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Four-Vector Notation
Adrian Down
January 25, 2006
1 Review: Vectors and dot products
1.1 Review
Last time, we saw that a tensor T transforms according to
T (^) ij′ = ΛikΛjlTkl
where the transformation matrix Λ is unitary,
ΛT^ Λ = (^113)
The summation indices run from 1 to 3.
1.2 Generalization to 4 dimensions
A vector a in 3 dimensions is written is
a = (a 1 , a 2 , a 3 )
The dot product is given by
a · b = a 1 b 1 + a 2 b 2 + a 3 b 3 = aibi
In 4 dimensions, vector is written without a vector sign ,
a =
a^0 , a^1 , a^2 , a^3
Note. The components 1 through 3 of the four-vector recover the three di- mensional vector.
We will see that the four-dimensional dot product requires negative signs,
a · b = a^0 b^0 − a^1 b^1 − a^2 b^2 − a^3 b^3
Note. The 4 dimensional dot product does not reduce to the 3 dimensional case if the 0th^ component is 0.
2 4 dimensional dot product
2.1 Derivation
Consider two frames of reference, a stationary frame S and a frame S′^ moving at velocity β 0 c.
Note. The velocity is written in terms of the dimensionless quantity β because it is more convenient to work with than the actual velocity v.
Define the 0 of the time component to be such that the origins of the two frames coincide at t = t′^ = 0. We want to describe the position in such a frame using the last three components of a four-vector,
r = (x 0 , x 1 , x 2 , x 3 )
The 0th^ component corresponds to time. To make the dimensions of the components of the vector consistent, we must take x 0 = ct. Consider an electromagnetic pulse emitted at t = t′^ = 0. A fundamental conclusion of Maxwell’s equations is that light must move at the same speed c in any frame. The expressions for the position of this pulse can be expressed in both frames,
c^2 t^2 = x^2 + y^2 + z^2 c^2 t′^2 = x′^2 + y′^2 + z′^2
This can be written using a dot product,
r · r = 0 = r′^ · r′
provided that we make the following definition to ensure that the length of the vector is the same in both frames. We must use negative signs in the definition of the dot product,
r · r = ct^2 − x^2 − y^2 − z^2
3D 4D
Indices 1 ≤ i, j, k ≤ 3 0 ≤ μ, ν, ρ, σ ≤ 0 Scalar product a · b = aibi = aiδij bj a^ a · b = aμgμν bν^ = aμbν^ = aμbν b Rotation a′ i = Λij aj c^ a′μ^ = Λμν aν^ d Length preservation a′^ · a′^ = a · a ⇒ ΛT^ Λ = 113 a · b = a′^ · b′^ = aμgμν bν^ = Λμρaρ ︸ ︷︷ ︸ a′μ
gμν Λνσbσ ︸ ︷︷ ︸ b′μ
e
Transformation ΛTikΛjk = δij f^ gρσ = gμν ΛμρΛνσ g aWe can include the δ to be pedantic and write the 3D case using notation similar to
that in 4D
bgμν is the metric tensor g =
1 − 1 − 1 − 1
cThe first index i refers to the row of the transformation matrix, and the second index j refers to the column dΛμν is an element of a spacetime “rotation” matrix that we have yet to determine eWe have used other random indices for summation to avoid repeating indices that we do not mean to sum over. fTo be pedantic and follow the four-dimensional rotation, we can write δklΛkiΛlj = δij gIt is this requirement on Λ that determines what its elements must be.
