# Scalar Product of displacement four vector

• Tony Stark
In summary, the scalar product of a displacement four vector with itself can be represented as (Δs)2= Δx.Δx or ds2=ηαβdxαdxβ. The flat space-time metric tensor ηαβ is mostly zero, except when α = β, simplifying the formula to ηααdxαdxα. Therefore, the scalar product can be used to calculate the square of the distance between two displacement four vectors.
Tony Stark

## Homework Statement

How does the scalar product of displacement four vector with itself give the square of the distance between them?

## Homework Equations

(Δs)2= Δx.Δx ( s∈ distance, x∈ displacement four vector)
or how
ds2αβdxαdxβ

## The Attempt at a Solution

Clearly I am completely new to the subject, thus I have no idea about it.

The flat space-time metric tensor (I love this sentence) ηαβ is a bunch of zero except when α = β so clearly we can reduce ds2 to some simpler formula, Cheers !

Last edited:
What actually do want to say? My question is how does the scalar product of 2 displacement four vectors gives the distance between them. I can not figure out the answer from your reply..

I mean that ηαβdxαdxβ = ηααdxαdxα summation over α,
[EDIT: In case I'm not clear enough η11 = 1, η22 = η33 = η44 = -1]

Last edited:
Tony Stark
I get the whole point. Thanks for all replies.

## 1. What is the definition of the scalar product of a displacement four vector?

The scalar product of a displacement four vector is a mathematical operation that calculates the dot product between two four-vectors. It is used to determine the magnitude of a vector in a specific direction.

## 2. How is the scalar product of a displacement four vector calculated?

The scalar product is calculated by multiplying the components of two four-vectors and then summing them together. The resulting value is a scalar quantity (i.e. a single number) rather than a vector.

## 3. What is the physical significance of the scalar product of a displacement four vector?

The scalar product has physical significance in relativity, as it is used to calculate the proper time interval between two events in space-time. It also has applications in quantum mechanics and electromagnetism.

## 4. How is the scalar product of a displacement four vector related to Lorentz transformations?

The scalar product is invariant under Lorentz transformations, meaning that it remains the same in all reference frames. This property is essential in special relativity, where the laws of physics must be the same for all observers.

## 5. Can the scalar product of a displacement four vector be negative?

Yes, the scalar product can be negative if the angle between two four-vectors is greater than 90 degrees. In this case, the magnitude of the resulting scalar is equal to the negative of the product of the magnitudes of the two vectors.

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