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Date Started: February 12, 2001

Last Modified: January 4, 2002

The paper presents non-quantum physics with emphasis on the general theory of relativity. The entire theory but not many applications or approximation techniques are included. Required mathematics is rigorously developed.

**Section 1: **Special Mathematical Notation

To keep formulas
simple, special notation is used. When a statement has **free indexes** (superscripts
and subscripts that are variable and not assigned a particular value), the
statement is assumed true for all values of the indexes. For example, "*x _{i}=c_{i}t*
where

**Summation
convention **states that when an index variable appears twice in an
expression, summation over the allowed values of the index is assumed. For
example, the chain rule simplifies to_{}. To avoid using the summation
convention, state "no summation" or enclose the indexes in parenthesis. The
expression must be expanded before applying the summation: (*x _{i}+y_{i}*)*(

**Section 2: **Tensors and Tensor Operations

A **tensor** T is defined as
an ordered collection of its components: _{}where each of the *i* and each of
the *j* can assume any integer value from *1* to *N* and all
components are real numbers. *N* is called **dimension**, *m* is
contravariant order, *n* is covariant order, and (*m, n*) is rank of *T*.
(The total number of components is N^{m+n}.) **Tensor field**
assigns a tensor to every point in a space. Tensors are often referred by
their components.

**Definition:** Let T be the
value of a tensor field at a certain point. Let T have components _{}_{}in
coordinates x^{1}, x^{2}, ..., x^{N}. Then in the
coordinates _{}the
components of T are _{}where *W* is a real number
(often a fraction or an integer) that is dependent only on the tensor field.
(Do not forget to use the summation convention on every *a* and every *b*.)
If *W=0*, the tensor is said to be absolute, otherwise the tensor is a
relative tensor of **weight** *W*. Unless stated otherwise, *W=0*.

**Theorem:** The definition is non-contradictory for
arbitrary tensor fields and coordinates provided that the expression is defined
and the Jacobian is not zero.

Tensors are added, multiplied by a scalar, and subtracted
component by component: *C=A+B* means_{}; B=c*A means_{}; *-B=-1*B*; *A-B=A+-B*.
The operations above are undefined if there is a mismatch in rank or
dimension. **Outer product**, *C=A**⊗B*, means_{}. **Contraction** is
setting a subscript equal to a superscript and invoking the summation
convention. For example, _{}. (The covariant and contravariant
orders are decreased by 1.) **Inner product** is an outer product followed
by contraction(s). Addition, subtraction, multiplication by a scalar,
contraction, outer product, and thus inner product of tensors can be shown to
transform under coordinate change as tensors would by the definition. For example,
*C=A**⊗B**⇔*_{} where the bars indicate a
new coordinate system. Tensor differentiation and integration, if done
component by component, does not transform properly: often, _{}. Section 6
describes how to differentiate tensors.

When a quantity with proper components is defined in multiple coordinate systems or in terms of tensors, the quantity is said to be a tensor if it always obeys the transformation equation. Tensors are usually preferred over "nontensors" because a tensor needs to be defined in only one coordinate system and because change of tensor components under coordinate transformations is predictable and depend only on the coordinates and the nature of the tensor.

**Section 3:** Examples of Tensors

**Kronecker
delta**, also called identity tensor, *δ ^{ij}=δ_{i}^{j}=δ_{ij}=1*
if

A tensor of rank
(0, 0) is called a **scalar** and has only one component. A tensor of rank
(1, 0) is called **contravariant vector** or simply **vector**. For
example, position and velocity are vectors. A contravariant vector can be
displayed as a sequence such as (*a _{1}, a_{2}, ..., a_{n}*).
The transformation equation implies that in smaller coordinates vector
components are larger. A tensor of rank (0, 1) is called

Let *E _{i}E_{j}...E_{k}E^{l}E^{m}...E^{n}*
be defined such that (

A **multilinear**
form is a function that takes *n* vectors and returns a scalar in a linear
way with respect to each of the vectors. For example, *f*(*cE _{1}*,

**Note:** Determinant
of a square matrix A, det A=_{}.

**Section 4:** Metric Tensor

An inner product between two
vectors **u** and **v**, a real number <**u**, **v**>, is defined
such that <**u**, **v**> = <**v**, **u**>, <**u + w**,
**v**> = <**u**, **v**> + <**w**, **v**>, **u**≠**0**⇒∃**w** <**u,
w**>≠0. Metric
tensor, **g**≡<E_{i},
E_{j}>E^{i}E^{j}. Conjugate metric tensor, g, is
such that g^{ij}g_{jk}=_{}. **Metric** is
a smooth (that is infinitely differentiable) tensor field made of metric
tensors. Geometry of any space with inner product--including distance, angles,
and curvature--is entirely determined by the metric. Length squared, ||**u|| ^{2}**≡<

