Hi, I wrote this tutorial to help me understand special relativity. I hope that it will work for you too. As you try it out just remember the following:

• Each of the round buttons opens an interactive figure or animation.
• Press them in any order and as often as you wish (though it makes sense to go left to right).
• Einstein derived this theory from the observation that light always travels at the same speed for all observers, regardless of their own motion or the motion of the light source.
• So what? Well, as you will see, this leads to some counterintuitive results - like the fact that time passes at different rates for observers moving relative to one another.
• This is very real. For example, GPS satellites move so fast that their onboard clocks must be adjusted for these relativistic effects to keep your location accurate. (Note: they must adjust for general relativity or gravity effects as well).

Enjoy, and feel free to contact me with questions or comments.

Actually, one last note: Though special relativity may become more intuitive after playing with these animations and interactive figures you definitely need math to understand it. So, I included some. The important math set up and final equations are highlighed. In between are derivations using simple algebra. You can either follow them, try them on your own or ignore them. .

RELATIVISTIC DOPPLER EFFECT

There is no medium necessary for light waves to propagate. Light travels through empty space at a constant speed c, for all observers regardless of how or how fast they are moving. However, the wavelength expands (red‑shifts) or contracts (blue‑shifts) for a receding or approaching light source, respectively. A good example is a galaxy billions of light‑years away. Because the universe is expanding, distant galaxies recede from us, and the light they emit is stretched to longer wavelengths. This is the same effect behind Hubble’s law: the farther away a galaxy is, the faster it appears to recede, and the more its light is red‑shifted.

Play with the figure sliders and notice how the wavelength and frequency change but the speed stays constant.

Note: because it is annoying to see constant movement on your screen, the animation will stop after 10 s. Just press the button again or use the sliders to restart it.

DERIVATION OF THE TIME DILATION FORMULA

The animation shows a box moving left to right with velocity v. It does not move vertically. A flash of light emited by the bulb on the top travels to a sensor on the bottom. In our stationary frame, the box moves horizontally a distance \(v \Delta t\) while the ligh flash traces a diagonal with length \(c \Delta t\). In the moving box's frame, the light travels vertically a distance of \(c \Delta \tau\), where \(\Delta \tau\) is the time interval measured by a clock in the moving box.

• \(\bbox[LightCyan]{c^2 \Delta \tau^2 + v^2 \Delta t^2 = c^2 \Delta t^2}\)
• \(c^2 \Delta \tau^2 = \Delta t^2 (c^2 - v^2)\)
• \( \frac{\Delta t^2 }{\Delta \tau^2 } = \frac{c^2} {c^2 - v^2} = \frac{1}{1 - \frac{v^2}{c^2}} \)
• \(\bbox[LightCyan]{\frac{\Delta t }{\Delta \tau } = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \ or \ \gamma = \frac{1}{\sqrt{1 - \beta ^2}}}\) End of proof

REF: lecture 11 Spacetime... Scott Hughes MIT dpt. Physics
Note: you will probably find descriptions stating that the Lorentz factor \(\gamma = \frac{\Delta \tau }{\Delta t}\), which will confuse you. So, let's clarify. The Lorentz factor \(\gamma\) is always equal or greater than 1. In this animated thought experiment \(\Delta t\) is what our clock measures, which is a greater interval than \(\Delta \tau\). \(\Delta \tau\) is what the clock in the box measures. In other words, if a device in the moving box sent us signals every second, we would receive them at intervals greater than a second.

DERIVATION OF THE SPACE CONTRACTION FORMULA

In our reference frame we see a box with length x moving to the right with speed v. A light bulb on the left end of the box flashes. The flash travels to the right taking a time interval of \(\frac{x}{c-v}\) to reach a mirror at the right end of the box. After reflecting on the mirror the flash travels back to the bulb taking a time interval of \(\frac{x}{c+v}\). Thus, our clock measures the total round trip time as \(\bbox[LightCyan]{\Delta t_{total} = \frac{x}{c-v} + \frac{x}{c+v}} \). But a clock in the moving box's frame measures the total round trip time as \(\bbox[LightCyan]{\Delta \tau_{total} = \frac{2\chi}{c}}\).

