Lesson 4 Web

Lesson 4: The Language of Tensors

Introduction

To describe the physical world accurately, we need a language that can handle direction, magnitude, and how quantities transform when we change our point of view. That language is the language of tensors.

In the first three lessons you built a solid foundation in classical mechanics using vectors and ordinary calculus. You learned how to describe motion, forces, and energy in a way that feels natural. Now we take the next important step where we learn a more powerful and general language.

Tensors let us describe physical quantities in a way that is independent of our choice of coordinates. They are the natural tools for talking about space and time together, for handling rotations and changes of frame, and for expressing physical laws in a form that remains true no matter how we look at the situation. This language will serve you well not only in classical physics but also when you later move into relativity and more advanced topics.

In this lesson we begin by carefully examining space and time in classical physics. We explore frames of reference, the geometric structure of space-time, and the principle of relativity. We then develop the ideas of scalars, vectors, tangent vectors, and one-forms, and finally introduce tensors themselves. By the end you will be able to describe particle motion using this elegant new language.

The journey from ordinary vectors to full tensors is challenging, but deeply rewarding. Each new idea builds directly on what you already know, and the payoff is a clearer, more powerful way to think about the physical world.

Space-Time in Classical Physics

In the physics you have learned so far, space and time have always been treated as separate, independent entities. Space is the arena in which things happen, and time is the parameter that tells us when they happen. This clean separation is one of the hallmarks of classical mechanics, and it is so natural that we rarely stop to question it. Yet understanding exactly how space and time work together—or rather, how they remain distinct—is essential before we can build more sophisticated descriptions of the world.

In this section we examine space and time as they appear in classical physics. We begin with space itself, then consider time, and finally explore how the two combine into the classical notion of space-time (written with a hyphen to emphasize that they are still fundamentally separate). We will look at frames of reference, how observers in different frames see the same events, and the geometric structure that underlies all of classical mechanics.

This careful look at the foundations will prepare you for the more powerful mathematical tools—vectors, one-forms, and tensors—that we introduce later in the lesson. By the end you will have a clear picture of the stage on which classical physics plays out, and you will be ready to describe motion and forces with greater precision and generality.

Space

We are basing what follows on the review of Lesson 1. Before we can describe the motion of a particle, we need a place for it to move. We begin by considering the set of all possible positions the particle could occupy during the motion we are studying. This collection of points is called the configuration space of the system—or simply its space.

For the moment we assume that the familiar rules of Euclidean geometry hold everywhere in this space. Such a space is therefore called Euclidean space.

For a system with one degree of freedom, the space is the Euclidean line, written .

For two degrees of freedom, the space is the Euclidean plane, written . This plane can be described using any convenient two-dimensional coordinate system (Cartesian, polar, etc.). Motion appears as a curve in the chosen coordinates, and any point on the plane is labeled by an ordered pair of real numbers ). We often denote such a point by an uppercase script letter, such as P.

For three degrees of freedom, the space is three-dimensional Euclidean space, written . Any point is specified by an ordered triple of real numbers . This space can be coordinatized in many ways—Cartesian, cylindrical, spherical, and others.

Exercise 4.1: Why do we choose to use the Euclidean space , instead of the real vector space, ?

Time

To describe the motion of any system—even something as simple as a single particle—we need to specify all the relevant variables that completely determine its condition at a given moment. This collection of variables is called the state of the system. Whenever the state changes, we say the system is a dynamical system.

Exercise 4.2: A particle’s state is described by its position and velocity . At t=0 the state is x=2 m, v=3 m/s. If the particle moves with constant velocity, write the state at t=5 s. Is this a discrete or continuous description?

The successive states of a dynamical system can be thought of as functions of points in a one-dimensional Euclidean space, . The points in this space are what we call moments of time, and they are simply real numbers.

If we observe the system only at specific, separated moments—from some starting time to a later time —the time interval is

(4.1)

In this case we say the dynamical system is discrete. Its states are discrete mappings of time, producing a sequence of values and giving us a stroboscopic, snapshot-like view of how the system evolves.

On the other hand, if the states change smoothly and continuously with time, we say the dynamical system is continuous. Its states are continuous flows with time, producing a smooth function that describes the evolution.

Exercise 4.3: You observe the position of a falling object only at integer seconds.
1) Is this a discrete or continuous description of the motion?
2) What information do you lose compared to a continuous description?

Discrete mappings are governed by the mathematics of difference equations. Continuous flows are governed by the mathematics of differential equations. When we need to find approximate solutions of differential equations, we look for a discrete mapping that closely approximates the smooth flow we are trying to model.

Exercise 4.4: A population grows according to the difference equation ..
1) Is this discrete or continuous?
2) Write the corresponding differential equation for continuous growth and explain the difference in physical meaning.

This distinction between discrete and continuous descriptions of time is fundamental. It will appear again when we discuss numerical methods and when we move from classical to more advanced formulations of physics.

Exercise 4.5: Reflect on the distinction between discrete and continuous descriptions of time. Give one example from everyday physics where a discrete description is more natural, and one where a continuous description is essential. Why does this distinction become especially important when we move to numerical simulations or more advanced theories?

Space-Time in Classical Mechanics: Frames of Reference

We have an interesting structure emerging based on our description of time. For each moment of time we have an entire configuration space.

Figure 4.1 The Configuration space for any moment of time.

In most cases this can be represented by the Cartesian product, . Note that the resulting space is of the class of spaces called a product space, since it is defined by the Cartesian product. This product space is what we call a frame of reference.

Figure 4.2 A frame of reference.

An element of this space is denoted by the time coordinate and a point in space . The thing to remember is that for this product space the configuration space is fixed, once chosen, so that for successive points in time the corresponding origin point in successive configuration spaces remains the same. In this way it is like a movie being put up on a screen where the points of the screen correspond in each successive frame. Thus the space is fixed while time is allowed to run forward.

Exercise 4.6: Consider a frame of reference with origin at the center of the Earth. A particle is at position m at time t=0. Write the position in this frame at t=5 s if the particle is at rest in this frame. What is the configuration space element at that time?

It is important to note the distinction between a frame of reference and a coordinate system. The frame of reference is an appeal to something that exists physically. The coordinate system is a mathematical construction that allows us to perform mathematical operations that we hope will mimic the reality of the reference frame.

Exercise 4.7: You are inside a train moving at constant velocity relative to the ground.
1) Is the train an inertial frame?
2) Describe the configuration space (product space) for an observer in the train versus an observer on the ground.

Exercise 4.8: Explain in your own words the difference between a frame of reference and a coordinate system. Give an example where changing the coordinate system does not change the frame of reference, and one where it does.

