Principle of Least Action

The Principle of Least Action, or Stationary Action Principle, forms the heart of physics. It encompasses the equations of motion and mathematically expresses how a physical system transitions over time.

To appreciate this principle, recall Newton's second law, which determines a system's trajectory by considering the masses of particles, the forces acting upon them, and their initial positions and velocities, defined by F=ma, where m represents mass and a acceleration. In contrast, the principle of least action calculates the system's trajectory using only the initial and final positions, masses, and velocities, omitting forces altogether. This principle subsequently selects the path that minimizes the action.

Before delving into the specifics of the action, it's essential to introduce an alternative formulation to Newton’s equations known as the Lagrangian.

The Lagrangian L is computed as the difference between kinetic energy T and potential energy V, where T is derived from the mass and the square of velocity, and V is computed based on mass, gravitational constant, and height above ground (the calculation of potential energy varies depending on the system).

Why is the Lagrangian expressed as the difference between kinetic and potential energy? Because as the system evolves, it transforms potential energy into kinetic energy, and this difference captures the dynamic relationship between these energy types. It's crucial to note that total energy is the sum of these two forms.

The inputs to the Lagrangian include positions x and velocities v, denoted by (dot{x}), where the dot signifies the first derivative, as velocity is the first derivative of position.

To compute the Lagrangian, one must minimally know the velocities, general coordinates, positions, and particle masses. Potential energy hinges on the positions of particles, as it denotes the potential work they can perform, whereas kinetic energy relies on particle velocities, reflecting their motion.

How does the action factor in? Imagine two points on a curved plane; multiple paths exist between them, but only one represents the shortest distance. The action parallels this scenario: to determine the system's trajectory, we must select the path that minimizes the action. Notably, the action remains stationary throughout the system's evolution.

Since the action must be stationary, its first-order partial derivative must equal zero:

[ frac{partial A}{partial x} = 0 ]

At a high level, the action is defined by the path integral of the Lagrangian over a specified time interval ([t_1, t_2]). While the integral of a function from point (t_1) to (t_2) is typically understood as the area under the curve, the path integral of the Lagrangian should be viewed as the integration of a functional—a function that takes other functions as input and produces a scalar output. The input is the Lagrangian, and the output defines the action. Across the potential paths between (t_1) and (t_2), the system will select precisely the path that minimizes the action.

Here's a simplified formula for the action as the path integral of the Lagrangian:

[ A = int_{t_1}^{t_2} L , dt ]

The definite integral can be computed as the Riemann sum of products of the Lagrangian's output and the change in time, denoted by (dt). Essentially, we compute the action by minimizing it over the time interval.

The action comprises the path integral of the Lagrangian between the system's initial and final positions. This means the path integral minimizes the action by evaluating the difference between potential and kinetic energy. The fundamental theorem of calculus allows us to compute the action as a continuous interval between (t_1) and (t_2), though it can also be calculated in discrete time steps (N). If we envision the action as a sum of discrete time steps (N), we compute it as the sum of products of the Lagrangian's value at each time step and the corresponding time (t).

The Lagrangian typically depends on positions and velocities but can also be time-dependent. A Lagrangian is termed time-dependent if it changes over time, even with constant positions and velocities. For a time-independent formulation, we substitute (L(x, dot{x})) into the equation to indicate dependence on positions and velocities:

[ frac{d}{dt} left( frac{partial L}{partial dot{x}} right) - frac{partial L}{partial x} = 0 ]

From the law of conservation of momentum, the derivative of the sum of all momenta in a system equals zero. In an isolated system, total momentum remains conserved. The derivative of a constant is zero, as the rate of change is held ceteris paribus.

Similarly, the law of conservation of energy asserts that an isolated system's total energy is conserved across transformations: the time derivative of total energy remains zero. Unlike momentum, energy exists in various forms. It is the total of these forms that is conserved, primarily kinetic and potential energy.

The Lagrangian, defined as the difference between these two energy forms, implies conservation of energy when invariant under time translations.