• we'll need this equation from the time dilation 💡 derivation: \(\bbox[LightCyan]{\frac{\Delta t}{\Delta \tau} = \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}}\)
• in our frame: \(\Delta t_{total} = \frac{x(c+v) + x(c-v)}{c^2 - v^2} = \frac{2xc}{c^2 - v^2} = \frac{\frac{2x}{c}}{1 - \frac{v^2}{c^2}} = \frac{2x \gamma^2 }{c} \)
• in the moving frame: \(\Delta \tau_{total} = \frac{2\chi}{c}\)
• Thus: \(\frac{\Delta t}{\Delta \tau} = \gamma = \frac{ \frac{2x \gamma^2 }{c}}{\frac{2\chi}{c}} \Rightarrow \bbox[LightCyan]{x = \frac{\chi}{\gamma}}\)

Which means that, moving objects are flattened in the direction that they are moving. While this not possible to verify by direct measurements, the length contraction indirectly explains certain observations in particle accelerators, properties of electromagnetism, and small corrections in GPS calculations.
REF: lecture 11 Spacetime... Scott Hughes MIT dpt. Physics

BONUS: DERIVATION OF THE LORENTZ-EINSTEIN COORDINATE TRANSFORMATION EQUATIONS

Look at the yellow highlighted section. The blue arrows represent space-time lengths traveled by the flash from the bulb to the mirror. The purple arrows are the same lenghts in the moving frame. Importantly the origins of both frames are the same. Looking at the blue arrows first: vt is the distance covered by the box, x is the distance covered by the flash, and \(\frac{\chi}{\gamma}\) is the length of the box as measured in the moving frame but converted to our frame using the space contraction formula derived above.

Adding these lengths we obtain the Lorentz coordinate transformation equation for space: \(x = vt + \frac{\chi}{\gamma}\quad\Rightarrow\quad \bbox[LightCyan]{\chi = \gamma(x - vt)}\)

If we do the same from the point of view of the moving frame we obtain a similar equation with a reversed sign: \(\frac{x}{\gamma} = \chi + v\tau \quad\Rightarrow\quad \bbox[LightCyan]{x= \gamma(\chi + v\tau)}\).

• Eliminate \(\chi\): \(\frac{x}{\gamma} - v\tau = \gamma x - \gamma vt\)
• \(v\tau = \gamma vt - \gamma x + \frac{x}{\gamma} \quad\Rightarrow\quad \tau = \gamma t + \frac{x}{v}\!\left(\frac{1}{\gamma} - \gamma\right)\)
• \(\tau = \gamma t + \frac{x}{v}\!\left(\sqrt{1-\frac{v^2}{c^2}} - \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}\right)\)
• \(\tau = \gamma t + \frac{x}{v}\!\left(\frac{-\frac{v^2}{c^2}}{\sqrt{1-\frac{v^2}{c^2}}}\right) = \gamma t - \gamma \frac{v x}{c^2}\)
• \(\bbox[LightCyan]{\tau = \gamma\left(t - \frac{v x}{c^2}\right)}\) which is the time coordinate transformation
• Alternatively we could eliminate x to obtain: \(\bbox[LightCyan]{t = \gamma\left(\tau + \frac{v \chi}{c^2}\right)}\)

REF: ...Lorentz transformation ... JM Levy U. Paris

MINKOWSKI OR SPACETIME DIAGRAMS

A CLOCK AND TWO MIRRORS DEFINING SIMULTANEOUS EVENTS

Meet our clock. A light bulb flashes in our stationary reference frame. The light travels in opposite directions toward mirrors placed at equal distances from the bulb. When the reflected light returns to the bulb, another flash is triggered. These intervals define the clock’s ticking. The two mirror reflections are simultaneous events, and in a spacetime diagram they lie on the same horizontal line. The bulb and mirrors trace three vertical worldlines. The photon paths always appear at 45° (as in the previous animation). Moving objects (none appear in this animation) have tilted worldlines, but they can never tilt beyond the 45° light line.