Exercise 4.9: Reflect on the idea that “space is fixed while time runs forward.” How does this product structure () reflect our everyday experience of the world? What would change if space and time were not separable in this way?

The Geometric Structure of Space-Time

When we want to describe situations where one kind of space is attached to every point of another kind of space, we encounter a very useful kind of mathematical structure. This structure has a “scaffolding” that holds everything together, which we call the base space. For classical mechanics, the base space is the one-dimensional Euclidean space , that represents time.

Attached to each point (each moment) in this base space is a complete copy of another space—in this case the three-dimensional Euclidean space . Each of these attached spaces is called a fiber.

The overall structure formed by the base space together with all of its fibers is known as a fiber bundle (or more generally a product space when the attachment is particularly simple).

Figure 4.3 Space-Time as a fiber bundle.

Exercise 4.10: Consider a base space (time) and fibers (space). At time t=0 a particle is at position (1, 2, 3) m. At time t=5
s it is at (4, 2, 3) m. Write the configuration space element at both times and describe the trajectory in the fiber bundle picture.

Exercise 4.11: You are inside a train moving at constant velocity.
1) Describe the base space and fibers for an observer in the train.
2) How does the fiber bundle picture explain why the laws of physics look the same in the train and on the ground?

Exercise 4.12: Explain in your own words the difference between the base space and the fibers in the classical space-time fiber bundle. Why is it useful to separate them in this way?

Exercise 4.13: Reflect on the idea that “each element of the fiber exists for only a moment before its contents are carried into the next fiber in time.” How does this picture of space-time as a fiber bundle match (or challenge) your everyday experience of the world? What would change if space and time were not structured this way?

Bundles are themselves a kind of space. If the fiber is a vector space, then that particular bundle is called a vector bundle. We can label the fiber bundle for classical mechanics the an upper-case script G, G.

So what? Why all of this formalism? What does it buy us? At one level it is just formalism, but it gives us precise language for the further mathematical formulation of physical theories. At another level it allows us to make a prediction right away. If space were not a bundle, then for every instant an entire universe would exist. In the next instant a new universe comes into being fully formed, but with no connection (other than time) to the previous universe. If the old universes are allowed to hang around, how does the new universe gets its contents from the old? On the other hand the idea of a fiber bundle is that the space of the bundle is what is preserved, that structure gives us the link from the past to present. Each element of the fiber exists for only a moment before its contents are carried into the next fiber in time.

The Geometric Principle of Physics

The character of field theory is its geometrical nature. By geometrical I mean coordinate-invariant, in other words it does not matter what coordinate system I choose to use. In fact, I can choose to use no coordinates at all. The easiest approach to this is to recast the well-known and familiar laws of classical physics in this new language—and that is the purpose of this lesson. The guiding principle will be that we can treat the relationships between physical quantities as geometrical relationships between geometrical objects.

Scalars and Vectors

In the language we are developing, scalars remain exactly what you already know them to be — simple numbers that carry no directional information. We will denote a scalar field as a function that assigns a single number to every point in three-dimensional Euclidean space. We usually write such a field using an italicized Latin letter with the point of evaluation shown explicitly, for example

A vector is likewise the same kind of object you have worked with before: a quantity that possesses both magnitude and direction. At each point in space we can have a vector field, which assigns a vector to every point.

This familiar starting point — scalars and vectors — will serve as the foundation for the more general objects (one-forms and tensors) that we will need to describe fields properly in classical field theory and general relativity.

Distance in Space

The most basic way to compare two points in three-dimensional Euclidean space is to ask how far apart they are. The distance between any two points is given by the Euclidean metric. If we let be the straight-line vector connecting the two points, then the distance (or length of that vector) is the norm ,

(4.2)

When we are interested in how a position changes from an initial state to a later state , we look at the displacement vector . The length of this displacement is

(4.3)

This quantity tells us the actual spatial distance traveled between the two positions, regardless of the path taken. It is the foundation for measuring lengths, areas, volumes, and changes in position throughout classical physics.

Exercise 4.14: Two points in have position vectors and . Compute the displacement vector and its length. What does this length represent physically?

Infinitesimal Displacement

When the two points connected by a displacement vector are extremely close to each other—so close that we can think of them as neighboring—we call the vector an infinitesimal displacement. In this situation we use the differential notation and write the infinitesimal distance as

(4.4)

These displacements are vanishingly small by themselves. To obtain any finite change in position, we must add together an infinite number of them, placed head to tail along a path. This idea of building up finite quantities from infinitely many infinitesimal pieces is at the heart of calculus and will be central to how we describe fields and motion in the coming sections.

Exercise 4.15: Two points in have position vectors and . Compute the displacement vector and its length. What does this length represent physically?

Translating Vectors

Once we have an infinitesimal displacement, the natural question is how to build a finite vector from it. The process is straightforward, and we repeatedly add the infinitesimal vector to itself, shifting it from one position to the next along a path. In other words, we perform vector addition an infinite number of times. This is exactly what integration does—it sums up infinitely many tiny contributions to produce a finite result.

This idea of carrying a small vector along a path by repeated addition is a simple example of what we will later call parallel transport. It shows how local, infinitesimal information can be extended over finite distances while preserving the vector’s character.

Exercise 4.16: Explain in your own words how translating an infinitesimal vector along a path produces a finite displacement. Give a physical example from mechanics where this idea is used (e.g., work along a path).

The Tangent Space of a Point

We have seen how an infinitesimal displacement can be translated from one location to another along a path. At any given point we can also imagine rotating such an infinitesimal vector around that point in any direction.

The collection of all possible infinitesimal displacements that can be attached to a particular point—obtained by translating and rotating them while keeping their tail fixed at that point—forms what we call the tangent space at that point.

This tangent space captures every possible “direction you can move infinitesimally” from the chosen point. It is the local arena in which vectors live, and it will become one of the central ideas as we develop the language of tensors.

Exercise 4.17: At a point P on a surface, describe in words what the tangent space consists of. Why is it useful to think of all possible infinitesimal displacements attached to that point?

Tangent Vectors

Suppose we have an arrow that begins at an initial point , and ends at a final point . We can define the vector representing this arrow as

(4.5)

In this form the vector is a directed line segment determined by two points, so we sometimes call it a bilocal object.

To make calculations easier, it is useful to think of the vector as a continuous path parameterized by a variable p that runs from 0 to 1,

(4.6)

where and .

If we now take the derivative of this path with respect to the parameter p, we obtain

(4.7)

This is exactly the vector we started with. In other words,

(4.8)

This perspective shows that a vector can be defined as the derivative of a parameterized curve. Because the result depends only on the local behavior at a single point (the tangent), we call such an object a tangent vector. Another common name for a tangent vector is a contravariant vector.