A similar phenomenon occurs concerning the action. Nature selects the trajectory that minimizes the action. This minimization resembles optimization problems, where the action encompasses numerous variables, including all coordinates at every moment. This extremizing nature is articulated by the Euler-Lagrange equation, which forms the equation of motion.

What are the Euler-Lagrange equations? They are differential equations guiding a system's movement from one moment to the next. While I won't derive the equations here, intuitively, we can set the derivative of the action (dA) with respect to position (dx) to zero. In other words, we consider a slight variation in the path and require that the partial derivative of the action equals zero.

This yields the two components of the Euler-Lagrange equation: the time derivative of the partial derivative of the Lagrangian concerning velocity and the partial derivative of the Lagrangian concerning position. These components represent changes in kinetic (momentum) and potential energy. Setting the difference between these two quantities to zero results in the action-minimizing Euler-Lagrange equation.

In a single coordinate or degree of freedom, the Euler-Lagrange equation appears as follows, where (L) indicates the Lagrangian, (dot{x}) denotes velocity, and (x) is position.

[ frac{d}{dt} left( frac{partial L}{partial dot{x}} right) - frac{partial L}{partial x} = 0 ]

In layman's terms, this conveys that the time derivative of the partial derivative of the Lagrangian with respect to velocity minus the partial derivative of the Lagrangian with respect to position equals zero. Intuitively, this can be rephrased as the instantaneous rate of change of the Lagrangian concerning velocity minus the instantaneous rate of change concerning position is stationary.

To distill further, the Euler-Lagrange equation implies that a physical system's motion corresponds to an extremum of the integral of the Lagrangian, which is the action.

The equation can be generalized to arbitrary coordinates ((x, y, z, ldots, n)):

[ frac{partial L}{partial q_i} - frac{d}{dt} left( frac{partial L}{partial dot{q}_i} right) = 0 ]

In practical instances, the action is a functional—a function of a function that maps from a function input (the Lagrangian) to a scalar output (the action's value).

While the stationary action principle allows for efficient trajectory calculations of a physical system, it requires knowledge of the initial and final positions. In its absence, we resort to the Newtonian framework, which mandates knowledge of particle positions and initial velocities.

The stationary action principle can also be adapted to quantum physics, albeit with significant caveats, where all potential paths between initial and final states are taken into account, and the action sums the probability amplitudes of each path to determine the system's probabilistic evolution.

Given this formulation, the classical stationary action principle can be viewed as a special case of the quantum formulation, where the stationary action paths dominate among all possible paths.

Lorentz Transformations

Understanding Lorentz Transformations serves as a gateway to Einstein's Special Theory of Relativity. They provide the mathematical framework for calculating relativistic spacetime transformations within inertial or uniform frames of reference—frames that exclude gravitational effects.

A key concept in special relativity is that motion can only be described concerning a specific frame of reference, rather than in absolute terms. For instance, while driving, I may feel stationary in relation to my car but am moving when considering my house.

The notion of relativistic motion is not new; it existed in classical mechanics and was initially articulated by Galileo.

The groundbreaking insight within special relativity is not about relativistic motion itself but rather identifying what remains constant across spatial translations. Classical mechanics typically views all motion as relative, while space and time coordinates only change additively, remaining static and independent for all observers.

The classical assumption regarding relative motion implied that light should follow relativistic laws. For example, if I stand still with a flashlight while you drive with one, the light from your flashlight should measure as the speed of light combined with your velocity.

However, experimental evidence contradicts this assumption. Regardless of the frame of reference, light maintains its constancy. Empirical findings confirm that the speed of light is absolute.

Rather than dismissing the observation as erroneous, Einstein proposed the constancy of light speed as a fundamental law of nature. If light consistently measures the same, then the coordinates of space and time must be represented differently.

To grasp how Einstein's theory of special relativity achieves this, one must first understand the simplified equations of motion as outlined in classical mechanics. These equations will be modified to ensure that relative motion among observers does not alter light's speed but transforms the interwoven metrics of space and time. This unique approach results in varying measures of time and distance across observers as velocities approach the speed of light.