THIS IS NOT TRUE BUT IT IS HOW WE INTUITIVELY THINK ABOUT TIME

Our stationary clock has blue worldlines (the thick worldline in the center for the bulb and the side thin worldlines for the mirrors). Odd numbers 1, 3, ... etc mark the simultaneous reflection events. A moving clock, shown in red, is also displayed. You can adjust its speed and zoom level with the sliders. The photons in the moving clock are fired at the speed of light, but in this intuitive (Galilean) picture their speed relative to us changes depending on the clock’s motion. IN REALITY, LIGHT SPEED IS THE SAME IN ALL FRAMES. But in this intuitive but wrong demonstration, if the clock moves to the right, the photon fired toward the left mirror appears to move slower than c (its worldline looks steeper) until it reaches the mirror. The reflection on the moving mirror causes it to move faster than c on its return trip. The opposite happens for the photon fired toward the right mirror. As a result, both clocks remain synchronized: you can always draw a horizontal line that intersects both bulbs and all four mirrors. This matches our everyday intuition — that the NOW we experience is shared everywhere in the Universe. But this intuition is not correct. Press the next 🚀 button to see why.

WHY MOTION DESYNCHRONIZES CLOCKS

This animation shows a moving light clock in our spacetime diagram. Our own identical clock is not drawn, but the gridlines correspond to its ticks, just as in the previous clock 🕓 animation. Unlike in the Galilean 🔭 interactive figure, here the speed of light is not affected by the clock’s motion, so the photons always trace 45-degree stippled lines. Because of this, the two reflections that are simultaneous in the moving clock’s own frame are not simultaneous in ours. From our perspective, the moving clock’s spacetime axes are Lorentz-transformed: its time axis becomes the tilted cτ worldline of the ship, and its space axis χ is reconstructed by connecting the two reflection events—the events that are simultaneous in the ship’s frame (connected by the thin red lines). For us, they reflections happen at different times.

PUTTING IT ALL TOGETHER: OBSERVE 😐 THINK 🤔 AND BE SURPRISED 😮

• Our blue clock does not move. Its light paths show simultaneous reflections off the left and right mirrors.
• You can change the speed (β = v/c) of the red moving clock with the slider on top (max. speed is β = 1, v = c, worldline at 45°).
• Play with the proper-time slider to rise and lower the hyperbola. Notice that for all moving clock speeds the hyperbola crosses our ct worldline and the moving worldline at exactly the same time marking.
• The same happens with space units if you play with the proper-space slider.
• This happens because the two clock worldlines have the same origin. In other words, the two clocks were together and perfectly synchronized and at that moment they were set to time = 0.
• The moving clock then moves away, and the hyperbolas trace a line in our space-time frame that shows where-when the moving clock and our clock read the same elapsed time.
• The proper-time hyperbola is not a "now" line. "Now" lines are respectively horizontal for us and parallel to the x' axis for the moving clock.
• You can see that the two reflections on the moving clock, which happen at the same now for them, happen at different times for us. So, the now in one of the frames already happened in another frame😮
• So, what the hyperbolas are telling us is that the people traveling with the moving clock do not feel that their clock is running slower than ours. They feel the time passing exactly as we do.
• If they turn around to meet our clock again then they would see that our clock ran longer than theirs. We aged more than they. You can turn the moving clock around and make it travel towards us by playing with the speed slider. Although there is not much room to the left of our ct axis you can see that time is still dilated for them

OK, let's do the math.

• Time-dilation (see 💡 button): \(c^2\tau^2 + v^2t^2 = c^2t^2\)
• Set \(vt = x\) and rearrange to find the invariant: \((ct)^2 - x^2 = (c\tau)^2\)
• Set the right-hand side to a constant to get the proper-time hyperbola: \(\bbox[LightCyan]{(ct)^2 - x^2 = \text{constant}}\)
• The proper-length hyperbola is similar but rotated toward the x-axis.
• The constant proper-length is a space-like interval where the signs are flipped relative to the proper-time hyperbola equation: \(\bbox[LightCyan]{x^2 - (ct)^2 = \text{constant}}\)
  Where the constant now represents the square of the proper length instead of proper time.