This way of thinking about vectors as derivatives along curves will be extremely useful as we develop the full language of tensors.

Exercise 4.18: A curve is parameterized by . Show that the tangent vector at any point is . What does this tell you about the nature of tangent vectors?

Space Curves

As we have seen, motion is described by a curve in space. This curve is determined by a position vector that is a function of time. Such a curve is called a trajectory, and we denote it by the symbol .

Figure 4.4 A trajectory in space-time.

But these components depend upon the coordinate basis vectors, right? Not so fast. We can write the basis vectors with complete generality. The position vector of classical mechanics is represented, where O is the origin,

(4.9)

and in terms of a linear combination with generic basis vectors

(4.10)

The velocity is then

(4.11)

Exercise 4.19: Go ahead and write out the coordinate-free vector interpretation of momentum, acceleration, and force.

The Principle of Relativity and Inertial Frames—What Makes Classical Physics Classical

Classical physics rests on a profound idea: the laws of motion should look the same to all observers who are moving at constant velocity relative to one another. This idea is the principle of relativity in its classical form. It tells us that there is no absolute rest frame—only relative motion matters.

In this section we explore what this principle really means. We begin with the concept of inertial frames, frames in which Newton’s first law (the law of inertia) holds true. We then examine how quantities such as position, velocity, and acceleration transform when we change from one inertial frame to another. This leads us to the Galilean transformation, the mathematical rule that connects different inertial frames.

We will also look at what happens when we consider rotating frames and the fictitious forces (centrifugal and Coriolis) that appear in them. Finally, we will see how all inertial frames are related through the Galilean group—a mathematical structure that encodes the symmetries of classical space and time.

Understanding these ideas is essential. They define what makes classical physics “classical” and set the stage for the more general language of tensors we are developing. By the end of this section you will have a clear picture of the framework that underlies all of Newtonian mechanics.

Inertial Frames

An inertial frame is a special kind of reference frame where the laws of physics take their simplest form. In particular, it is a frame where Newton’s first law—the law of inertia—holds true, “A body with no net force acting on it moves in a straight line at constant speed.”

Such a frame has the following properties:

The frame is the product space , combining time and three-dimensional space.

Clocks can be synchronized throughout the frame.

Distance intervals are the same everywhere.

Angles are the same everywhere.

Time intervals are the same everywhere.

When we consider systems with more than one particle, we naturally work with more than one frame. Inertial frames have important relationships to one another:

Any frame whose spatial points are all moving at constant velocity relative to an already known inertial frame is itself inertial.

You can always choose to view one inertial frame as being at rest while another moves relative to it. The frame you choose to regard as at rest is called the rest frame.

The laws of physics are the same for all inertial frames.

This last statement is known as the Principle of Relativity. It was first clearly noted by Galileo in 1638 and later used by Christian Huygens (whose work on the subject began in 1656 and was published in 1703).

This principle is one of the deepest and most powerful ideas in classical physics. It tells us that no inertial frame is preferred over any other — there is no absolute state of rest.

Exercise 4.20: A particle is at rest in frame f. Frame f' moves with velocity m/s relative to f. Write the position of the particle in frame f' as a function of time.

The Law of Inertia and Mach’s Principle

Newton’s first law of motion can be stated in a particularly clear way using the language of inertial frames, “If a particle is far removed from all other influences in the universe, that particle will move with constant velocity with respect to any inertial frame.”

There are deeper questions behind the law of inertia, “Why does a body resist acceleration? Why do inertial frames exist at all?”

Ernst Mach suggested that inertia is not an intrinsic property of a body, but arises from its interaction with all the distant matter in the universe. In other words, the local inertial behavior we observe is determined by the global distribution of mass. A body moves in a straight line at constant speed when it is “at rest” relative to the average motion of the distant stars and galaxies.

This idea, known as Mach’s principle, is not part of the strict mathematical structure of Newtonian mechanics, but it offers a powerful physical intuition about why the laws take the form they do. It will become especially important when we study general relativity.

Exercise 4.21: You are inside a smoothly moving train (constant velocity). A ball is dropped from rest relative to the train. Describe the motion of the ball as seen from the train and from the ground. Which frame is inertial? Explain using the law of inertia.

The Galilean Transformation

We can write down the mathematical rule that lets us relate the coordinates of an event in one inertial frame to the coordinates of the same event in another inertial frame.

Suppose we have two inertial frames, where frame f that we regard as at rest, and frame f' that is moving with constant velocity V relative to f. If we denote the position vector in frame f by , then the position vector in the moving frame f' is given by

(4.12)

while time is the same in both frames

(4.13)

Let us look at what this is telling us. If you are moving at constant velocity V and you observe an object that is at rest in frame f at some initial position , you will see its apparent position change over time by the amount -V t. This makes perfect sense since the relative position shifts at a constant rate equal to your own velocity.

Equations (4.12) and (4.13) are known as the Galilean transformations. They tell us how to convert positions (and therefore velocities and accelerations) from one inertial frame to another.

Exercise 4.22: How do we express the position in the f frame?

Exercise 4.23: Derive the transformation for the velocity of a particle when changing from frame f to frame f' moving with velocity relative to f.

Motion in Inertial Frames

We can calculate the velocity of a particle relative to an inertial frame using the relation

(4.14)

where is the velocity of the moving frame (e.g., the ship) relative to the inertial frame.

Exercise 4.24: If we use the Cartesian coordinate frame (x,y,z) as our configuration space. And if we are aboard a ship at sea, moving with the velocity in the x direction, (we are in the rest frame of the ship) how do we represent the position of a rock in the following situations:
The rock is directly in front of us?
The rock is off to one side of us by a distance ?

Exercise 4.25: Again using the Cartesian frame (x,y,z) as our configuration space, if we are aboard a ship moving with the velocity vector , and there is another ship with velocity vector , how do we represent the motion of the other ship from our ship?

Exercise 4.26: Again using the Cartesian frame (x,y,z) as our configuration space. How do we represent the relative motion of a falling body from our rest frame?

Exercise 4.27: Again using the Cartesian frame (x,y,z) as our configuration space. How do we represent the relative motion of a particle experiencing projectile motion?

The Relative Motion of Inertial Frames

Consider two inertial reference frames, R and R'. Let the velocity of frame R' relative to frame R be the constant vector . If is the position vector of a point (or particle) as measured in frame R, while s the position vector of the same point as measured in frame R', then the two position vectors are related by

(4.15)

together with the identification of time in both frames

(4.16)

This Galilean transformation assumes the validity of the following statement, which we formalize as an axiom of classical physics:

Classical Physics Axiom I: There is no limit to the speed that a particle can attain with respect to some inertial frame.

