Question

We have seen that the scalar product \(x \cdot x\) of any four-vector \(x\) with itself is invariant under Lorentz transformations. Use the invariance of \(x \cdot x\) to prove that the scalar product \(x \cdot y\) of any two four-vectors \(x\) and \(y\) is likewise invariant.

Short answer

The scalar product \(x \cdot y\) is invariant under Lorentz transformations.

Step-by-step solution

Define the Scalar Product of Four-Vectors

The scalar product of two four-vectors \( x \) and \( y \) is given by \( x \cdot y = x_0 y_0 - x_1 y_1 - x_2 y_2 - x_3 y_3 \). Here, \( x_i \) and \( y_i \) represent the components of the four-vectors, where \( i = 0, 1, 2, 3 \) and \( x_0 \) and \( y_0 \) typically correspond to the time components while the others correspond to the spatial components.

Understand the Invariance of a Single Four-Vector

For a single four-vector \( x \), the scalar product with itself is given by \( x \cdot x = x_0^2 - x_1^2 - x_2^2 - x_3^2 \). The exercise tells us this quantity is invariant under Lorentz transformations, meaning after transformation, its value remains the same, despite changes in the vector's components.

Consider Two, Rather Than One, Four-Vectors

Now consider two different four-vectors \( x \) and \( y \). The invariant nature of their scalar products with themselves, \( x \cdot x \) and \( y \cdot y \), is known. Our goal is to use this information to prove the invariance of \( x \cdot y \).

Use Additivity of Scalar Product

Recall the property that scalar products distribute over addition: \( (x+y) \cdot (x+y) = x \cdot x + 2x \cdot y + y \cdot y \). Since we know \( x \cdot x \) and \( y \cdot y \) are invariant, we investigate the quantity \( (x+y) \cdot (x+y) \).

Apply Lorentz Invariance to \( (x+y) \cdot (x+y) \)

The combined vector \( x+y \) is also a four-vector. Thus, \( (x+y) \cdot (x+y) \) must be invariant under Lorentz transformations. This implies that each component of the resulting expansion must remain unchanged when transformed.

Analyze Result to Conclude Invariance of \( x \cdot y \)

Since \( (x+y) \cdot (x+y) = x \cdot x + 2x \cdot y + y \cdot y \) is invariant, and \( x \cdot x \) and \( y \cdot y \) are invariant by themselves, the same in these differences on either side must hold; thus, \( 2x \cdot y \) and subsequently \( x \cdot y \) is Lorentz invariant.

Key concepts

Four-vectors

Four-vectors are a fundamental concept in the realm of special relativity, crucial for describing physical phenomena. They extend the idea of three-dimensional vectors to four dimensions, incorporating time as the fourth component alongside the three spatial components. A four-vector, often denoted as \( x \), typically takes the form \( (x_0, x_1, x_2, x_3) \). Here,

• \( x_0 \) is the time component
• \( x_1, x_2, x_3 \) are spatial components

This structure allows a unified treatment of space and time, proving useful in describing events in spacetime.
A critical aspect of four-vectors is that they transform according to Lorentz transformations, which preserve the form of the spacetime interval between events. Therefore, any quantity derived from a four-vector with fundamental physical significance should also remain unchanged under these transformations. Such invariance is at the core of understanding how we interpret the laws of physics in different inertial frames.

Scalar Product

The scalar product of two four-vectors is a crucial operation that measures the "length" of a four-vector or the angle between two four-vectors in spacetime. For four-vectors \( x \) and \( y \), the scalar product is defined as:

• \( x \cdot y = x_0 y_0 - x_1 y_1 - x_2 y_2 - x_3 y_3 \)

In this expression, the temporal components are multiplied, and the spatial components are subtracted, aligning with the requirement for Lorentz invariance.
This Minkowski metric ensures that the scalar product is negative definite in spacetime, meaning its value remains consistent regardless of the observer's inertial frame. Importantly, this property signifies the underlying Lorentz invariance of such measurements, stating that physical laws should hold the same forms in all inertial frames, exemplified by the invariance of scalar products.

Lorentz Transformations

Lorentz transformations are the mathematical relationships that describe how measures of space and time change for observers moving relative to each other at constant velocities. They are derived from Einstein's theory of special relativity, ensuring that the speed of light and the laws of physics remain consistent across different inertial reference frames.
The transformation for a generic four-vector \( x \) into another four-vector \( x' \) is given by a specific linear transformation matrix, which modifies both the temporal and spatial components of \( x \) according to the observer's relative motion.
A remarkable feature of Lorentz transformations is that they preserve the interval between two events in spacetime, expressed by the scalar product \( x \cdot x \). Hence, even though the components of a single four-vector may change when moving between reference frames, quantities such as the scalar product remain invariant. This invariance ensures that physical phenomena have consistent descriptions, a foundation necessary for both theoretical and experimental physics.