**Note:** Outside of relativity,
inner product is defined so that <**u**, **u**> ≥ **0**.

**Theorem:** **(a)** g_{ab}
is an absolute tensor of rank (0, 2); **(b)** g_{ab}=g_{ba};
**(c)** if metric tensor exists, conjugate metric tensor also exists; **(d)**
g_{ab}**u**^{a}**v**^{b} = <**u**, **v**>.
**(e)** For every symmetric tensor g such that the matrix g_{ij} is
invertible, there exists an inner product such that g_{ab}**u**^{a}**v**^{b}
= <**u**, **v**>. **(f)** If metric is positive definitive, <**u**,
**u**> = 0⇔**u**=**0**.

**Section 5:** Manifolds

**Definition:** n-dimensional smooth manifold is a set
of points combined with a set of coordinate systems such that:

1. Every
coordinate system (also called smooth chart) consists of a subset R of the
manifold and (*x _{1}, x_{2}, ..., x_{n}*) for each
point in R. All

2. In every coordinate system, different points have different coordinates.

3. Every
coordinate system is open and connected. Open means that for each (*x _{1},
x_{2}, ..., x_{n}*) in the system there exists

4. Union of all R is the set of points that make up the manifold.

5. For every 2 coordinate systems that share some of the points, the transformation between the coordinate systems must be smooth (that is infinitely differentiable) and thus have a nonzero Jacobian.

6. For
every 2 points A and B, there exists A_{1}, A_{2}, ..., A_{n}
such that A_{1}=A, A_{n}=B, and for every integer i (0<i<n)
there exists a coordinate system such that A_{i} and A_{i+1}
belong to the system. In other words, the manifold is connected.

Manifold is, informally, a generalization of a smooth surface or a space. Once a manifold is defined, more coordinate systems can be added provided that the Jacobian of all coordinate transformations exists and is nonzero at all points where the coordinates apply. Two manifolds are called topologically equivalent if there is a one-to-one transformation between the manifolds that preserves continuity. For example, all 2-dimensional ellipsoids are topologically equivalent, but a sphere is not topologically equivalent to a torus. Tensor field on a manifold is specified by specifying the field for every coordinate system such that the tensors obey the transformation equation.

**Definition:** Pseudo-Riemannian manifold is a smooth
manifold with a metric.

**Definition:** Riemannian manifold is a pseudo-Riemannian manifold whose
metric is positive definitive.

**Section 6:** Covariant Differentiation and Curvature

The actual
change with location of a quantity depends not only on the change its
coordinate components but also on the change of nature (size) of coordinates
with location. For example, decrease of length of *x ^{1}*
(derived from

_{}

where Christoffel symbol of the
second kind, _{}=g^{ia}
*[*jk*, *a*] and Christoffel symbol of the first kind, [*ac*, *b*]
≡_{}. **Covariant** (the
adjusted differentiation is called covariant) derivative of an absolute tensor
of rank *(m, n)* is an absolute tensor of rank *(m, n+1)*. The
derivatives are denoted by adding a comma after the last subscript (such as A_{i,j})
and placing the number of the axis/coordinate after the comma. Second and
higher derivatives do not require additional commas: A^{i}_{,jk}.
The derivative is a linear operator. Covariant derivative, especially of a
scalar, is sometimes called gradient: B=∇A
means B_{i}=A** _{,i}**. Theorem:

A smooth curve
is defined by a smooth function **r**(t)=(x^{1}(t), x^{2}(t),
..., x^{n}(t)). The curve is called normalized if ||d**r**/dt||=1 or
||d**r**/dt||=-1 for every t on an open interval. **Intrinsic derivative **along
a curve, _{}and
is used instead of the non-adjusted derivative to compensate for the change of
coordinate sizes and angles.

Curvature (more precisely intrinsic
curvature of the manifold) at a point is fully described by the **Riemann
Christoffel Tensor: **_{}.

R_{abcd}≡g_{au}R^{u}_{bcd}.
**Theorem:** R_{abcd }=-R_{bacd}=R_{dcba}; R_{abcd}+R_{adbc}+R_{acdb}=0;
R_{abcd,e}+ R_{abec,d}+ R_{abde,c}=0.

**Theorem:** For every point in every pseudo-Riemannian
manifold, there exists a coordinate system (called **geodesic coordinate
system**) for which ∂g_{ij}/∂x^{k} (and thus all
of the Christoffel symbols) are zero at the point.