Rotating Frames

The time rate of change of a physical quantity as measured in frame R is not necessarily the same as the rate measured in frame R'. This difference becomes especially important when frame R' is rotating relative to frame R.

Theorem 4.1: Let Q, be an arbitrary scalar quantity. Denote its time derivative in frame R by and its time derivative in frame R' by . Then

(4.17)

Exercise 4.28: Verify this for a simple scalar (e.g. temperature).

Theorem 4.2 Rotating Derivative Rule: Let be an arbitrary vector quantity. Denote its time derivative in frame R by and its time derivative in frame R' by . Let as the angular velocity of frame R' relative to frame R. Then

(4.18)

Exercise 4.29: Verify this for the position vector.

These results are known as the rotating-frame derivative rule (or transport theorem).

Exercise 4.30: A person stands on a rotating platform. Explain why they feel a centrifugal force even though no real force is acting. How does this relate to inertial frames?

Properties of Angular Velocity

Angular velocity obeys several simple but useful algebraic and differential properties when switching between rotating frames.

Theorem 4.3 Opposite Sign Rule for Angular Velocity: If is the angular velocity of frame R' with respect to frame R and is the angular velocity of frame R with respect to frame R'. Then

(4.19)

Theorem 4.4 Addition of Angular Velocities: If is the angular velocity of frame R'' with respect to frame R, is the angular velocity of frame R with respect to frame R', and is the angular velocity of frame R' with respect to frame R''. Then

(4.20)

Theorem 4.5 Time Derivative of Angular Velocity: If is the angular velocity of frame R' with respect to frame R, is the time rate of change of (as measured in frame R), and is the time rate of change of as measured in frame R'). Then

(4.21)

Exercise 4.31: Using Theorems 4.3–4.5, show that the angular acceleration transforms in a manner consistent with the opposite-sign rule.

Exercise 4.32: Consider three nested rotating frames (e.g., a rotating platform on a rotating ship). Apply Theorem 4.4 to find the total angular velocity relative to the inertial frame.

Transformation Equations for Particle Kinematics

Consider the two frames R and R' introduced earlier, with origins O and O', respectively. Let a particle be located at point P. We define the position vector from O to O', as and the position vector from O' to P, as . By vector addition, the position vector from O to P is

(4.22)

Taking the time derivative of this relation gives the velocity transformation

(4.23)

Applying the rotating-frame derivative rule (Theorem 4.2 / equation (4.18)) to the relative position vector yields

(4.24)

Exercise 4.33: Derive (4.24) in detail.

where the prime denotes quantities measured in the rotating frame R'. Substituting this into (4.23) produces the full velocity relation

(4.25)

Differentiating again with respect to time gives the acceleration. After applying the rotating-frame derivative rule, we obtain

(4.26)

We can again apply the rotating-frame derivative rule and get

(4.27)

We can also write it this way,

(4.28)

We can substitute this into (4.26) to get

(4.29)

This is the general transformation for particle acceleration between an inertial frame and a rotating frame. The terms (Coriolis term), (centrifugal term), and (Euler term) appear naturally when working in the rotating frame.

These extra terms are the origin of fictitious forces in non-inertial frames and will be central when we discuss dynamics in rotating frames.

Exercise 4.34: Explain the steps in the derivation above (particularly the second time differentiation and the origin of each fictitious acceleration term).

The Equation of Motion in Arbitrary Frames

We can make a direct application of (4.29) to get,

(4.30)

where is the translational acceleration term, is the Coriolis term, is the centrifugal force, and is the Euler force.

Simple, right? We can often “knock down” (i.e., set to zero) some of these terms depending on the specific situation. All of the forces on the left-hand side other than the real force arise from the use of a noninertial frame and are collectively referred to as fictitious forces.

Exercise 4.35: Identify which terms vanish (and why) when
    1) the frame R' is translating but not rotating,
    2) the frame rotates with constant angular velocity about a fixed axis,
    3) we are on the surface of the Earth (approximating it as a rotating frame).

Fictitious Gravitational Forces

Recall from the discussion of accelerating frames that the term arises from the transformation of the second time derivative. The fictitious force term appears only when frame R' is undergoing translational acceleration relative to frame R. Comparing this term to gravity gives

(4.31)

Thus, a fictitious gravitational force is experienced whenever the reference frame undergoes uniform translational acceleration. A familiar example is the sensation of being pushed back into your seat when a car accelerates forward in a straight line.

This observation leads to an interesting equivalence. If frame R' is in free fall within a gravitational field, then , so the fictitious force exactly cancels the real gravitational force. The result is weightlessness, as observed by occupants of the frame.

Exercise 4.36: An elevator accelerates upward with acceleration . What fictitious gravitational force does a passenger of mass m experience? What is the apparent weight?

Centrifugal Force

The fictitious force term

(4.32)

is called the centrifugal force. Here, the position vector points from the origin of the rotating frame R' to the particle at point P. Note that is perpendicular to both and . Consequently, the centrifugal force is directed perpendicularly outward from the axis of rotation (the line through the origin O' parallel to ). If the perpendicular distance from the particle to this axis is r sin θ, the magnitude of the centrifugal force is .

Exercise 4.37: Derive the expression for the length of this segment.

This outward force is fictitious—it appears only in the rotating frame and arises from the coordinate transformation.

We experience this force whenever we turn a corner in a car (or any time we move in a curved path in a rotating frame).

Exercise 4.38: In the rotating frame of the Earth, calculate the centrifugal acceleration at latitude φ and discuss its effect on effective gravity.

Coriolis Force

When deriving the effective equation of motion in the rotating frame ( R’ ), several fictitious forces appear. One of the most important is the following, known as the Coriolis force. It is the only fictitious force that depends on the velocity of the particle with respect to the rotating frame R'.

Exercise 4.39: Consider a particle moving with constant velocity in the rotating frame. Show that the Coriolis force is perpendicular to both and and explain its effect on the trajectory (e.g., deflection to the right in the Northern Hemisphere).

Exercise 4.40: Compare the magnitude of the Coriolis force to the centrifugal force for typical Earth-based velocities.

The Galilean Group

We can generalize the set of operations that transform one inertial frame into another inertial frame. If we have the position vector from a point P in to the origin O then we can make a collection of the transformations that map a point of an inertial frame to a point in another inertial frame (a point transformation). We call this group the Galilean group. It has the following four parts.

Special Orthogonal Group

A length-preserving transformation is a rotation. If A is the corresponding transformation matrix, then the position vector transforms as

(4.33)

Equivalently, A satisfies,

(4.34)

with det A=1.

