# Orthogonal

## Glossary of Terms

This is a glossary of some of the terminology used in the notes on the physics of *Orthogonal*. Although the notes generally link to Wikipedia for widely used mathematical and scientific terms, in some cases
a shorter and less technical explanation is provided here instead.

This is *not* a glossary of terms used in the novel. If you’re wondering how many scants there are in a stride or how many scrags in a heft, there’s an appendix for that at the back
of the book.

**Basis**. A basis for an *n-*dimensional vector space V is a set of *n* vectors, {**e**_{1}, **e**_{2}, ... **e**_{n}}, such that any vector
**v** in V can be written as a sum of multiples of the basis vectors:

**v** = *v*^{1} **e**_{1} + *v*^{2} **e**_{2} + ... + *v*^{n} **e**_{n}

The numbers *v*^{1}, *v*^{2}, ... *v*^{n} are called the *coordinates* or *components* of the vector **v** with respect to the basis.
The same vector will have different components with respect to different bases.

In these notes, we are mostly concerned with four-dimensional vector spaces, and instead of numbering the basis vectors 1, 2, 3, 4 we’ll often label them with letters:
**e**_{x}, **e**_{y}, **e**_{z} and **e**_{t}.

Most bases we use will be **orthonormal**: the “normal” parts means all the vectors have length 1, and the “ortho” part means they are all mutually perpendicular.

As an example, in the vector space of four-tuples of real numbers, R^{4}, with the standard dot product, one orthonormal basis is the **standard basis**:

**e**_{x} = (1, 0, 0, 0)

**e**_{y} = (0, 1, 0, 0)

**e**_{z} = (0, 0, 1, 0)

**e**_{t} = (0, 0, 0, 1)

**Determinant**. The determinant of an *n*×*n* matrix A, written det(A), gives the factor by which the linear function
corresponding to A changes the volume of an *n*-cube. The determinant will be zero if A squashes the *n*-cube down to less than *n* dimensions. The determinant will be negative if A reverses the orientation of the *n*-cube.

For example, in the case of two dimensions, the points {(0,0), (1,0), (1,1), (0,1)} give the vertices of a square of size 1, listed in *counter-clockwise* order. If we apply A to these points, we
get four new points; the absolute value of det(A) will be the area of the parallelogram whose vertices are those points, and det(A) will be negative if the new points end up in *clockwise* order.

Suppose A is the 2×2 matrix:

Two edges of the parallelogram will be the vectors **v**=(*a*, *c*) and **w**=(*b*, *d*).

Since
the vector **p**=(–*c*, *a*) is perpendicular to **v**, the “height” *h* of the parallelogram — if the edge
**v** is taken as the “base” — is the length of the projection of **w** onto the unit vector **p**/|**p**|:

*h* = **w** · **p**/|**p**|

Here *h* will be negative if the tip of **w** falls below the base **v**, i.e. if the original square has been mapped onto the parallelogram clockwise.
The area of the parallelogram is the height times the length of the base, and:

det(A) = *h* |**v**| = *h* |**p**| = **w** · **p** = *a**d*–*b**c*

In general, to find the determinant of an *n*×*n* matrix A, you list every possible permutation of the numbers
{1,2,...*n*}. Associated with each permutation is a number known as its sign, which is +1 if the permutation involves an even number of swaps from the original
order, and –1 if it involves an odd number of swaps. For each permutation, you multiply its sign by *n* matrix components, one from each row,
with the column number determined by the corresponding number in the permutation. The sum of all these products is the determinant:

det(A) = ∑_{p a permutation of {1,2,...n}} sign(*p*) A^{1}_{p(1)} A^{2}_{p(2)} ... A^{n}_{p(n)}

For our example of a 2×2 matrix, the permutations of {1,2} are just {1,2} itself, with sign 1, and {2,1}, with sign –1, so the determinant is:

det(A) = A^{1}_{1} A^{2}_{2} – A^{1}_{2} A^{2}_{1} = *a**d*–*b**c*

One very useful special case is the determinant of a *diagonal matrix*, D, a matrix whose components are 0 except on the diagonal. In this case, the only permutation whose product
sticks to the diagonal is the one that leaves all the numbers {1,2,...*n*} in their original order, so we have:

det(D) = D^{1}_{1} D^{2}_{2} ... D^{n}_{n}

**Dot product**. In four-dimensional Riemannian geometry, the dot product of two vectors **v** and **w**, written as **v** · **w**, is given by:

**v** · **w** = *v*^{x} *w*^{x} + *v*^{y} *w*^{y} + *v*^{z} *w*^{z} + *v*^{t} *w*^{t}

where *v*^{x} etc. and *w*^{x} etc. are the components of the vectors in an orthonormal basis.

