a spacetime diagram, indicating that the object approaches but never reaches the speed of light. The term 'hyperbolic' relates to the shape of the world line that the motion describes when plotted on such a diagram. This hyperbolic path reflects that, as objects accelerate and approach the speed of light, the effects of time dilation and space contraction become more pronounced.

The equation derived from the exercise, \( (\mathrm{d} / \mathrm{d} \tau) \mathbf{A}=\alpha^{2} \mathbf{U} \), characterizes this type of motion. Demonstrating that the motion is rectilinear within an appropriate inertial frame substantiates that objects under a constant proper acceleration do indeed follow hyperbolic paths. Understanding hyperbolic motion is crucial for interpreting trajectories in high-speed travel where relativistic effects cannot be ignored, such as in spaceflight around massive stars or near the speed of light.