To fully specify a rotation in , three parameters are required. For example, we may choose the angle θ in the plane, the angle φ in the plane, , and the rotation angle ψ about the vector itself. The set of all three-parameter rotations that satisfy (4.33) is called the special orthogonal group and is denoted SO(3). A space with this group structure is called isotropic—the same in all directions. This builds on the rotating-frame results of Theorems 4.1–4.5 and provides the group-theoretic foundation for rotations.

Exercise 4.41: Show that the composition of two rotations is another rotation (this demonstrates the closure of SO(3)).

Exercise 4.42: Parameterize a simple rotation (e.g., about the z-axis) and write the explicit 3×3 matrix.

SO(3) will be important when discussing angular momentum and symmetries later in the course.

The Translation Group

Imagine you can pick up any point in space and slide it to a new location without changing the structure around it. The collection of all possible such shifts forms a mathematical structure known as the translation group.

Figure 4.5 The Translation of a vector.

A translation moves the position vector by a fixed amount

(4.35)

The three components of the translation vector serve as the parameters of this group.

A space that admits such a group of translations—meaning you can shift any point to any other point while preserving the structure—is called homogeneous. It looks the same at every location. This is one of the reasons we work with the Euclidean space rather than the more general , Euclidean space is homogeneous under translations.

This property of homogeneity is fundamental. It tells us that no point in space is special—the laws of physics do not prefer one location over another.

Relative Velocity Transformations

We have already encountered the Galilean transformations that relate coordinates between two inertial frames. When we focus specifically on how velocities transform, we obtain a particularly simple and important set of relations.

If frame f' is moving with constant velocity relative to frame f, the position in the moving frame is given by

(4.36)

When we are interested in a finite time interval Δ t instead of the parameter t, we can write

(4.37)

Since the relative velocity V has three independent components, this gives us three corresponding(4.35) transformation equations—one for each spatial direction.

These relations tell us how the apparent position of an object changes when viewed from a moving frame. They are the foundation for understanding relative motion in classical mechanics and will appear again when we discuss the full Galilean group.

Exercise 4.43: Derive the transformation for the velocity in the moving frame from the above equations.

Homogeneity of Time

Imagine shifting the zero of your clock by some amount. The laws of physics do not change—the motion of planets, the swing of a pendulum, the fall of an apple all behave exactly the same no matter when you decide to start counting time. This is the homogeneity of time. You can shift the origin of time without changing the structure of the laws of physics. This is why we prefer the Euclidean line over the more general for time.

We can express this homogeneity as a simple transformation

(4.38)

where shifting the origin of time by Δ t in one frame corresponds to the same shift in the other

(4.39)

Together with the three parameters from translations and the three from rotations, these ten parameters (three translations, three rotations, three boosts, and one time shift) define the full set of transformations that take one inertial frame into another. They are the parameters of the larger structure known as the Galilean group.

This homogeneity of time—the fact that the laws of physics do not depend on when you start your clock—is one of the deep symmetries of classical physics. It will become especially important when we later explore Noether’s theorem and conservation laws.

Exercise 4.44: Derive the transformation for the velocity in the moving frame from the above equations.

Exercise 4.45: Reflect on the Principle of Relativity. Why is it one of the deepest ideas in classical physics? How does it set the stage for special relativity? Give an example from everyday life that illustrates the principle.

The Language of Tangent Vectors, One-Forms, and Tensors

Scalars

At the most basic level, a scalar is simply a number. It carries no directional information and has no index structure.

While the idea may not seem important yet, we can already think of a scalar as a rank-0 tensor—the simplest kind of indexed object, with no indices at all. This perspective will become very useful as we build up to more complicated tensors later in the lesson.

Exercise 4.46: A scalar field is given by . Evaluate this scalar at the point (1,2,3) in Cartesian coordinates. Is this value the same in every coordinate system? Why or why not?

Tangent Vectors as Indexed Objects

Suppose you have a set of three quantities that change under a coordinate transformation in exactly the same way as the components of a vector. Such a set is what we call a tangent vector. Sometimes these are called a contravariant vector, because when the basis vectors transform, the components transform in the inverse way to keep the overall vector the same.

The invariant quantity formed from these components,

(4.40)

is called the squared norm of the tangent vector.

Tangent vectors have a single contravariant index (superscript), thus they are rank one tensors.

This indexed way of writing vectors will become increasingly important as we build the full language of tensors. It gives us a compact, coordinate-independent way to describe directional quantities while making the transformation properties explicit.

Exercise 4.47: A tangent vector has components , , . Compute its squared norm . What does this quantity represent physically?

The Kronecker Delta

One of the most useful symbols we will encounter repeatedly in tensor analysis is a simple object called the Kronecker delta, named after Leopold Kronecker (1823–1891). We will see that the Kronecker delta acts like a selector that picks out matching components and ignores the rest.

To see where it comes from, consider the scalar product of two vectors expressed in components:

(4.41)

This raises an interesting question, “What is the scalar product of the basis vectors?” The answer is

(4.42)

This is the definition of the Kronecker delta.

We can represent the Kronecker delta as a matrix. In three-dimensional Euclidean space it looks like the identity matrix:

(4.43)

Using this, the scalar product simplifies beautifully

(4.44)

Exercise 4.48: Explain this result.

What happens when we take the product of two Kronecker deltas?

(4.45)

Again, this is equal to 0 for every case where i≠j, so we rewrite it

(4.46)

How do we evaluate ? This depends on how many possible dimensions there are, it will be a sum consisting of 1 as a term for every dimension. Thus in we will have

(4.47)

and similarly

(4.48)

(4.49)

Now consider a product with a shared index

(4.50)

There is a useful rule where contracting a Kronecker delta with another indexed object replaces the repeated index:

(4.51)

We can expand this in to make sure it is true,

(4.52)

(4.53)

Because the repeated index j is summed over, we see that

(4.54)

This operation is called contraction of indices (or simply contraction). It is used extensively in field theory and tensor analysis. The pattern extends to more Kronecker deltas

(4.55)

We will hear more about contraction as we proceed.

Exercise 4.49: Consider the expression .
    1) Use the contraction rule to simplify it.
    2) Expand it fully in to verify your result.
    3) Interpret physically what this simplified expression represents (think about the scalar product of two vectors).

Exercise 4.50: Simplify the expression using the contraction rule. Expand it fully in to verify your result. What physical quantity does this represent?