This generalises to the *n*-dimensional case in the obvious way. Using the Einstein summation convention:

**v** · **w** = *v*^{i} *w*^{i}

The dot product of two vectors is related to their lengths and the angle between them, θ:

**v** · **w** = |**v**| |**w**| cos θ

The definition we’ve given here, which starts from an **orthonormal** basis, is appropriate for physical applications,
where we take it for granted that we can measure the length of a vector, and determine on physical grounds whether or not it’s perpendicular
to another vector.

In pure mathematics, though, we need to *start* with a dot product in order to define an orthonormal basis;
it’s the fact that **e**_{x} · **e**_{y} = 0 that tells you **e**_{x} and **e**_{y} are orthogonal, and **e**_{x} · **e**_{x} = 1 tells you
**e**_{x} has a length of 1. So for example, when dealing with the vector space of four-tuples of real numbers, R^{4}, we would generally use the **standard dot product**, which we define as:

(*v*^{x}, *v*^{y}, *v*^{z}, *v*^{t}) · (*w*^{x}, *w*^{y}, *w*^{z}, *w*^{t})
= *v*^{x} *w*^{x} + *v*^{y} *w*^{y} + *v*^{z} *w*^{z} + *v*^{t} *w*^{t}

which leads to the standard basis for R^{4} being orthonormal.

Details in the notes on Dot products.

**Dual**. If V is a real vector space, then a **dual vector** on V is a linear function *f* from V to the real numbers R.

The **dual space** to V, written V*, is the set of all dual vectors on V, made into a vector space in its own right.

If {**e**_{1}, **e**_{2}, ... **e**_{n}} is a basis for V, then the **dual basis** for V*, {**e**^{1}, **e**^{2}, ... **e**^{n}}
is a set of functions on V that satisfies the condition:

**e**^{i}(**e**_{j}) = δ^{i}_{j}

where δ^{i}_{j}, known as the Kronecker delta symbol,
is 1 if *i*=*j* and 0 if *i*≠*j*.

Details in the notes on Dual vectors.

**Einstein summation convention**. The Einstein summation convention is a convenient short-hand for writing the sums of terms in an expression involving
the components of vectors and matrices. Using the summation convention, whenever the same index variable appears twice in a product, the implication is
that the product is actually a sum over all applicable values of that index. So if we write:

M^{i}_{j} *v*^{j}

where *v*^{j} indicates a component of an *n*-dimensional vector **v** , the index *j* appears twice and the summation convention tells us to expand this as:

M^{i}_{j} *v*^{j} = M^{i}_{1} *v*^{1} + M^{i}_{2} *v*^{2} + ... + M^{i}_{n} *v*^{n}

Note that when we write a term like *v*^{x} *w*^{x}, the repeated “x” here is labelling a *particular* component of each of these vectors, so the summation
convention does not apply. Whenever *x*, *y*, *z* or *t* appear as superscripts or subscripts, they are simply meant as labels corresponding to the traditional coordinate axes,
not as index variables that need to be summed over.

**Euclidean group**. The Euclidean group E(*n*) is the group of all symmetries of *n*-dimensional Euclidean space, including translations, rotations and reflections.
In these notes, we are mostly concerned with E(4), the symmetries of 4-dimensional Euclidean space.

The subgroup of the Euclidean group that includes only translations and rotations, ruling out reflections, is known as the **special Euclidean group**, SE(*n*).

Details in the notes on Symmetries.

**Euclidean universe**. The term “Euclidean universe” is used in these notes to describe an idealised four-dimensional Riemannian universe
that is *infinite* and *flat*, and hence obeys the postulates of Euclidean geometry.

In our own universe, the geometry of small regions of *space* far from any strong gravitational fields
is very close to Euclidean, but the geometry of *space-time* in such regions isn’t, because intervals in time do not obey Pythagoras’s Theorem.
In our universe, nearly-flat space-time is approximated by Minkowski spacetime.

**NB**: In the physics literature, the term “Euclidean” is frequently applied to the versions of physical laws
produced by Wick rotation. These are **not** the laws applicable in the universe of *Orthogonal*.