The equations of motion are often summarized with the acronym SUVAT (s = distance, u = initial velocity, v = velocity, a = acceleration, t = time):

[ s = ut + frac{1}{2}at^2 ]

Minkowski Metric

To clarify Lorentz transformations, we will utilize spacetime diagrams that reverse the distance and time axes, representing time on the x-axis and distance on the y-axis. The y-axis will depict large distance intervals, as we aim to explain motion relative to light speed. Light travels at (3 times 10^8 , text{m/s}^2). In our spacetime diagrams, one second will correspond exactly to this distance, resulting in the straight diagonal of our diagram forming a 45-degree angle between the axes, symbolizing the constancy of light speed over time. The diagonals across a Cartesian grid denote the asymptotic limits of light speed, constraining our translations of time on the y-axis and space on the x-axis.

Any straight line diagonal to our Cartesian grid at angles other than 45 degrees indicates uniform motion at subluminal speeds. In the Newtonian view, the speed of light is akin to any other speed, meaning an obtuse angle greater than 45 degrees suggests faster-than-light velocity. Moreover, the speed of light will appear relative to a frame of reference. If I travel at half the speed of light in the same direction, I will perceive light as moving at half its speed since I am approaching it at half its velocity. This model rests on the assumptions that time and distance units remain constant for all reference frames.

Transitioning from viewing space and time as independent measures to integrating them into a continuum known as spacetime involves transforming the time variable into a measure of distance. We achieve this by multiplying the time variable by (c), the speed of light constant. When we multiply (c) by (t), we obtain (ct), which corresponds to (1 , text{light m/s}^2).

In the Newtonian-Galilean framework, two reference frames (S) and (S') are represented by coordinates ((x, t)) and ((x', t')) respectively, where the apostrophe indicates relative frames of reference (not to be confused with differentiation). These frames can be inverted, and the inverses are equivalent within Galilean relativity. From frame (S), the coordinates of (S') are given by:

[ x' = x - vt, quad t' = t - frac{vx}{c^2} ]

Conversely, from frame (S') to (S):

[ x = x' + vt', quad t = t' + frac{vx'}{c^2} ]

These transformations ultimately render light relative rather than spacetime invariant. The challenge is to translate from (S) to (S') while preserving (c) (the speed of light) and proportionally scaling the time and distance variables within the spacetime continuum.

To derive these transformations, we can utilize the spacetime diagrams, scaling time by the constant (c approx 299 times 10^6). The desired translation is expressed as follows:

[ x' = frac{x - vt}{sqrt{1 - frac{v^2}{c^2}}}, quad t' = frac{t - frac{vx}{c^2}}{sqrt{1 - frac{v^2}{c^2}}} ]

We will leverage this symmetry between frames of reference to derive the gamma factor, which serves as the common scaling factor for spacetime transformations while reflecting light's constancy. The Galilean symmetry of relative motion is illustrated by the graphs below, depicting the two reference frames as inverses of each other:

Since light's speed remains constant across all reference frames, if we start both frames from the origin ((x = 0, t = 0)), the path of light will satisfy the following equations:

[ x = ct, quad x' = ct' ]

The conversion from (x) to (x') is expressed by the equation below, where (x') is the difference between (x) and the product of velocity and time. To derive the Lorentz transformation, we need a scaling factor (beta), which equals (frac{v}{c})—the ratio of velocity to the speed of light—used to scale (ct) (light-speed scaled time). Expanding this expression reveals that it algebraically reduces to the Newtonian transformation in the brackets. As we will see, as the Lorentz factor approaches 1, the Lorentz transformations become equivalent to their Newtonian counterparts, corresponding to our everyday perceptions of event simultaneity.