Cartesian Scalars and Vectors

Before we move on to more general objects, it is worth pausing to connect the scalars and vectors we already know to the specific coordinate systems we use in calculations. You might wonder—”Aren’t we trying to develop coordinate-free notation?” Yes, we are. But the practical truth is that when we need to perform detailed calculations in physics, we almost always choose a particular coordinate basis. So we will occasionally step from the elegant coordinate-free view back to the concrete methods needed for actual computation.

Cartesian Scalar

A Cartesian scalar is a quantity that can be represented as a function in any Cartesian coordinate system. If we change from one Cartesian frame R to another Cartesian frame R’, the value of the function may change, but the scalar itself does not

(4.56)

The scalar is an invariant—its geometric meaning remains the same no matter which Cartesian coordinates we choose. Only the numerical values of the components adjust with the coordinate system.

Cartesian Vector

A Cartesian vector is a collection of components associated with a particular Cartesian frame. When we change from frame R to frame R', the components transform according to

(4.57)

For a pure rotation the transformation is

(4.58)

where is the rotation matrix. For a pure translation the components remain unchanged

(4.59)

Once again we see that while the specific components depend on the coordinate system we choose, the vector itself exists independently of those coordinates. It is a geometric object that transcends any particular basis.

This distinction—between the invariant geometric entity and its coordinate-dependent components—is a central theme that will run through our development of tensors.

Exercise 4.51: Why is it useful to have both coordinate-free and coordinate-dependent descriptions available?

Exercise 4.52: Explain the difference between a Cartesian scalar and a general scalar. Why is it useful to have both coordinate-free and coordinate-dependent descriptions?

The Dual of a Vector

Suppose you have a way to take any vector and produce a single real number from it in a linear fashion. This kind of object—a real-valued linear function on vectors—is the key idea we need next.

In earlier lessons we learned to write a vector as a linear combination of basis vectors

(4.60)

Now we consider a different kind of object, where a real-valued linear function that takes a vector as input and returns a number. We denote such a function by .

Because it is linear, it satisfies

(4.61)

and

(4.62)

If we define , , then for any vector we have

(4.63)

Notice that the indices on appear “downstairs,” in contrast to the upstairs indices on the components of the vector. This reversal of index position is the hallmark of what we call the dual of the vector.

We can think of the linear function itself as an object that pairs with vectors to produce numbers. If belongs to the vector space V, then belongs to another vector space called the dual space of .

The components with lower indices are called covariant because a change in scale of the basis vectors affects them in the opposite way compared to the contravariant components of ordinary vectors.

One-Forms

Suppose we have a particle moving through a gravitational field. As it moves a tiny distance , in the ith direction, a certain amount of work is done. The sign of that work depends on whether the motion is with or against the field. If the particle moves away from the gravitating body the work is positive; if it moves toward the body the work is negative.

For simplicity, assume the gravitational force is uniform and constant and that the motion is linear. In this case the work done is proportional to the number of displacement elements traversed in each direction. We can think of this work as a function that takes a displacement vector as input and returns a scalar (the total work) as output. This is very much like a scalar product, but what exactly is being “multiplied” by our displacement?

Let w be the work done when the particle moves across one displacement element in the ith direction . If the particle traverses such elements, the total work in that direction is . We denote this quantity by .

The total work done Wdone by the field on the particle is then given by the sum over all directions

(4.64)

Here Φ represents the force (or field strength) corresponding to the components . What does this mean? Each displacement element can be thought of as piercing a local “surface” perpendicular to the ith direction. The total work is therefore the sum of the number of such surfaces penetrated in each direction.

This object—that takes a vector as input and returns a scalar—is called a is called a one-dimensional differential form, or a one-form, or just a 1-form. Sometimes it is called a covector or covariant vector. This is the same situation we saw above with the dual vector.

We can write the infinitesimal displacement in terms of a basis

(4.65)

A general one-form is then expressed as

(4.66)

By convention we often use Greek letters to symbolize differential forms.

A 1-form has a single covariant index, and it is thus a rank one tensor.

This dual relationship between vectors and one-forms is fundamental. It will allow us to build the full language of tensors in a natural and powerful way.

Exercise 4.53: A one-form pairs with a vector to give . If , , and , compute the value of . What does this number represent physically in the context of work in a field?

Exercise 4.54: Reflect on the dual relationship between tangent vectors and one-forms. Why is it useful to have both contravariant and covariant objects? How does this prepare us for the full language of tensors?

An Introduction to Tensors

We have already met scalars (rank zero) and vectors (rank one). Now we take the natural next step and combine vectors to form more general objects.

From the two tangent vectors and in . From them we can form the nine quantities . These nine quantities form the components of a tensor of the second rank. This is sometimes called the outer product of the vectors and . The outer product is sometimes called the tensor product and is denoted .

The tensor is a rather special one because its components are related in a particular way. However, if we add together several such products, we obtain a general second-rank tensor

(4.67)

The crucial property of a general tensor is that under a change of coordinates its components transform in exactly the same way as the quantities .

We may lower one of the indices in by the same process we used for vectors. This gives us tensors such as or . We can lower both indices to get .

Exercise 4.55: What happens when we lower the indices of to get either , , or ?

If we set the two indices equal in (i.e. m=n) ) and sum over the repeated index, we obtain the scalar . This is equal to .

Exercise 4.56: Show that is a scalar C.

We can continue this process and multiply more than two tangent vectors or one-forms together, making sure all indices are distinct. In this way we can construct tensors of higher rank. If all the vectors are contravariant we get a tensor with all upper indices. We can then lower any of the indices to obtain a general tensor with any combination of upper and lower indices.

We may set a covariant index equal to a contravariant index. We then sum over all values of that repeated index. The repeated index becomes a dummy index. The resulting tensor has two fewer effective indices than the original one. We saw this earlier for vectors; for general tensors the process is also called contraction. For example, starting with the fourth-rank tensor , one way of contracting it is to put p = s, this gives the second rank tensor having only nine components, arising from the three values of m and n. We could contract again to get the scalar .

We can now appreciate the important rule of balancing of indices. In any valid equation, every free index must appear once (and only once) in each term, and always in the same position (all upper or all lower). A repeated index in a term is a dummy index and must appear once upper and once lower. It can be replaced by any other letter not already used in that term. Thus . An index must never occur more than twice in a term.

This balancing rule is one of the most useful guides when working with tensors. It helps you catch mistakes and understand how quantities transform.

Tensors are the natural language for describing fields in a coordinate-independent way. You will see their power more clearly as we apply them to physical situations in the coming sections.

Tensors in Coordinate-Free Notation

Another powerful way to think about tensors is to focus on what they actually do, rather than on the arrangement of their indices. Imagine a machine that takes vectors and one-forms as inputs and produces a single number as output. That machine is a tensor. The rank of the tensor is simply the number of inputs it requires. A second-rank tensor, for example, takes two inputs (which can be tangent vectors or one-forms) and returns a scalar.