**Group**. In mathematics, a group is a set G along with an “operation”: a way of combining two elements of G to yield a third element.
This group operation is sometimes called “multiplication”, and the element we get by combining the elements *g* and *h* is usually written *g**h*.
It must
obey similar rules to the ordinary multiplication of positive numbers, but it need not be **commutative**: that is, it’s possible that *g**h*≠*h**g*.

- Group “multiplication” must be
**associative**: (*g**h*)*k* = *g*(*h**k*).
- The group must contain an
**identity**, usually called *e* (but sometimes called 1 or I for matrix groups) with
*g**e* = *e**g* = *g*.
- Every element
*g* of the group must have an inverse, *g*^{–1}, which satisfies *g**g*^{–1} = *g*^{–1}*g* = *e*.

Some examples of commutative groups:

- The positive real numbers, R
^{+}, with ordinary multiplication as the group operation.
- The vector space of
*n*-tuples of real numbers, R^{n}, with vector addition as the group operation and the zero vector as the identity.

Some examples of non-commutative groups:

- The set of
*n*×*n* matrices of real numbers with non-zero determinant, written GL(*n*),
with matrix multiplication as the group operation.
- The set of
*n*×*n* matrices of real numbers whose transpose is equal to their inverse, written O(*n*),
with matrix multiplication as the group operation.

**Identity matrix**. The *n*×*n* identity matrix, written I_{n}, is the matrix whose components are all zero except along the diagonal, where
they are 1. If *n* is clear from the context, we will just write I rather than I_{n}.

For example, I_{4} is:

1 | 0 | 0 | 0 |

0 | 1 | 0 | 0 |

0 | 0 | 1 | 0 |

0 | 0 | 0 | 1 |

Multiplication with the identity matrix leaves any other matrix unchanged; if A is any *n*×*n* matrix, then:

A I_{n} = I_{n} A = A

**Inverse matrix**. The inverse of an *n*×*n* matrix A is the *n*×*n* matrix, written A^{–1} such that:

AA^{–1} = A^{–1}A = I_{n}

where I_{n} is the *n*×*n* identity matrix. The inverse of A will exist if and only if the determinant of A is not zero.

The inverse of the product of matrices is the product, *in reverse order*, of the inverses of the individual matrices:

(AB)^{–1} = B^{–1} A^{–1}

**Linear function**. A linear function is a function from one vector space to another (perhaps the same one), such that it makes no
difference whether you apply the function before or after adding vectors or multiplying them by numbers. That is, the function *f* is linear if:

*f*(**v** + **w**) = *f*(**v**) + *f*(**w**)

*f*(*s* **v**) = *s* *f*(**v**)

If we have a linear function *f* from an
*m*-dimensional vector space V to an *n*-dimensional vector space W,
and we pick bases {**e**_{1}, **e**_{2}, ... **e**_{m}} for V and
{**e’**_{1}, **e’**_{2}, ... **e’**_{n}} for W,
we can describe *f* in terms of an *n*×*m* matrix M. The matrix component M^{i}_{j} is the *i*th component, with respect to our basis for W,
of the vector *f*(**e**_{j}), and using the Einstein summation convention:

*f*(**v**) = *f*(*v*^{j} **e**_{j}) = *v*^{j} *f*(**e**_{j}) = M^{i}_{j} *v*^{j} **e’**_{i}

**Lorentzian**. The adjective “Lorentzian” is used in these notes to distinguish **the physics that applies in our own universe** from that which
applies in the universe of *Orthogonal*.

Details in the notes on Lorentzian and Riemannian Geometry.

[Why “Lorentzian”?
Hendrik Antoon Lorentz was a Dutch physicist who developed the mathematical framework that Einstein
employed in special relativity. The term “Lorentzian” is used to refer to a geometry where one of the dimensions is distinguished from the others
as a *time* dimension. In contrast, in a Riemannian space, all dimensions are fundamentally the same.]

**Matrix**. An *n*×*m* matrix M is a grid or table of numbers (in these notes, either real or complex numbers), with *n* rows of *m* numbers.
The individual numbers are known as the *components* of the matrix; we will write M^{i}_{j} for the *j*th number in the *i*th row of the matrix M.

If A is an *n*×*m* matrix and B is an *m*×*p* matrix, we can multiply A by B to form an *n*×*p* matrix C=AB, with:

C^{i}_{j} = A^{i}_{1} B^{1}_{j} + A^{i}_{2} B^{2}_{j} + ... + A^{i}_{m} B^{m}_{j}

or using the Einstein summation convention, simply:

C^{i}_{j} = A^{i}_{k} B^{k}_{j}

Note that in order for the matrix product AB to exist, the size of the *rows* of A must equal the size of the *columns* of B. If A and B are square matrices
of the same size then both AB and BA will exist, but they will not generally be equal.

