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312 lines
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<h1 class="center">Relativity Introduction</h1>
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<p>First, a little story:</p>
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<div class="fun">
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<p>Alex and Sam are two immortal super-beings</p>
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<p>Alex: Hey Sam, I am going to create an elegant Universe.</p>
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<p>Sam: Sounds cool. How does it begin?</p>
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<p>Alex: I will make it expand from a point.</p>
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<p>Sam: So ... two types of point then? One central point and all the other points? Not very elegant. And does it expand into already created space?</p>
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<p>Alex: Darn.</p>
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<p><i>(...some time later...)</i></p>
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<p>Alex: Hey Sam, I've figured it out. It is <b>space itself</b> that will expand.</p>
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<p>Sam: Right, so everything just gets pulled apart, and all points have the same properties.</p>
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<p>Alex: Yes. And things can move around inside the space making wonderful patterns.</p>
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<p>Sam: And this all happens instantly?</p>
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<p>Alex: I thought about that. I will set a maximum speed inside space. And time will just happen. It is going to be cool.</p>
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</div>
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<p>Now, let's learn about our (real) Universe!</p>
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<h2>Relative</h2>
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<p>There is no "central reference point" in the Universe. So we can only measure speed <b>relative</b> to us or something else.</p>
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<div class="example">
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<h3>Example: when we say a car is going 100 km/h (about 60 mph) we mean <b>relative to the ground</b></h3>
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<p>But think about this:</p>
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<ul>
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<li>The Earth spins on its axis at 1670 km/h at the equator</li>
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<li>The Earth orbits the Sun at 30 km/s</li>
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<li>The Sun moves around the Galaxy at 220 km/s</li>
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<li>Our Galaxy is moving relative to local galaxies at about 300 km/s</li>
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</ul>
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<p>But we don't notice any of that.</p>
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<p>We only notice relative speeds.</p>
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</div>
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<div class="example">
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<h3>Example: you are in a moving Train.</h3>
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<p class="center"><img src="images/train-table-tennis.svg" alt="playing table tennis in a train" height="125" width="398" ></p>
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<p>Have a game of table tennis!</p>
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<p>The ball bounces back and forth just like you were at home.</p>
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<p>If there were no windows, and the train was running on a smooth track there is actually no way you could tell how fast you were going in relation to the ground.</p>
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</div>
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<div class="def">
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<p class="center larger">The laws of Physics are not affected by your <b>speed</b>.</p>
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</div>
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<p>(Imagine how weird it would be if light, electricity, magnetism behaved differently at different speeds!)</p>
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<h2>Speed of Light</h2>
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<p>Light travels at almost <b>300,000,000 meters per second</b> (to be exact: 299,792,458 meters per second) in a vacuum. That speed is called <b>c</b>.</p>
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<p class="large center">c = speed of light in a vacuum</p>
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<p>And <b>c</b> is the same in all directions!</p>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/earth-orbit-speed-light.svg" alt="earth orbit speed of light" height="175" width="236" ></p>
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<p> </p>
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<p>The Michelson–Morley experiment in 1887 measured the speed of light "forward" and "backward" of Earth's orbit and found <b>no difference</b>. Despite Earth moving quite fast through space.</p>