For an arbitrary rank-two tensor you can use a coordinate-free notation with boldface type and empty slots to show where the inputs go

(4.68)

You can shorten this to just C when there are no inputs specified. You can place either a tangent vector or a one-form in each slot. For example,

(4.69)

You can see that the tensor is linear by examining how it behaves when we scale or add inputs. For a rank-three tensor we can introduce the scalars e and f along with the new vector to obtain

(4.70)

In this geometrical language we can also rewrite the scalar product in a revealing way

(4.71)

This coordinate-free viewpoint emphasizes that tensors are fundamentally multilinear maps. It frees us from any particular coordinate system and makes the geometric meaning clearer. You will find this way of thinking extremely useful as we apply tensors to physical fields in the coming sections.

The Tensor Product

Suppose you have three vectors , , and . You can combine them to create a new mathematical object that takes three other vectors , , and as input and returns a single scalar as output. This operation is called the tensor product and is written, in coordinate-free notation,

(4.72)

Because the result must be a scalar, the natural definition is

(4.73)

In this way we have taken three ordinary vectors and constructed a single object of rank three—a rank-three tensor.

This construction is the foundation for building more general tensors. It shows how we can combine simpler objects to create powerful tools that can act on multiple vectors at once. Such tensor products will be essential when we describe fields and their interactions in classical field theory and general relativity.

Tensors in Component Notation

We have already seen how tangent vectors and one-forms can be expressed using basis elements. The same idea extends naturally to tensors. What this really means is that we are working with a product space built from vector spaces.

Consider the situation we had earlier in (4.66). We have the Cartesian product of two vector spaces, , where the second space is the dual space of . A mapping that is linear in each argument is called a multilinear mapping.

The set of all multilinear mappings defined on a product of vector spaces forms a tensor space. In analogy with vectors being elements of vector spaces and one-forms being elements of dual spaces, tensors are elements of tensor spaces.

It turns out that multilinear mappings having the same vector space structure are isomorphic. So tensor spaces such as and are isomorphic. An element of one tensor space is an element of any of them.

For our rank two tensor C from above, we can write

(4.74)

In the situation described by (4.66) we write

(4.75)

This component notation gives us a practical way to work with tensors while keeping the underlying geometric meaning clear.

Tensor Addition

Imagine you are adding two forces or two contributions to a field. You can only add them meaningfully if they are “the same kind of thing”—same number of directions, same way of transforming. The same rule applies to tensors.

It is only possible to add tensors if they have exactly the same index structure—that is, the same number and position of upper and lower indices.

For example, if we have two tensors with the same index configuration, their sum is simply

(4.76)

This is perfectly acceptable.

On the other hand, you cannot add tensors that have different index structures, such as

(4.77)

The reason is that the two objects live in different tensor spaces and do not have a natural way to be combined component by component. Addition is only defined within the same tensor space.

This rule keeps our mathematics consistent and physically meaningful. It ensures that when we add tensors, the result is still a well-defined tensor of the same type—ready to be used in field equations or conservation laws.

Exercise 4.57: Given two tensors and , write their sum. Now try to add and . Is this allowed? Why or why not?

The Metric Tensor

We can generalize the familiar scalar product into a more powerful object—a tensor. In coordinate-free notation we write

(4.78)

This object is called the metric tensor. It is very important!

The metric tensor is symmetrical, any tensor is symmetrical if the order of its inputs don’t matter (this is very similar to matrices)

(4.79)

In components we write

(4.80)

How does this relate to the Euclidean metric we examined way back at (4.2)? Recall that the distance between two points is

When we make this continuous and write it in terms of components, it becomes the line element

(4.81)

We can make a transformation to a new set of coordinate axes, where each of the of (4.81) becomes a linear function of of the new set of axes so that the quadratic form (4.80) is a general quadratic form,

(4.82)

A generic tangent vector, , has the form

(4.83)

as we saw in (4.59) when we discussed Cartesian vectors.

Suppose we have another tangent vector . We can form the linear combination for any scalar λ. The length of this general vector is

(4.84)

As you can see, expanding and requiring that this expression is a scalar (invariant) for all λ shows that each coefficient must separately be an invariant. Examining the term you can see that we have

(4.85)

We can interchange the indices in the second term,

(4.86)

Exercise 4.58: Why can we do this?

so that we can evaluate (4.86)

(4.87)

From this we see that the second term in (4.84) is indeed an invariant, it is the inner product of and .

We define g as the determinant of . If this determinant were to vanish (g=0), then the axes would not provide independent directions in space (the basis vectors would not span independent directions). We thus assert that the determinant must not vanish, g≠0. If we have orthogonal axes, the diagonal elements of become 1 each and the off-diagonal elements are all 0. From this we can calculate g=1.

Exercise 4.59: Perform this calculation.

We now define a one-form by lowering the index,

(4.88)

Since g≠0, we can solve these equations for (we can raise indices with the inverse),

(4.89)

We calculate each as the cofactor of the corresponding in its corresponding determinant, divided by the determinant itself.

If we substitute the value of from (4.83) with that in (4.88)

(4.90)

This equation must be true for any three quantities and we can make the inference,

We can use (4.88) to lower any index in a tensor. We can use (4.89) to raise any index in a tensor. Raising and lowering indices are operations that are collectively called index gymnastics. They are all applications of contraction and allow us to move freely between contravariant and covariant indices while preserving the tensorial character.

Exercise 4.60: In a Cartesian coordinate system the metric is . Compute the squared length of the vector with components (3, 4, 0). What happens to this length if we rotate the coordinate system?

Non-Orthogonal Coordinate Frames

So far we have mostly worked with orthogonal coordinate systems, where the basis vectors are at right angles and the familiar Pythagorean theorem holds directly. But there are situations where the coordinate axes are not orthogonal—they are skew, or non-orthogonal. In such cases the basis vectors are not generally orthonormal. They must, however, still be linearly independent.

Geometrically, this means the basis vectors cannot all lie in the same plane. From an algebraic point of view it means that no nontrivial linear combination of them can vanish.

You can still expand any vector in terms of these basis vectors in the usual way. The squared length element is then given by

(4.92)

While it is possible to define such a metric tensor , it will not necessarily be Pythagorean in the familiar sense because the axes are skew.

This more general framework becomes important when we work with curvilinear coordinates or when we need the full power of tensor methods in curved spaces. It prepares us for the richer geometry we will encounter later.

Exercise 4.61: Explain in your own words why the metric tensor is needed when the basis vectors are not orthogonal. Give a physical example where non-orthogonal coordinates might be useful.