The formulas below illustrate how we transition from the initial formula to the gamma-scaled transformation formula for relative position:

[ x' = gamma (x - vt) ]

Similarly, we can derive the time transformation from frame (t) to frame (t') with the equation below. Using spacetime diagrams, we start with (ct'). Notably, (ct') can be computed through the difference between (ct) and the scaled (vx), with the entire expression adjusted by the Lorentz factor (gamma):

[ t' = gamma left(t - frac{vx}{c^2}right) ]

When speeds are negligible, the term (frac{vx}{c^2}) approaches 0, and (gamma) simplifies to 1, yielding (t' = t). This result aligns with our everyday Newtonian experience, where 1 second for a stationary observer is roughly equal to 1 second for a moving observer at a constant velocity.

As previously observed, the transformation from (x) to (x') includes (ct) as a term, while the transformation to (t') includes (x). Incorporating these terms into each other's reference frame transformations interweaves time and space into a co-dependent continuum, where a unit change in one variable corresponds to a unit change in the other. This interrelationship accounts for the proportionality of time dilation and space contraction described by Lorentz transformations.

How do we determine the Lorentz factor's value? One method is to multiply our transformation equations and solve for the common factor. Remember, we can substitute (x) and (x') with (ct) and (ct') respectively, allowing us to cancel out like terms and derive (gamma):

[ gamma = frac{1}{sqrt{1 - frac{v^2}{c^2}}} ]

Now we can express the (x') reference frame through the following substitution:

[ x' = gamma (x - vt) ]

And for the (t') reference frame:

[ t' = gamma left(t - frac{vx}{c^2}right) ]

In each equation, as the velocity (v) approaches the speed of light, (frac{v^2}{c^2}) approaches 1, and the denominator approaches 0. According to (E=mc^2), objects with rest mass cannot, as a matter of physical principle, achieve luminal speeds. Hence, it is not physically feasible for the denominator to equal 0. The 0 limit signifies infinite rapidity (denoting the transformation's angle). As rapidity approaches infinity, time converges to rest, and length measurements approach zero.

Conversely, when the velocity is minimal, (frac{v^2}{c^2}) is a very small number, and the denominator approaches 1. When the denominator (the Lorentz factor) equals either 1 or nearly 1, it becomes insignificant, and the equation approximates Newtonian motion. Essentially, the equations of motion are determined by the numerator, reducing to Newton's equations.

The Lorentz factor is central to comprehending Lorentz transformations. Recall that in Galilean relativity, the interchangeability of inertial reference frames is achieved through rotations, described by trigonometric functions. These functions preserve Euclidean distance. Specifically, rotations maintain the radius, meaning that length units remain constant during transformations.

In contrast, Lorentz transformations conserve the spacetime metric. Unlike the Euclidean metric, the spacetime metric makes all transformations relative to the speed of light as an absolute value. Therefore, the speed of light serves as an asymptote that Lorentz transformations approach but cannot equal. In the spacetime diagram, the speed of light is denoted by the equations (x = ct) and (x' = ct'). The asymptotes consist of diagonals that intersect both axes. Since the range of spacetime transformations is infinite (indicating a range of (-infty) to (+infty)) yet asymptotic to the diagonals, they are expressed by hyperbolic functions or rotations. Hyperbolic rotations are analogous to trigonometric functions, employing hyperbolas instead of circles. Unlike finite circles, hyperbolic rotations can extend to infinite ranges. Their equivalents in trigonometric functions can be expressed as exponential operations on the special number (e) (approximately 2.718), with analogues to sin(x) denoted by (sinh(x)) and cos(x) denoted by (cosh(x)):

Just as points on a unit circle are defined by (sin x, cos x), points on the right half of a unit hyperbola are characterized by (cosh x, sinh x). The angle of hyperbolic rotations in the context of special relativity is termed rapidity, denoted by the symbol (eta). Below are the hyperbolic rotations that correspond to the previously derived Lorentz transformations:

The relationship between the Lorentz factor (gamma) and the rapidity of hyperbolic rotations is expressed as follows:

[ gamma = e^{eta} ]

If Galilean rotations maintain the radius or Euclidean distance, what do Lorentz transformations conserve? They preserve the Minkowski metric, represented by the following equality, analogous to Euclidean distance:

[ s^2 = x^2 - c^2t^2 ]