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<p>It was a surprise at the time.</p>
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<p>So the <b>speed of light is constant</b> and is not affected by any relative speed.</p>
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<p class="large center">c is the same for all observers, <span class="large">independent of the motion of the source</span>!</p>
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<div style="clear:both"></div>
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<p>So, if I zoom towards you at 0.9c and shine a light ahead of me:</p> <p><img src="images/relativity-towards.svg" alt="earth orbit speed of light" height="150" width="369" ></p>
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<ul>
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<li>I will see the light moving at <b>c</b></li>
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<li>You will <b>also</b> see the light moving at <b>c</b> (not 1.9c)</li>
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</ul>
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<p>You might think we should add <b>c</b> and <b>0.9c</b> together to get 1.9c, but the Universe does not work like that. Many, many experiments have proven that.</p>
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<p>(We are talking speed here, <b>no acceleration</b>. Technically it is called an "inertial frame".)</p>
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<h2>A Moving Box</h2>
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<p>To learn how this works, imagine you are <b>inside</b> a moving box.</p>
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<p>You measure the time it takes light to go from a torch to a detector, and get the answer <b>t</b></p>
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<p class="center"><img src="images/relativity-light.svg" alt="relativity and speed of light" height="310" width="360" ></p>
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<p>But the box is <b>moving past someone else</b> at speed <b>v</b>. They also measures how long the light takes.</p>
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<p>They see the light take a <b>longer (slanted) path</b>, BUT it still travels at the speed of light, so it <b>must take more time</b>, and they measure time <b>r</b></p>
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<ul>
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<li><b>t</b> is the time for you, the <b>inside</b> observer</li>
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<li><b>r</b> is the time for them, the <b>outside</b> observer</li>
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</ul>
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<p><i>That is the key point. Having <b>c</b> the same for everyone means that time can be different.</i></p>
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<p> </p>
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<p> </p>
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<p>Having accepted this, let us do some mathematics! What are the distances?</p>
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<ul>
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<li>The distance for the <b>inside</b> observer is the time t by the speed of light: <b>ct</b></li>
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<li>The distance for the <b>outside</b> observer is the time r by the speed of light: <b>cr</b></li>
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<li>And the <b>outside</b> observer sees the box move this far: <b>vr</b></li>
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</ul>
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<p>So these are the distances:</p>
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<p class="center"><img src="images/relativity-distances.svg" alt="relativity and speed of light" height="310" width="360" ></p>
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<p>Which we can put into one diagram like this:</p>
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<p class="center"><img src="images/relativity-triangle.svg" alt="relativity and speed of light" height="150" width="125" ></p>
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<p>It is a right-angled triangle that we can solve using <a href="../pythagoras.html">Pythagoras formula</a>:</p>
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<div class="tbl">
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<div class="row"><span class="left">Start with:</span><span class="right">(cr)<sup>2</sup> = (vr)<sup>2</sup> + (ct)<sup>2</sup></span></div>
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<div class="row"><span class="left">Move (vr)<sup>2</sup> to left:</span><span class="right">(cr)<sup>2</sup> − (vr)<sup>2</sup> = (ct)<sup>2</sup></span></div>
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<div class="row"><span class="left"> (ab)<sup>2</sup> = a<sup>2</sup>b<sup>2</sup>:</span><span class="right">c<sup>2</sup>r<sup>2</sup> − v<sup>2</sup>r<sup>2</sup> = c<sup>2</sup>t<sup>2</sup></span></div>
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<div class="row"><span class="left"> Simplify left:</span><span class="right">r<sup>2</sup>(c<sup>2</sup>−v<sup>2</sup>) = c<sup>2</sup>t<sup>2</sup></span></div>
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<div class="row"><span class="left"> Divide by c<sup>2</sup>−v<sup>2</sup>:</span><span class="right">r<sup>2</sup> = <span class="intbl"><em>c<sup>2</sup>t<sup>2</sup></em><strong>c<sup>2</sup>−v<sup>2</sup></strong></span></span></div>
