Special Relativity

Prerequisites: Newtonian Mechanics. Full path (5 pages)

• Intuitive
• Concrete
• Abstract
• Why is it interesting?
• FAQ
• History

Intuitive

Special relativity is based on two postulates:

1. the laws of physics are the same for all (inertial) observers
2. the speed of light in a vacuum has the same value for all (inertial) observers.

These two postulates taken together tell us that space and time are not absolute but must be relative, i.e. change depending on the observer. Otherwise, it isn't possible that different moving observers always measure exactly the same value for the speed of light. This was discovered by the famous Michelson Morley experiment.

• A particularly nice intuitive introduction is the free online book Einstein for Everyone by John D. Norton
• The basic idea of relativity is nicely explained here: The First Principle of Relativity | Of Particular Significance
• Of Galileo, Groups and What’s So Special about the Speed of Light by Rod Vance
• It's About Time: Understanding Einstein's Relativity by Mermin is a great book for laymen on special relativity.
• Relativity visualized - a website by Ute Krause and Corvin Zahn

Special relativity is Einstein’s description of how some of the basic measurable quantities of physics – time, distance, mass, energy – depend on the speed of the measuring apparatus relative to the object being studied. It shows how they must change in order to guarantee that Galileo’s principle of relativity (that the laws of physic should be the same for every experimenter, regardless of speed) should hold even at speeds near that of light.

Gravity from the Ground Up by Bernard Schutz

Concrete

The best beginner Special Relativity Textbooks are:

• Special Relativity by French
• Special Relativity for Beginners by Jürgen Freund

The standard textbook is Spacetime Physics by Taylor, Wheeler

Recommended Further Reading:

• Notes on Special Relativity for High-School Students by Field Medalist Terrence Tao
• Nothing but Relativity by Palash B. Pal
• Derivation of the Lorentz Transformation by Yakovenko
• A simple method for illustrating the difference between the homogeneous and inhomogeneous Lorentz groups by Y. S. Kim
• Representing arbitrary boosts for undergraduates by Charles P. Frahm

Distance in Space vs. Space-Time:

The diagram below illustrates the difference between distance in Euclidean space (left-hand side) and "distance" (actually proper time) in space-time according to Einstein's special theory of relativity (right-hand side). Time dilation is shown in red. For a more detailed explanation of this diagram see Fun with Symmetry.

• No-Nonsense Electrodynamics by Jakob Schwichtenberg, chapter "Special Relativity"
• No-Nonsense Quantum Field Theory by Jakob Schwichtenberg, chapter "Spacetime"

Abstract

Mathematically, special relativity is the statement that all laws of physics are invariant under the Poincare group.

In special relativity spacetime denotes the configuration space of a single point particle. We denote the Minkowski spacetime by $Q$, i.e., $\mathbb{R}^{3+1}\ni(x^0,\ldots,x^n)$ with the Minkowksi metric

$$\eta_{\mu\nu} = \left( \begin{array}{ccccc} 1 & 0 & 0 & \ldots & 0 \\ 0 & -1 & 0 & \ldots & 0 \\ 0 & 0 & -1 & & 0 \\ \vdots & \vdots & & \ddots & \vdots \\ 0 & 0 & 0 & \ldots & -1 \end{array}\right).$$

Free Particle

What interests us most of the time is path of a particle $$q:\mathbb{R}(\ni t) \longrightarrow Q$$ where $t$ is a parameter for the path, not necessarily $x^0$, and not necessarily the proper time either.

To derive the equations of motion for such a free particle, we need the corresponding action $$S(q) = \int_{t_0}^{t_1} L\Bigl(q^i(t),\dot{q}^i(t)\Bigr)\,dt$$ for $q\colon [t_0,t_1]\rightarrow Q$. A useful hint towards the correct Lagrangian is that it should be independent of the arbitrary parametrization parameter $t$. The obvious candidate for the action $S$ is mass times the "proper time" $$S = m\int_{t_0}^{t_1} \sqrt{\eta_{ij}\dot{q}^i(t)\dot{q}^j(t)}\,dt .$$

Using $$\begin{equation} L = m \sqrt{\eta_{ij}\dot{q}^i\dot{q}^j} \end{equation}$$ and the Euler-Lagrange equations, we get $$\begin{align*} p_i = \frac{\partial L}{\partial\dot{q}^i} &= m\frac{\partial}{\partial\dot{q}^i}\sqrt{\eta_{ij}\dot{q}^i\dot{q}^j} \\ &= m\frac{2\eta_{ij}\dot{q}^j}{2\sqrt{\eta_{ij}\dot{q}^i\dot{q}^j}} \\ &= m\frac{\eta_{ij}\dot{q}^j}{\sqrt{\eta_{ij}\dot{q}^i\dot{q}^j}} \, = \frac{m\dot{q}_i}{|{\dot{q}}|}. \end{align*}$$ The Euler–Lagrange equations say $$\dot{p}_i = F_i = \frac{\partial L}{\partial q^i} = 0.$$

We can understand this better when we use the proper time as our parameter $t$ so that $$\int_{t_0}^{t_1}|{\dot{q}}|dt = t_1-t_0, \quad \forall\,t_0,t_1 .$$ This fixes the parametrization up to an additive constant. Thus $|{\dot{q}}|=1$, so that $$p_i = m\frac{\dot{q}_i}{|{\dot{q}}|} = m\dot{q}_i$$ and the Euler–Lagrange equations say $$\dot{p}_i =0 \Rightarrow m\ddot{q}_i = 0 .$$ We can therefore conclude that our free particle moves unaccelerated along a straight line.

Why is it interesting?

We believe that special relativity at the present time stands as a universal theory describing the structure of a common space-time arena in which all fundamental processes take place. All the laws of physics are constrained by special relativity acting as a sort of ”super law”.

One more derivation of the Lorentz transformation by Jean-Marc Levy-Leblond

The Michelson-Morley experiment demonstrated that the speed of light is the same in all frames of reference. Special relativity is the correct theory that incorporates this curious fact of nature.

Special relativity is an extension of classical Newtonian mechanics. In particular for objects that move very fast (close to the speed of light) predictions that we get using Newtonian mechanics turn out to be wrong and only special relativity yields the correct results. For slowly moving objects both, Newtonian mechanics and special relativity yield approximately the same results.

Special relativity predicts many effects like the relativity of time or of the dilation of fast moving objects which are all in agreement with experiments. An important application of special relativity is the GPS system. Without the correct equations of special relativity, the clocks on GPS satellites wouldn't show the correct time and thus precise navigation would be impossible.

FAQ

History

• Who Invented Relativity? from Reflections on Relativity by Kevin Brown

Path to this page

1. Calculus
2. Linear Algebra
3. Newton's Second Law
4. Vector Calculus
5. Newtonian Mechanics

Leads to: Anti Particles, Canonical Quantum Field Theory, Classical Field Theory, General Relativity, Helicity, Klein-Gordon Equation, Locality, Lorentz Group, Michelson-Morley Experiment, Minkowski Metric, Wick Rotation.