Given that actual Lorentz transformations occur in four dimensions—one of time and three of space, or analogously four spacetime dimensions—the four-dimensional Minkowski interval is expressed by:

[ s^2 = g_{munu} dx^mu dx^nu ]

The animated diagram below visualizes these hyperbolic transformations as distortions in spacetime across two dimensions, approaching the diagonal asymptotes as velocity nears the speed of light. The distortions on the grid illustrate changes in the spacetime metric due to the relative speeds of observers. As speeds approach the luminal limit, space (the horizontal axes hyperbolas) contracts, and time (the vertical axes hyperbolas) dilates. These intertwined transformations conserve the Minkowski metric (s^2), which proportionally scales these transformations against the invariance of light speed.

Space contraction and time dilation can be interchanged between observers at rest and those moving at uniform or inertial speeds. If you're moving uniformly at speeds close to the speed of light relative to a stationary observer, it's equally valid to describe you as stationary while the other person moves at near-light speed.

The Metric Tensor: Geometry of Curved Surfaces

Lorentz transformations in special relativity occur in flat pseudo-Euclidean space. What constitutes flat space? It is a geometry where the metric—or distance measure between points—is consistent. The most recognized flat metric is defined by the Pythagorean Theorem. Another flat metric includes the Minkowski spacetime metric discussed earlier.

The Euclidean metric determines the distance between two points as the square root of the sum of the squared lengths of the shortest sides of a right triangle, following the Pythagorean Theorem: (a^2 + b^2 = c^2).

Geometrically, the Euclidean distance between two points is represented as the square root of the sum of the squared differences between each coordinate ((x,y)).

The Pythagorean theorem can be generalized to (n) dimensions:

Thus, Euclidean distance in three dimensions can be expressed by the following formula:

This generalization preserves distance as a property of Euclidean flat space, meaning the metric remains constant.

To understand the metric tensor, we should recognize the Pythagorean theorem as a specific case of flat or Euclidean space.

In other words, we need to establish a value-neutral space from which the Euclidean distance defined by the Pythagorean theorem can be derived.

Before we can do this, we must explore why the differences between coordinates are squared in the Pythagorean theorem. This can be elucidated in various ways, but a geometric explanation suffices. They are squared because it generates geometric areas of equal lengths, given that areas result from the product of length and width. This enables us to calculate the hypotenuse as the square root of the sum of the squares of the right-angled sides. This outcome is articulated by the metric tensor defined by the Kronecker delta, which outputs 1 if (i=j) and 0 if (ineq j).

We can also demonstrate this result through the generalized metric of a space, where the metric tensor comprises a smoothly varying inner product on the tangent space.

What is a tangent space? A tangent space comprises all vectors tangent to a point on a manifold.

The general form of the equation is given below, where (g) represents the metric tensor and (delta_v) denotes the index of each metric tensor value per coordinate term, while (dX) signifies infinitesimal displacements per coordinate:

Using the above equation, we can express the squared distance between two points in two dimensions as follows:

In the above formula, the zeros and ones beside the (g) coefficients and the (x) variables indicate indices, specifically representing the permutation matrix of 0 and 1: 01, 00, 11, 10.

The (dx^0) and (dx^1) coefficients denote infinitesimal displacements of two different coordinates, where 0 and 1 are indices. The product of the displacements for each coordinate is multiplied by the corresponding value of (g), the metric tensor.

Consequently, in the above formula, (g) serves as a coefficient of the metric tensor for each index. Why are there four terms in this formula? Because two points are characterized by four coordinates or scalar values. In Euclidean geometry, the implicit basis vectors are the tangent vectors (0,1) and (1,0). These tangent vectors span the entirety of Euclidean space. The (g) value defines the inner product between tangent vectors at any point on the vector space, with values obtained through the inner product of all possible combinations of the basis vectors.

