### Special Relativity

A theory developed by Albert Einstein stating that the laws of motion are the same for all inertial (non-accelerating) frames of reference and that the speed of light (in a vacuum) is the same for all inertial reference frames. This leads to the equivalence of mass and energy, time dilation, and length contraction.

There are many ways to discuss special relativity (SR); one is to start with empirical data

#### 1.‘Time dilation’ and the “downfall of the present”

Ideal clocks can be imagined as a pack of firecrackers of identical size, composition, and fuse. If they are all lit at once, we would expect that the firecrackers would explode at the same time, let’s say, one second later. Well, they do, when they are at rest with respect to each other. But what if we throw them to the left and right so they’re moving at different velocities v with respect to each other, and keep one at rest at the origin? Stunningly, the actual result is that they do not explode simultaneously! Instead, identical clocks (i.e., ‘firecrackers’) in relative motion run at different rates than identical clocks at rest, a fact called “time dilation”, and thoroughly verified throughout the twentieth century. So the ordinary idea of a ‘present moment’ that moves equally into the future for everyone just doesn’t hold! But there’s more: the faster they move away from us, say along the x axis, the more time t passes before they explode. In fact the events in space x and time t where they explode are all related to each other and to the time (here one second) when the firecracker we kept at rest exploded. Let’s call this time “proper time” τ. Then τ2 = t2 - x2/c2, where c is the speed of light. Still these are identical clocks, so what’s happening? Perhaps we should say that they each tick at the same rate in their own reference system, but the way we measure time and space itself must be reconsidered. Physicists refer to the proper time as an “invariant spacetime interval” since it represents an identical ‘distance’ or ‘interval’ between the origin of the experiment and the events in space and time where the one-second proper time ticks occurred (i.e., the firecrackers exploded).

#### 2. Synchronization and the “downfall of the present”

Time dilation leads inexorably to the “downfall of the present”. Suppose, instead, that there were a physically significant global present, a universal “now” as classical physics and common sense hold. How would we specify it, i.e., how would we synchronize clocks A and C in relative motion to tell what event along the worldline of clock C corresponds to the “now” along the worldline of clock A? An obvious answer would be synchronize a third clock at rest with respect to A, then move it from A to C, set C’s time to match it, and thus to match A. The problem is time dilation: if we move identical and synchronized clocks around to different positions as just described, they will no longer be synchronized! In fact, there is no physically significant way of determining a global present according to SR. Instead of a universal, unique “present”, there is only a “present” defined by each moving observer in an equivalent way.

#### 3. Implications

The immediate implications are a variety of ‘paradoxes’, most of which represent variations on the themes of time dilation and what is its converse, ‘length contraction.’ In effect, all such “paradoxes” arise because we so naturally look at the world as “3+1", i.e., as a 3-dimensional spatial universe changing in time, a perspective lodged in both ordinary human experience and the classical physics of Newton and Galileo. Instead, SR invites us to look at the union of space and time in “spacetime”, often referred to as “3+1 --> 4". Here, though time and space measurements vary between moving observers, the measure of the ‘spacetime interval’ between events is invariant.

#### 4. The invariance of causality

The speed of light is not only a constant in SR; it functions as a limitation on which events in my future I can effect, namely those which I can reach with light signals or slower-moving phenomena. This means that the order of events along any worldline moving past me is invariant: all observers in relative motion will agree with this order. (See General Relativity.)

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Contributed by: CTNS