Question

Newton's first law can be stated: If an object is isolated (subject to no forces), then it moves with constant velocity. We know that this is invariant under the Galilean transformation. Prove that it is also invariant under the Lorentz transformation. [Assume that it is true in an inertial frame \(\mathcal{S}\), and use the relativistic velocity-addition formula to show that it is also true in any other \(\mathcal{S}^{\prime} .\) ]

Short answer

Constant velocity in one frame remains constant in another frame, proving invariance.

Step-by-step solution

Understanding the Lorentz Transformation

The Lorentz transformation relates coordinates and time in one inertial frame \(\mathcal{S}\) to those in another \(\mathcal{S}'\), moving at velocity \(v\) relative to \(\mathcal{S}\). The transformations are given by: \[ x' = \gamma (x - vt), \, t' = \gamma (t - \frac{vx}{c^2}) \] where \(\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}\). This transformation assumes that the speed of light \(c\) is constant in all inertial frames.

Introducing Newton's First Law

Newton's first law states that if an object is isolated (no forces act on it), it will move with a constant velocity if viewed from any inertial frame. Mathematically, if \(\textbf{v} = \frac{d\textbf{r}}{dt}\) in frame \(\mathcal{S}\), then for an isolated object \(\textbf{v}\) is constant with zero acceleration \(\textbf{a} = \frac{d\textbf{v}}{dt} = 0\).

Relativistic Velocity Addition Formula

The velocity of an object in \(\mathcal{S}'\), \(u'\), can be found from its velocity in \(\mathcal{S}\), \(u\), using the relativistic velocity addition formula: \[ u' = \frac{u - v}{1 - \frac{uv}{c^2}} \] This formula accounts for the effects of special relativity, ensuring that the speed of light remains constant.

Prove Constant Velocity in \(\mathcal{S}'\)

If the object's velocity \(u\) is constant in \(\mathcal{S}\) (i.e., \(u = u_0\) and doesn't change with time), we expect \(u'\) also to be constant in \(\mathcal{S}'\). With \(u = u_0\), calculate \(u'\) using the velocity addition formula: \[ u' = \frac{u_0 - v}{1 - \frac{u_0v}{c^2}} \] Since \(u_0\) and \(v\) are constants, \(u'\) is also constant with respect to time.

Conclude Invariance

Both \(\mathcal{S}\) and \(\mathcal{S}'\) are inertial frames, and we have shown that a constant velocity in \(\mathcal{S}\) transforms to a constant velocity in \(\mathcal{S}'\). Thus, Newton's first law, which asserts constant velocity in the absence of force, holds true in all such relativistic frames, proving its invariance under Lorentz transformation.

Key concepts

Lorentz Transformation

The Lorentz Transformation is a fundamental concept in Einstein's theory of Special Relativity. This transformation helps us understand how the measurements of time and space change between two different "inertial frames" (which are essentially reference frames moving at a constant velocity relative to each other). Imagine two observers, each in their own train car moving with constant speed but relative to each other. The Lorentz Transformation allows us to translate physical quantities measured by one observer to those measured by the other.

Key to this transformation is the constant speed of light, denoted by \(c\), which remains unchanged in all inertial frames. The equations are as follows:

• \[ x' = \gamma (x - vt) \]
• \[ t' = \gamma \left(t - \frac{vx}{c^2}\right) \]

Here, \(x\) and \(t\) are the space and time coordinates in the original frame \(\mathcal{S}\), and \(x'\) and \(t'\) are those in the moving frame \(\mathcal{S}'\). The term \(\gamma\) is called the Lorentz factor, given by \(\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}\), where \(v\) is the relative velocity of the two frames.

Through the Lorentz Transformation, we see that an object's velocity and the timing of events can appear differently depending on the observer's motion, but the physics remains consistent.

Relativistic Velocity Addition

Relativistic Velocity Addition is a formula used in Special Relativity to calculate how velocities add up when objects are moving at speeds close to that of light. It ensures that the resulting velocity never exceeds the speed of light, maintaining the constancy of light speed in all inertial reference frames.

Under classical, non-relativistic physics, if two objects are moving towards each other at velocities \(u\) and \(v\), their relative velocity is simply \(u + v\). However, in the realm of relativistic speeds, this simple addition does not hold.

The relativistic velocity addition formula is given by:

• \[ u' = \frac{u - v}{1 - \frac{uv}{c^2}} \]

Here, \(u'\) is the velocity observed in the moving frame \(\mathcal{S'}\), and \(u\) is the velocity in the original frame \(\mathcal{S}\). This equation accounts for the time dilation and length contraction effects of relativistic speeds.

By using this formula, you maintain consistent physics across reference frames. For instance, if an object is moving with constant velocity \(u_0\) in one frame, the formula shows that \(u'\), the velocity in a different moving frame, is also constant, demonstrating the invariance of motion.

Inertial Frames

Inertial Frames are a key concept in both classical and modern physics. They refer to reference frames where an object either stays at rest or moves at a constant velocity unless acted upon by an external force. Newton's First Law of Motion, which states that an isolated object will move at constant velocity in such frames, is grounded on this principle.

Inertial frames admit a simplification of physics laws because they do not accelerate, unlike non-inertial or accelerated frames, which require additional forces (like fictitious forces) to describe motion. The idea of inertial frames makes it easier to apply physics principles, especially when transitioning between different observers, like in problems involving special relativity.

Both the Lorentz Transformation and the Relativistic Velocity Addition rely on the concept of inertial frames to preserve consistent physics laws across different perspectives. They help ensure that Newton's First Law holds true even when observed from a frame moving at relativistic speeds. Whether viewed in frame \(\mathcal{S}\) or \(\mathcal{S'}\), an object's lack of acceleration and constant velocity remain invariant, maintaining the core aspect of Newton’s motion law across all inertial frames.

Thus, understanding inertial frames is crucial for grasping how motion and relativistic effects are observed in the vast universe.