When the coefficients represent an orthonormal relationship between two points, the (g) values reduce to the identity matrix:

In two dimensions, we can express the Euclidean distance as the product of the metric tensor and the squared vector of the distance between each coordinate. Since, for right angles in flat Euclidean space, the metric tensor is an identity matrix, the squared distance between two points reduces to the Pythagorean theorem as shown below:

This formula can also be articulated as a linearly weighted combination represented in our initial formulation:

As illustrated, when (g=0), we eliminate the latter two terms, simplifying the equation to the Euclidean distance. Thus, we have demonstrated how the generalized form of the metric tensor implies Euclidean distance as a special or limiting case.

What happens when the shortest distance cannot be expressed through Euclidean distance? Our everyday intuitions assume the existence of right angles for the lengths of opposite and adjacent lines to validate the Pythagorean theorem as a distance measure of the hypotenuse. In linear algebra, this is akin to presuming orthonormal bases as the metric of the space. Bases represent the set of linearly independent vectors spanning that vector space. Orthonormal bases are perpendicular unit vectors, or unit vectors whose inner product is zero.

However, this a priori assumption may lack empirical justification. In fact, the underlying geometry may be curved or skewed in various ways. If that is the case, how do we express the shortest distance between two points? To define a non-Euclidean space, we must select a different set of basis vectors for our metric. The inner product of the permutation space of those basis vectors will yield the metric tensor that delineates distance and angles within that metric through linear combinations of any infinitesimal displacements between two points, expressed by the formula:

Now, let's consider an example using polar coordinates ((r, theta)), where (r) denotes the radius and (theta) signifies the angle. The (g) metric tensor is obtained through the inner products of the permutation space of ((r, theta)) as shown below:

In the case of Euclidean polar coordinates, the metric tensor yields the following matrix:

This occurs because distance is computed through:

The distance between two points ((r_1, theta_1)) and ((r_2, theta_2)) is determined by calculating the distances (r_2 - r_1) and (theta_2 - theta_1), and plugging them into the following formula:

Up to this point, all our examples have been in two-dimensional space. Naturally, we can extend these principles to three or (N) dimensional spaces. The metric tensor for three-dimensional space would take the form of a 3x3 matrix, and so forth.

Grasping the metric tensor represents a crucial step toward comprehending general relativity and Einstein's field equations.

In general relativity, Einstein's field equations utilize the metric tensor to describe the curved geometry of spacetime.

Specifically, Einstein's field equations rely on three tensors: 1) Einstein's Tensor (G), which depicts spacetime curvature derived from the metric tensor's derivatives, 2) the energy-stress tensor (T), which illustrates the distribution of matter and energy in the universe, and 3) the metric tensor (g), which delineates the measures of lengths and angles within the curved geometry. Einstein's field equations are typically summarized by the equation:

In general relativity, the metric tensor consists of a 4x4 matrix comprising 16 components. Just as in our two-dimensional example, the metric tensor embodies the permutation space of all dimensions—3 of space and 1 of time, combined into 4 spacetime dimensions. However, since the matrix is inherently symmetric, only 10 of these components are independent.

The generic form of the metric tensor is displayed below:

The values of the metric tensor change with the curvature of spacetime, encoding the mass-energy distribution. Therefore, unlike Euclidean distance, which maintains length across all transformations, curved geometry does not. This is why the metric tensor is a vital aspect of understanding general relativity.

Having journeyed through these concepts, you may find yourself less intimidated by complex ideas and mathematical formalism in physics!

Sources

Susskind, Leonard, and George Hrabovsky. The Theoretical Minimum: What You Need to Know to Start Doing Physics. Basic Books, 2014.

Susskind, Leonard, and Art Friedman. Special Relativity and Classical Field Theory: The Theoretical Minimum. Penguin Books, 2018.

Susskind, Leonard, and Art Friedman. Quantum Mechanics: The Theoretical Minimum. Penguin Books, 2015.

Susskind, Leonard, and André François Cabannes. General Relativity: The Theoretical Minimum. Basic Books, 2023.

Wolfram Demonstrations Project. Understanding Special Relativity: The Lorentz Transformation, Time Dilation, and Length Contraction. (n.d.). https://demonstrations.wolfram.com/UnderstandingSpecialRelativityTheLorentzTransformationTimeDi/