We can treat an *n*×*m* matrix M as a linear function from an
*m*-dimensional vector space V to an *n*-dimensional vector space W,
by thinking of any vector **v** in V as an *m*×*1* matrix, consisting of a single column of numbers which are the *m* components of **v** in some chosen basis.
Then the matrix product M**v** gives us an *n*×*1* matrix, which we can think of as the *n* components of a vector in W in some chosen basis.
So M, along with a choice of bases in V and W, gives us a linear function *f* from V to W.

If the basis for V is {**e**_{1}, **e**_{2}, ... **e**_{m}}
and the basis for W is {**e’**_{1}, **e’**_{2}, ... **e’**_{n}}, we can write, using the Einstein summation convention:

*f*(**v**) = *f*(*v*^{j} **e**_{j}) = *v*^{j} *f*(**e**_{j}) = M^{i}_{j} *v*^{j} **e’**_{i}

**Null Vector**. A vector that points along the world line traced out by light in ordinary space-time.

Details in the notes on Lorentzian and Riemannian Geometry.

**O(4)**. The set of 4×4 matrices of real numbers whose transpose
is equal to their inverse is known as O(4).
It is a group, with matrix multiplication as the group operation, and another name for it is the 4-dimensional *orthogonal group*.

If these matrices are treated as linear functions on four-space, they describe all rotations and reflections in that space.

Details in the notes on Symmetries.

**Riemannian**. The adjective “Riemannian” is used in these notes to distinguish **the physics that applies in the universe of ***Orthogonal* from that which
applies in our own universe.

Details in the notes on Lorentzian and Riemannian Geometry.

[Why “Riemannian”?
Georg Friedrich Bernhard Riemann was one of the pioneers of the field of differential geometry,
a subject that generalises Euclidean geometry to apply to curved surfaces and spaces. Within that field, mathematicians use the term “Riemannian” to describe
geometries that we’d normally think of as kinds of *space*,
whether flat or curved, where all the dimensions are treated as fundamentally the same. In contrast, in the Lorentzian *space-time* of our own universe, one of the dimensions, *time*, is singled out for
special treatment.]

**Riemannian Scalar Wave (RSW) Equation**. This is the equation for a scalar wave *A* moving through the vacuum of the Riemannian universe:

∂_{x}^{2}*A* + ∂_{y}^{2}*A*
+ ∂_{z}^{2}*A* + ∂_{t}^{2}*A* + ω_{m}^{2} *A* |
= |
0 |
(RSW) |

Details in the notes on Scalar Waves.

**Riemannian Vector Wave (RVW) Equations**. These are the equations for a vector wave **A** moving through the vacuum of the Riemannian universe:

∂_{x}^{2}**A** + ∂_{y}^{2}**A**
+ ∂_{z}^{2}**A** + ∂_{t}^{2}**A** + ω_{m}^{2} **A** |
= |
**0** |
(RVW) |

∂_{x} *A*^{x} + ∂_{y} *A*^{y} + ∂_{z} *A*^{z} + ∂_{t} *A*^{t} |
= |
0 |
(Transverse) |

Details in the notes on Vector Waves.

We can extend the first equation to include a source term:

∂_{x}^{2}**A** + ∂_{y}^{2}**A**
+ ∂_{z}^{2}**A** + ∂_{t}^{2}**A** + ω_{m}^{2} **A** + **j** |
= |
**0** |
(RVWS) |

Details in the notes on Riemannian Electromagnetism.

**Scalar**. A number that all observers will agree on, because it does not rely on their choice of coordinate system.
For example, air pressure is a scalar, but “the component of the electric field in the *x* direction” isn’t.

**SO(4)**. The set of 4×4 matrices of real numbers whose transpose is equal to their inverse
and whose determinant is 1 is known as SO(4).
It is a group, with matrix multiplication as the group operation, and another name for it is the 4-dimensional *special orthogonal group*.

If these matrices are treated as linear functions on four-space, they describe all rotations in that space.

Details in the notes on Symmetries.

**Subspace**. A subspace of a vector space V is a subset of V that is itself a vector space.

For example, in a vector space of two or more dimensions any line that passes through the origin will be a 1-dimensional subspace of the original vector space.
But not all subsets of a vector space are *subspaces*:
the sphere of all vectors of length 1 does *not* form a subspace because you can add two vectors of length 1 to get another vector
of a different length, outside the chosen subset.