**Part II:**
General Theory of Relativity

**Section 1:** Geometry of space-time.

Our world is
modeled by a 4-dimensional pseudo-Riemannian manifold called **space-time**.
For every point, there exists a frame of reference, that is a coordinate system,
such that in the frame at that point, the metric, g=η where η≡E^{1}E^{1}+E^{2}E^{2}+E^{3}E^{3}-E^{4}E^{4}.
Such frame of reference is called **locally normal**. If g=η and ∂g_{ab}/∂x^{c}=0, the frame
of reference (for each point, such frame can be proven to exist) is called **locally
inertial**. **MCRF** is a locally normal frame of reference where the
point object is currently at rest.

To be a complete
theory, the mathematical model must be related to human perception. If g= η,
x^{4}=A^{4}E_{4} refers to a point in time and A^{i}E_{i}
= (*x _{1}, x_{2}, x_{3}*) (i=1, 2, 3) refers to a
location in space. Let an observer be at rest in a locally inertial frame of
reference. The observer will locally define distance in space between (

If g→η at *x ^{i}* and

**Section 2:** Dynamics of Point Objects

A point object is a
useful approximation for objects whose size that is small compared to the
distance between the objects. It is described by its smooth curve: **r**(*t*).
(*t* is a parameter, *r ^{4}* is time; two curves

For a normalized
curve, **acceleration** (**a**) is the intrinsic derivative of the velocity.
When a particle does not interact, its **a** is **0**, and it is said to
move on a geodesic. (If the curve cannot be normalized, **a**=**0** when
for an appropriate invertible s(t), δr(s(t))/δs=0.) Light has
undefined 4-velocity and zero acceleration. Since an event cannot cause
itself, for every smooth curve r(s) with ||**u**||=-1 at all points, s≠t ⇒ r(s)≠r(t).

**Rest mass**
(an absolute scalar *m _{0}*) is mass in MCRF.

**Section 3:** Relativistic Fluids

Since all objects have sizes,
point objects are impossible. Instead, the universe is modeled by tensor
fields (sometimes called **fluids**) such as the **stress-energy** tensor
(**T ^{ab}**). If a part of the world is modeled by tensor field

**Section 4:** Gravitation and the Foundations of Relativity

**Gravitation** is represented by curvature of
space-time. Ricci's tensor, R_{bd} = R^{a}_{bad}; R^{ab}=g^{au}g^{bv}R_{uv}.
Einstein's tensor is G^{ab}=R^{ab}-½g^{ab}g^{cd}R_{cd}.

(Postulate) **Einstein's Field Equation:** G=8πT.

**Theorem:** **(a)** G^{ab}=G^{ba}; **(b)**
G^{ab}_{,b}=0.

**Corollary:** **(a)** T^{ab}=T^{ba};
**(b)** T^{ab}_{,b}=0.

Interpretation of part (b) of the corollary:

1. Locally, energy and momentum are conserved.

2. Whenever two elements interact, the change of T for element A due to element B is exactly opposite to the change of T for element B due to element A.

General relativity is a template model--it specifies how interactions can be described but does not state all equations for all interactions. An interaction is specified by stating the type of tensor fields matter and the interaction have, the equations for the tensor fields, stress-energy tensor of the interaction field, and force density on the element by the field. All equations are expressed in terms of the tensors (including the metric g) through addition, contraction, products, covariant differentiation, and operations derived from these operations. Index variables but not specific index values may be used, and no variable in an expression may appear twice as a subscript or a superscript (except when separated by '+' in the expanded form).

**Theorem:** (It is
stated informally. It is also called **Principle of Equivalence**)

1. Physical laws are the same in all frames of reference.

2. Other operations on tensors may not be included in the equations

3. Physical laws are local (no actions at a distance)

Justification of part (1): The expressions transform as the appropriate tensors do.

An implication of part (2): Riemann Christoffel tensor (derived from the metric through non-adjusted differentiation and other operations) cannot be included.

**Section 5:** Laws of Electromagnetism

Let *j _{i}* be
electric current density in

**Note:** Coulomb's constant,
k=0.007297...ħc/e^{2}= c^{2}*10^{‑7}(kilogram*meter)/(coulomb)^{2}.

**Theorem:** **J**^{a}_{,a}=0.
Interpretation: Electric charge can move but cannot be created or destroyed.

**Section
6:** Concluding Remarks

A non-quantum
physical model is called complete if, given an initial state of the system, its
development in time can be (theoretically) fully predicted. Theorem: Gravitation
and electromagnetism, as presented above, are complete. In addition to
electromagnetism, user-defined interactions can be added to avoid using quantum
mechanics: Such interactions usually describe macroscopic predictions of
quantum mechanics. If the result of a calculation is in different units than
you want the result to be, multiply and/or divide by the following conversion
constants: c = 299792458 meter/second, G ≈
6.67259*10^{‑11} (meter)^{3}/(kilogram*(second)^{2}).
From this paper, you have learned the entire non-quantum physics in the sense
that the knowledge that is not part of the paper can be derived from knowledge
that is. However, theorems, relevant approximations in real-world situations,
and practice are required to effectively solve problems.