General Cartesian Systems and Affine Transformations

So far we have mostly worked with orthogonal coordinate systems and simple translations. But we can generalize further. The study of the invariants under a certain broad class of linear transformations is known as affine geometry.

A general affine transformation in three dimensions can be written as

(4.93)

If we set the constant terms to zero (), the transformations preserve the origin of the coordinate system. Such transformations are called centered affine transformations. They form a group under composition.

Exercise 4.62: Show that centered affine transformations form a group.

The covariant and contravariant bases are reciprocal. From the raising and lowering relations we have

(4.94)

and

(4.95)

Combining these gives

(4.96)

This reciprocity is a fundamental property that allows us to move freely between covariant and contravariant indices while preserving the tensorial character. It will be extremely useful as we work with general coordinate systems and more advanced tensor operations.

Exercise 4.63: Show that the composition of two centered affine transformations is another centered affine transformation. What does this tell you about the group structure?

Transformation of Coordinates

So far we have mostly worked in a single coordinate system. But in practice we often need to change from one set of coordinates to another. The rules that tell us how basis vectors and tensor components transform under such changes are essential for working consistently in different frames.

We can relate a new set of basis vectors to an existing set by applying some transformation matrix Λ,

(4.97)

The inverse relation is

(4.98)

We can express a tangent vector in the new basis as

(4.99)

which gives the transformation rule for the components

(4.100)

The same formulas apply to a one-form. For the dual basis (for one-forms) we have

(4.101)

and

(4.102)

For the components of a one-form

(4.103)

(4.104)

When we compose two successive transformations, the product of the two transformation matrices is called a concatenation. The combined transformation for contravariant components is

(4.105)

while for covariant components it is

(4.106)

These rules allow you to change coordinate systems in a consistent way while preserving the tensorial character of your objects. They will be extremely useful as we work with more general curvilinear coordinates and when we move into curved space-time.

The Levi-Civita Tensor

This is an important idea pronounced (Levy-Chiveeta). It is sometimes called the Levi-Civita symbol, or the permutation symbol. In older literature it is sometimes called an indicator. It is written .The indices i,j,k represent the three directions of space. The ε symbol takes on one of three values: 0, 1, or –1.

First of all, if any two indices are the same.—for example, and are both zero.

The only time is not zero is when all three indices are different.

There are six possibilities: , , , , , .

The first three have value 1, and the second three have value −1.

What is the difference between the two cases? Here is one way to describe it: Arrange the three numbers 1, 2, 3 on a circle, like a clock with only three hours.

Figure 4.5 A representation of the permutations for the Levi-Civita tensor.

Start at any of the three numbers and go around clockwise. You get (123), (231), or (312), depending on where you start. This is called an even permutation.

If you do the same going counterclockwise, you get (132), (213), or (321). This is called an odd permutation.

The rule for the Levi-Civita symbol is that , for the clockwise sequences, and for the counterclockwise sequences.

It turns out that the Levi-Civita tensor describes the volume of a parallelepiped in space. This makes it an extremely useful object when we need to compute volumes, determinants, or vector products in a coordinate-independent way. You will see it appear again when we work with volume elements and the full machinery of tensor calculus.

Exercise 4.64: Compute for and . What physical quantity does this represent?

Properties of the Levi-Civita Tensor

The volume of a parallelepiped spanned by three vectors has two possible states.

The volume vanishes when the three legs are not linearly independent.

If the volume is determined for one parallelepiped, then it is determined for them all.

With these facts in mind we require only a number and antisymmetry to determine ε.

Here is the relationship between the Levi-Civita tensor and the Kronecker delta.

(4.107)

Second-order determinants can be found

(4.108)

A third-order determinant is written

(4.109)

These identities show how the Levi-Civita tensor encodes the idea of oriented volume and allows us to compute determinants in a coordinate-independent way. You will see it appear again when we work with volume elements, vector products, and the full machinery of tensor calculus.

The Vector Product

The vector product of two vectors is one of the most useful operations in classical physics—it gives a vector perpendicular to both inputs whose magnitude is the area of the parallelogram they span.

In tensor notation the vector product of two vectors can now be written as

(4.110)

The triple scalar product is written

(4.111)

These expressions give us a compact and coordinate-independent way to write the familiar vector product and scalar triple product. You will see them appear again when we work with angular momentum, torque, volume elements, and many other physical quantities.

Exercise 4.65: Reflect on how the language of tensors unifies scalars, vectors, one-forms, and higher-rank objects. Why is this language so powerful for describing physical fields? Give an example of a physical law that becomes clearer or more elegant when expressed in tensor form.

Particle Motion in the Language of Tensors

You have already learned to describe the motion of a particle using ordinary vectors. Now we can express the same physical quantities using the more powerful and general language of tensors.

The position vector is also written or .

The trajectory is a time-dependent curve whose location is determined by the time-dependent position vector , or or .

The velocity vector is the time-derivative of the time-dependent position vector,

(4.112)

The acceleration vector is the time-derivative of the velocity vector

(4.113)

The momentum vector is the scalar multiple of the mass and the velocity vector.

(4.114)

The kinetic energy is one half of the product of the mass and the square of the velocity

(4.115)

Newton’s equation of motion is written

(4.116)

Using tensor notation, all these familiar quantities become components of well-defined tensors. This makes it much easier to change coordinate systems and prepares us for more advanced descriptions of fields and motion in curved space-time.

Doing This Stuff in Mathematica

For the tensor analysis we will need to load xAct. This is a free add-on package available from the web site: http://xact.es/. We launch it:

We next need to establish our space, in this case . We will use the DefManifold command:

DefManifold[M, dim, {a, b, c, ...}] defines M to be the label we will use for an dim - dimensional space. This label can be anything we want it to be. We then write dim as the number of dimensions in the space. This can be a positive integer, or it can be a constant symbol. Then we list the abstract indices a, b, c, ... that will be used by our tensors. In our case we will define a manifold called M4, it will have four dimensions, and it will use a number of lower - case Greek letters. I will use the Greek letters, because most symbols I will want to use will not interfere with the chosen indices.

We can define a tangent vector,

We can define a one-form,

We can define a general tensor

α
	β

We can define the metric tensor,

DefMetric[sign of the determinant, the label of the metric and its indices, the label for the covariant derivative (we will worry about what this means a little later, for now we can think of it as our semi-colon symbol), {the symbols associated to the covariant derivative}, and how to print the metric]

Most of this stuff is gobbledygook right now.

We can demonstrate how the metric tensor products produce the Kronecker delta by contraction

	μ
ρ

we can demonstrate metric contraction.

α
	γ

α	γ