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<div class="row"><span class="left">Simplify right:</span><span class="right">r<sup>2</sup> = <span class="intbl"><em>t<sup>2</sup></em><strong>1−v<sup>2</sup>/c<sup>2</sup></strong></span></span></div>
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<div class="row"><span class="left">Square root:</span><span class="right">r = t <span class="intbl"><em>1</em><strong>√(1−v<sup>2</sup>/c<sup>2</sup>)</strong></span></span></div>
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</div>
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<p>So now we can calculate how much time for an outside observer (r) compared to an inside observer (t).</p>
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<p>That last term is so important it gets called gamma (the Greek letter <span class="large">γ</span>) or "the Lorentz factor":</p>
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<p class="center large">γ = <span class="intbl"><em>1</em><strong>√(1−v<sup>2</sup>/c<sup>2</sup>)</strong></span></p>
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<div class="def">
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<p class="larger">That increase in time that an outside observer experiences is called "Time Dilation", dilation means getting larger.</p>
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</div>
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<h2>Gamma Examples</h2>
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<p>Here are some <span class="large">γ</span> values for different speeds (you can do the calculations yourself, but will need the <a href="../calculator-precision.html">Full Precision Calculator</a>):</p>
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<p class="center"><i>Remember that <b>c</b> is about 300,000,000 m/s</i></p>
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<h3>Highway speed of 100 km/h (28 m/s): γ = 1.000 000 000 000 004 29...</h3>
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<p>There are 14 zeros in there, so almost exactly 1. That is why we never notice these time effects at the speeds we normally go at.</p>
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<h3>Jet speed of 2,000 km/h (556 m/s): γ = 1.000 000 000 001 72...</h3>
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<p>So even at jet speed there are 11 zeros in there.</p>
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<h3>GPS satellite orbit speed of 14,000 km/h (≈4000 m/s): γ = 1.000 000 000 084 1...</h3>
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<p>Still almost exactly 1. A clock on a satellite would be affected by about:</p>
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<p class="center">0.0000000000841×24×60×60 = 0.000 007 seconds every day</p>
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<p>That is <b>7000 nanoseconds</b>, but GPS needs about 20 nanoseconds accuracy, so we actually <b>need relativity calculations to make GPS work</b>. (Note: there is also a gravity effect, but we are not looking at that here)</p>
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<h3>10% of the speed of light: γ = 1.005...</h3>
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<p>Even going at 10% the speed of light (about <b>100 million km/h</b>) there is only a .005 difference</p>
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<h3>50% of the speed of light: γ = 1.155...</h3>
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<h3>90% of the speed of light: γ = 2.294...</h3>
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<h3>99% of the speed of light: γ = 7.089...</h3>
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<p>At 99% of the speed of light, for every <b>day inside</b> the box someone outside experiences <b>a week</b>.</p>
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<p>In the style of the triangle we saw earlier it looks like this:</p>
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<p class="center"><img src="images/relativity-triangle-7.svg" alt="relativity and speed of light" height="65" width="422" ></p>
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<p>You can check for yourself: does <b>1<sup>2</sup> = 0.99<sup>2</sup> + 0.141<sup>2</sup></b> ?</p>
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<h3>99.9% of the speed of light: γ = 22.4...</h3>
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<div class="example">
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<h3>Example: Muons</h3>
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<p>Muons are special particles with a half of life of only 2.2 microseconds. Light travels only 700 meters in that time.</p>
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<p>But we get lots of muons from the upper atmosphere, thousands of meters away!</p>
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<p><b>Why?</b></p>
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<p>Because they are moving so close to the speed of light that <b>gamma is about 20</b></p>
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<p>So for them <b>2 microseconds</b> have passed, but as outside observers we see about <b>40 microseconds</b> passing, which is enough time for them to go about 14,000 m</p>
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<p>Another demonstration of relativity.</p>
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</div>
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<p> </p>
