Penrose
Penrose
Penrose
Contents
1 Review of tensor algebra 1
2 Penrose diagrammatic notation 2
3 Demonstrations 5
4 References 7
Figure 1: Einstein eld equations in natural units with zero cosmological constant with
the diagrammatic notation
multilinear map which feeds on p vectors and q dual vectors to give out a number.
Examples:
Any scalar is a rank (0, 0) tensor.
∈V∗
It has one slot for a dual vector, so for w ∈ V ∗ , v(w) is a scalar. In matrices representation,
we know this evaluation is just multiplication of the vector v by the row vector w from the
1
left; wv .
∈V
It has one slot for a dual vector, so for v ∈ V , w(v) is a scalar. In matrices representation,
we know this is also wv . So we see that a dual vector acting a vector w(v) is the same as the
vector acting on the dual v(w).
∈V∗ ∈V
) ∈ V ⊗ V∗ is a rank 2 tensor
z}|{ z}|{
• A( ,
Filling the rst slot A(v, ) with v ∈ V makes it a rank (0, 1) tensor (a dual vector). In
matrix representation, it's equivalent to multiplying the column vector v from the right; the
result of Av is indeed a column vector. Filling the second slot A( , w) with w ∈ V ∗ makes
it a rank (1, 0) tensor (a vector). In matrix represention, it's multiplying the row vector w
from the left of the matrix; wA. Filling both slots A(v, w) give a rank (0, 0) tensor which is
a scalar. In matrix representation, it's equivalent to multiplying the matrix from the left and
the right wAv , and this indeed gives a scalar.
Alternatively, one can ll two slots with one rank-2 tensor. Or three slots with one rank-1
tensor and one rank-2 tensor, and so forth.
2
Application/composition of tensors is represented by connecting the free lines of the symbols
together:
Figure 2: We see in order: w(v), m(t), n(r), Q(m, v, w, t, n) and the matrix product AB
It follows that scalars are objects without lines projecting upwards nor downwards. So as can
be seen in the gure above, w(v), m(t), n(r) and Q(m, v, w, t, n) would all be scalars (rank (0,
0) tensors). The matrix product AB however has two lines projecting out of it still, so it's a
rank (1, 1) tensor (product of two matrices is a matrix).
tensor Q yields:
In einstein's notation, the partial trace over two slots is represented by equating the respective
upper and lower indices (e.g. Aµµ ); or equivalently contracting the indices with the identity
(Aµν δ νµ ). In penrose notation, it is thus represented by connecting the upper slot with the
3
Figure 3: The antisymmetrisation opeartion is the sum of all possible even permu-
tations (achieved by an even number of swaps on the slots) minus all possible odd
permutations (achieved by an odd number of swaps) multiplied by the total number of
permutations 1/n! for normalisation. The symmetrisation operation is similar, except
that all permutations, even and odd, are summed over.
Figure 5: Any rank 2 tensor can be decomposed into a symmetric part and an antisym-
metric part.
Figure 6: The inverse and the determinant of an invertible matrix A written in the
diagrammatic notation.
It is then convenient to represent the metric (0, 2) tensor with a hoop symbol g =
ˆ and
4
the inverse metric (2, 0) tensor with an inverted hoop g −1 = ˆ: . Joining the two hoops
(composition of the metric with its inverse) then naturally give a vertical line (i.e. identity):
.
Figure 7: The transpose of A is simply composing hoops (in einstein's notation: lower
the upper index and raise the lower index). The second line demonstrates how (AB)T =
B T AT
The covariant derivative of a tensor is represented by a circle around the tensor's symbol with
one lower slot line projecting down out of the circle: . For instance, the covariant
3 Demonstrations
5
Figure 9: The rst expression is the Ricci identity dening the Riemann tensor
[∇µ , ∇ν ]rα = Rµναβ rβ . The second expression is the antisymmetry of the Riemann tensor
in its rst two indices. The third expression is the 1st Bianchi identity R[µνα]β = 0 and
the fourth expression is the 2nd (or dierential) Bianchi identity ∇[β Rµν]αβ = 0.
6
Figure 12: Leibniz rule expansion ∇µ (B β[κ Q(α|β| γ)δ] ν Aην )v δ = ∇µ (B β[κ )Q(α|β| γ)δ] ν Aην v δ +
(α γ) ν (α γ) ν
B β[κ ∇µ (Q η δ
|β| δ] )A ν v + B β[κ Q η
|β| δ] )∇µ (A ν )v
δ
4 References
Roger Penrose, "The Road to Reality"
Wikipedia: https://en.wikipedia.org/wiki/Penrose_graphical_notation