**Transpose**. The transpose of an *n*×*m* matrix A is the *m*×*n* matrix, written A^{T}, whose column are the rows of the original matrix.

The transpose of the product of matrices is the product, *in reverse order*, of the transposes of the individual matrices:

(AB)^{T} = B^{T} A^{T}

**Vector space**. A vector space V is a set of objects that we can add to each other, and multiply by numbers of some kind (in these notes, the numbers will either be real numbers or complex
numbers).

- Every vector space includes a
*zero vector*, **0**, that can be added to any vector without changing it: **v**+**0** = **v**.
- Every vector
**v** has an opposite (or *additive inverse*), –**v**, such that **v**+(–**v**) = **0**.
- Multiplication
*distributes* over addition: *s* (**v**+**w**) = *s* **v** + *s* **w** and (*r* + *s*) **v** = *r* **v** + *s* **v**.
- We can also define subtraction of one vector from another:
**v**–**w** = **v**+(–**w**).
- A vector space must be closed under all these operations. If we’ve chosen a set V and defined these operations on it, they must never yield results that lie
*outside* V.

For example, let V be the set of all four-tuples of real numbers, R^{4}. Then we define addition, multiplication, the zero vector, and the opposite of a vector in the obvious ways:

- (
*x*, *y*, *z*, *t*) + (*a*, *b*, *c*, *d*) = (*x*+*a*, *y*+*b*, *z*+*c*, *t*+*d*)
*s* (*x*, *y*, *z*, *t*) = (*s**x*, *s**y*, *s**z*, *s**t*)
**0** = (0, 0, 0, 0)
- –(
*x*, *y*, *z*, *t*) = (–*x*, –*y*, –*z*, –*t*)

**Wick rotation**. Wick rotation is a procedure employed in modern physics in order to analyse certain problems *in our own universe*.
In Wick rotation,
time *t* is replaced by an imaginary number *i*τ and
*nothing else is changed*.

In contrast to this, the physics of the novel *Orthogonal* involves changing the signs of some parameters in various equations,
leading to very different results than Wick rotation.

For example, the ordinary, Lorentzian
Klein-Gordon equation in natural units is:

∂_{x}^{2}ψ + ∂_{y}^{2}ψ + ∂_{z}^{2}ψ – ∂_{t}^{2}ψ – *m*^{2} ψ |
= |
0 |
(1) |

(You will often see this written as ∂^{μ}∂_{μ}ψ + *m*^{2} ψ = 0,
summing over μ with the Einstein summation convention, but the positive
sign for the *m*^{2} ψ term comes about because the author is using the (+–––)
signature
for the
Minkowski
metric, i.e.
∂^{μ}∂_{μ}ψ =
∂_{t}^{2}ψ
– ∂_{x}^{2}ψ
– ∂_{y}^{2}ψ
– ∂_{z}^{2}ψ.)

The Wick-rotated version, where *t* → *i*τ, is:

∂_{x}^{2}ψ + ∂_{y}^{2}ψ + ∂_{z}^{2}ψ + ∂_{τ}^{2}ψ – *m*^{2} ψ |
= |
0 |
(2) |

The Riemannian version that applies in the universe of *Orthogonal* is similar, but the sign of the *m*^{2} term is changed:

∂_{x}^{2}ψ + ∂_{y}^{2}ψ + ∂_{z}^{2}ψ + ∂_{t}^{2}ψ + *m*^{2} ψ |
= |
0 |
(3) |

The ordinary Klein-Gordon equation **(1)** has bounded plane-wave solutions:

ψ(**x**) |
= |
sin(*k*^{x} *x* + *k*^{y} *y* + *k*^{z} *z* – *k*^{t} *t*) |
(4) |

where **k** is any timelike vector with (*k*^{x})^{2} + (*k*^{y})^{2} + (*k*^{z})^{2} – (*k*^{t})^{2} = –*m*^{2}.

The version **(3)** that applies in the universe of *Orthogonal* also has bounded plane-wave solutions:

where |**k**|^{2} = *m*^{2}.

But *all* solutions of the Wick-rotated version, **(2)**, grow exponentially in at least one direction.
If we substituted the solution **(5)** into equation **(2)** it would give |**k**|^{2} = –*m*^{2}, which can only be true if at least one component of **k**
is imaginary — and the sine of an imaginary quantity grows exponentially.

Orthogonal / Glossary / created Wednesday, 6 April 2011
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