Question

Show that the four-velocity of any object has invariant length squared \(u \cdot u=-c^{2}\)

Short answer

The four-velocity has invariant length squared \\(-c^2\\).

Step-by-step solution

Understand the Objective

The goal is to show that the four-velocity vector, when dotted with itself, equals \(-c^2\). This is an invariant length, meaning it is the same in any frame of reference.

Define Four-Velocity

The four-velocity \( \mathbf{u} \) of an object is defined as \[ \mathbf{u} = \left( \gamma c, \gamma \mathbf{v} \right) \]where \( \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \) is the Lorentz factor, \( c \) is the speed of light, and \( \mathbf{v} \) is the three-velocity of the object.

Compute the Dot Product

Calculate the dot product of the four-velocity vector with itself:\[ \mathbf{u} \cdot \mathbf{u} = (\gamma c)^2 - (\gamma \mathbf{v}) \cdot (\gamma \mathbf{v}) \] Simplify to: \[ \mathbf{u} \cdot \mathbf{u} = \gamma^2 c^2 - \gamma^2 v^2 \]

Factor out Gamma Squared

Factor \( \gamma^2 \) from both terms:\[ \mathbf{u} \cdot \mathbf{u} = \gamma^2 (c^2 - v^2) \]

Substitute for Gamma

Substitute \( \gamma^2 \) using its relation:\[ \gamma^2 = \frac{1}{1 - \frac{v^2}{c^2}} \]Then:\[ \mathbf{u} \cdot \mathbf{u} = \frac{c^2 - v^2}{1 - \frac{v^2}{c^2}} \]

Simplify the Expression

Multiply numerator and denominator:\[ \mathbf{u} \cdot \mathbf{u} = c^2 \frac{1 - \frac{v^2}{c^2}}{1 - \frac{v^2}{c^2}} = c^2 \]

Apply the Negative Sign

Remember that in spacetime, the metric signature leads to the four-velocity squared being negative for time-like vectors:\[ \mathbf{u} \cdot \mathbf{u} = -c^2 \] Thus, the four-velocity magnitude squared is invariant and equals \(-c^2\).

Key concepts

Lorentz Factor

In special relativity, the Lorentz factor plays a crucial role in understanding how time and space are perceived differently by observers in different inertial frames. The Lorentz factor, denoted by \( \gamma \), helps account for the effects of relativistic speeds where the velocity of an object is a significant fraction of the speed of light, \( c \).

The Lorentz factor is given by \[ \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \]where \( v \) is the velocity of the object.

This formula shows that as an object's speed approaches the speed of light, the Lorentz factor increases dramatically.

• When \( v = 0 \), \( \gamma = 1 \). Space and time appear 'normal'.
• As \( v \) gets closer to \( c \), \( \gamma \) approaches infinity, reflecting significant relativistic effects such as time dilation.

The Lorentz factor modifies time intervals and lengths from one reference frame to another.

Understanding \( \gamma \) is also essential for working with four-vectors like four-velocity, where different elements of these vectors are linked through the Lorentz factor.

Invariant Length

The concept of invariant length is a fundamental principle in the framework of special relativity. It refers to a quantity that remains constant irrespective of the observer's inertial frame of reference.

For four-velocity, which combines time and space components, its invariant length squared is particularly important. The invariant length squared of the four-velocity \( \mathbf{u} \) is given by\[ \mathbf{u} \cdot \mathbf{u} = -c^2 \].This value remains unchanged regardless of how fast an observer moves or which direction they are traveling.

• The minus sign in the equation arises due to the nature of space-time metrics in relativity, distinguishing time-like components from space-like ones.
• It highlights an essential property of relativistic space-times, where time appears differently across frames, but certain quantities, like the invariant length, remain perpetual.

The invariant length of vectors, such as four-velocity, is foundational.

It ensures that physical laws remain consistent across different reference frames.

Special Relativity

Special relativity, introduced by Albert Einstein in 1905, revolutionized the way we understand space, time, and energy. It comprises two main postulates:

• The laws of physics are the same in all inertial frames of reference.
• The speed of light, \( c \), is constant in a vacuum for all observers, regardless of their motion relative to the light source.

These principles lead to a number of surprising implications, such as time dilation and length contraction.

Special relativity indicates that at high velocities, close to the speed of light, time appears to slow down and lengths contract along the direction of motion. The equations of special relativity, particularly through the Lorentz transformations, allow us to translate physical laws across different frames of reference.

Four-vector formulation, including quantities like four-velocity, is a direct result of special relativity, providing a consistent way to handle dynamics across different frames.

• This includes the notion of invariant quantities, which remain constant despite changes in perspective.
• The theory paves the way for modern physics by explaining a wide range of phenomena that classical physics could not.

In short, special relativity is vital for understanding the universe at relativistic speeds, significantly impacting physics' theoretical and practical aspects.