Exercises

**1.** Show that C=A*B⇔C_{ik}=A_{ij}B_{jk}
for all matrices A, B, and C.

**2.** For a tensor of rank (*l, m*) and weight *W*
in n-dimensional space increase of coordinates in '*a*' times increases
the tensor components in '*b*' times. Prove: *W* = (*l-m+*log_{b}*a*)/*n*.

**3.** How does coordinate change affect

**(a)** scalars **(b)** contravariant
vectors **(c)** covariant vectors

**(d)** tensors of rank (0, 2) **(e)** tensors of rank (2, 0) **(f)**
tensors of rank (1, 1).

**4.** Prove that **(a)** _{}=det A where A_{ij}=_{} **(b)**_{}.

**5.** Show that ||u + v||^{2} = ||u||^{2} + ||v||^{2}
+ 2 <u, v>.

**6.** In parts (a) and (b), show that the space is a
Riemannian manifold.

**(a)** Euclidian n-dimensional space with metric g_{ij}=δ_{ij}.

**(b)** The set of all x (0≤x<1)
such that coordinate system 1: 0<x<1, y_{1}=x, and system 2:

0≤x<½, y_{2}=x,
or ½<x<1, y_{2}=x-1. For metric, g_{11}=1 in both
coordinate systems.

Note: The manifold can be represented as a circle of circumference 1.

**(c)** Show that no appropriate coordinate system
can span the entire manifold in part (b).

**7.** For tensors in problem **3**, find **(a)** covariant
derivative **(b)** intrinsic derivative.

**8.** For an arbitrary 2-dimensional orthogonal (g_{12}=0)
coordinate system find

**(a)** All of the Christoffel symbols.

**(b)** Riemann Christoffel tensor in terms of the
metric.

Hint: Use symmetry in the equations to save time.

**9.** Lorentz transformation (of coordinates) is

(*x ^{1}, x^{2}, x^{3}, x^{4}*)→((

**(b) **Use your answer to
**(a)** to prove that if g=η, then g=η after a Lorentz
transformation.

**10.** Find T for a uniform element of momentum density
(p_{x}, 0, 0, E) where g=η.

**11.** Find Einstein's tensor **(a)** for metric in problem **7**

**(b)** for an
arbitrary locally inertial (4-dimensional) coordinate system.

Note: Part **(b)** may be omitted because it may require a long
calculation.

**Appendix:
Preliminary Mathematics**

Let *x _{1},
x_{2}, ..., x_{n}* be independent variables related to

Let a space have
a measure based on a region (length, area (A(R)), and volume (V(R)) are
measures), *m*(R); and an additive function whose argument is a region:
F(R). Then, f(**r**) = dF/d*m*=limit as size of R→0 of F(R)/*m*(R) when
such limit exists, such that **r** always belongs to R. Then,_{}. Properties of
measure: *m*(R)≥
0; measure is additive, that is measure of two non-intersecting regions is sum
of measures of the regions. **Note:** *dF/dm* is sometimes called *F*
density. *F(x)* is continuous if as change in *x*→0, change in *f*(*x*)→0. *df/dt* ≡
*f**′(t)* ≡ limit of *(f(t+a)-f(t))/a*
as *a*→0 for any
*f(t)* for which the limit exists. *d ^{n+1}f/dt^{n+1}=(d^{n}f/dt^{n})/dt*
with

'√' is a symbol for square
root. |*x*|=√(*x*)^{2}.
*a*≡*b*
means *a* is defined to be equal to *b*.

π ≡ 1/1-1/3+1/5-1/7+1/9-.... ∃*x* means 'there
exists *x* such that'. ∀*x*
means 'for all *x*'. When ∃
or ∀ are used,
parenthesis after the variable, if present, indicate restrictions on the values
of the variable. **0** is a zero vector (or matrix or tensor), that is a
vector whose all components are zero. A permutation is any rearrangement
(including no rearrangement at all). Permutation is called even if it can be
done using an even number of interchanges and odd if it can be done using an
odd number of interchanges. Theorem: no permutation is both even and odd.

'A⇒B' means 'A implies B', and 'A⇔B' means 'A⇒B and B⇒A'.