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<h2>Adding and Subtracting Relative Velocities</h2>
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<p>So, how <b>do</b> we add velocities, then?</p>
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<p>For speeds that we normally experience it is OK to just add them, you won't notice anything wrong.</p>
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<p>But for very fast speeds we need to think of relativity.</p>
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<p>In the case of velocities heading in the same direction we can use this fomula:</p>
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<p class="center large">v<sub>new</sub> = <span class="intbl"><em>v<sub>1</sub> + v<sub>2</sub></em><strong>1 + v<sub>1</sub>v<sub>2</sub>/c<sup>2</sup></strong></span></p>
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<div class="example">
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<h3>Example: A spaceship going at 0.6c launches a rocket (relative to it) at 0.5c, what does an outside observer see?<br>
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<img src="images/relativity-add.svg" alt="rocket launches rocket" height="127" width="459" ></h3>
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<p> </p>
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<p>Let's use the formula above:</p>
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<p class="center large">v<sub>new</sub> = <span class="intbl"><em>0.6 + 0.5</em><strong>1 + 0.6×0.5/1<sup>2</sup></strong></span></p>
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<p class="center large">v<sub>new</sub> = <span class="intbl"><em>1.1</em><strong>1 + 0.3</strong></span></p>
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<p class="center large">v<sub>new</sub> = 0.846...</p>
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<p>The two speeds combine to make 85% of the speed of light. Neat, huh?</p>
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</div>
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<div class="example">
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<h3>Example: Earlier we looked at this:<br>
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<img src="images/relativity-towards.svg" alt="earth orbit speed of light" height="150" width="369" ><br>
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What <b>does</b> happen when we add 0.9c to c?</h3>
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<p>Let's use the formula above:</p>
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<p class="center large">v<sub>new</sub> = <span class="intbl"><em>0.9 + 1</em><strong>1 + 0.9×1/1<sup>2</sup></strong></span></p>
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<p class="center large">v<sub>new</sub> = <span class="intbl"><em>1.9</em><strong>1 + 0.9</strong></span></p>
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<p class="center large">v<sub>new</sub> = 1</p>
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<p>So the outside observer sees the combined speed of <b>0.9c</b> and <b>c</b> as exactly <b>c</b></p>
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<p>So we never get above c</p>
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</div>
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<h2>Mysterious?</h2>
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<p>This may seem strange, but not really mysterious. It is just a direct result of the speed of light being a constant for all observers.</p>
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<p>The strangeness is likely because in our everyday world we never have to deal with it. Maybe if the speed of light was only 100 km/h we would have a better "feel" for it?</p>
|
||
<p> </p>
|
||
|
||
|
||
<h2>Footnotes</h2>
|
||
|
||
<p>That was an introduction into "Special Relativity" which does not include any effects of acceleration or gravity.</p>
|
||
<p>Albert Einstein released his special theory of relativity in 1905, but it took until 1915 for him to release his <b>general</b> theory of relativity that does include the effects of gravity.</p>
|
||
<p> </p>
|
||
<p>Note also that when we mention the speed of light, or <b>c</b>, please remember:</p>
|
||
<div class="def">
|
||
<p>Light <b>only travels that speed in a vacuum</b>!</p>
|
||
<p>It can travel <i>slower</i>, read more at <a href="light.html">Light</a>.</p>
|
||
</div>
|
||
<p>And even though it is called the speed of <b>light</b>, it applies to the whole <a href="electromagnetic-spectrum.html">Electromagnetic Spectrum</a>, <a href="gravity.html">Gravity</a> Waves and more (basically any particle without mass).</p>
|
||
<p> </p>
|
||
<p>And remember: we all experience the same laws of physics.</p>
|
||
|
||
|
||
<h2>Faster than c?</h2>
|
||
|
||
<p>Space can expand faster than c, but as far as we know <b>no wave/particle can go faster than c</b>.</p>
|
||
<div class="fun">
|
||
<p>Except for this <b>fictional</b> lady:</p>
|
||
<p style="padding: 0 0 0 20px;"><i>There was a young lady named Bright<br>
|
||
who traveled much faster than light.<br>
|
||
She set out one day<br>
|
||
in a relative way,<br>
|
||
and came back the previous night.</i></p>
|
||
</div>
|
||
<p> </p>
|
||
<div class="questions">17776, 17777, 17778, 17779, 17770, 17771, 17772, 17773, 17774, 17775</div>
|
||
|
||
<div class="related">
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<a href="index.html">Physics Index</a>
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