Last time we saw that Em non-paradoxically time-travels over three years into Al’s future by flying **12** *light years* at half the speed of light for just over two decades. Her journey completed, Em has aged only **20.8** years while Al has aged **24**.

That may not seem like much of a gain, but Em was only moving *really* fast — not *really*, ** really** fast. If she travels at

**99%**of light-speed, her round trip shortens to

*years while Al doesn’t wait much longer than it takes light to make the six light-year round trip:*

**1.7****12.12**years! And at

**99.9%**

**, Em’s whole trip takes her only half a year!**

*c*Today we break down dime tilation. I mean, time dilation!

By “break down” I mean provide some charts and talk about it some more. Mainly about why time has to slow down when you look at a moving frame.

Let’s start with the charts:

The first one is a time-space diagram of Em’s flight (at**/**

*c***2**as usual) from Al’s point of view and the years that elapse. For these diagrams we assume Em flies continuously and does not stop. The second one is the same thing from Em’s point of view. Remember that she sees space as moving and therefore contracted — that’s why the marked distances are less.

By the way: The scale of *diagram 2* sets each square at **0.866** rather than **1.0**. That doesn’t change anything, it just lets the diagram lines live neatly on the graph paper lines. Think of it as a zoom.

Certain reference events are marked on each chart to show how things shift due to the frame of reference change. For example, notice that events **A** & **B** are simultaneous to Al (*diagram 1*), but **B** & **C** are simultaneous to Em (*diagram 2*).

Are you’re wondering what Em’s chart would look like if her years were sliced in whole years (1, 2, 3,…) like Al’s are in his chart? It’d look like a mirror image of Al’s chart — same numbers.

The numbers are based on the *gamma factor* for moving at **0.5** ** c** (aka

**/**

*c***2**). At that speed, the

*gamma*(γ) is

**0.866**. It’s how much the length contracts and time dilates (expands) —

**86.6%**!

Which may have you wondering, what is the *gamma* at different speeds?

%c | m/s | MPH | gamma |
1/gamma |
---|---|---|---|---|

0.0000 | 0 | 0 | 1.00000000 | 1.00000000 |

0.0001 | 300 | 671 | 1.00000000 | 1.00000000 |

0.0010 | 2,998 | 6,706 | 1.00000000 | 1.00000000 |

0.0100 | 29,979 | 67,062 | 1.00000001 | 0.99999999 |

0.1000 | 299,792 | 670,616 | 1.00000050 | 0.99999950 |

1.0000 | 2,997,925 | 6,706,164 | 1.00005000 | 0.99995000 |

10.0000 | 29,979,246 | 67,061,642 | 1.00503782 | 0.99498744 |

20.0000 | 59,958,492 | 134,123,284 | 1.02062073 | 0.97979590 |

25.0000 | 74,948,114 | 167,654,105 | 1.03279556 | 0.96824584 |

30.0000 | 89,937,737 | 201,184,927 | 1.04828484 | 0.95393920 |

33.3333 | 99,930,719 | 223,538,584 | 1.06066004 | 0.94280916 |

40.0000 | 119,916,983 | 268,246,569 | 1.09108945 | 0.91651514 |

50.0000 | 149,896,229 | 335,308,211 | 1.15470054 | 0.86602540 |

60.0000 | 179,875,475 | 402,369,853 | 1.25000000 | 0.80000000 |

66.6666 | 199,861,439 | 447,077,168 | 1.34163971 | 0.74535659 |

70.0000 | 209,854,721 | 469,431,495 | 1.40028008 | 0.71414284 |

75.0000 | 224,844,344 | 502,962,316 | 1.51185789 | 0.66143783 |

80.0000 | 239,833,966 | 536,493,138 | 1.66666667 | 0.60000000 |

85.0000 | 254,823,589 | 570,023,959 | 1.89831599 | 0.52678269 |

90.0000 | 269,813,212 | 603,554,780 | 2.29415734 | 0.43588989 |

91.0000 | 272,811,137 | 610,260,944 | 2.41191535 | 0.41460825 |

92.0000 | 275,809,061 | 616,967,108 | 2.55155182 | 0.39191836 |

93.0000 | 278,806,986 | 623,673,272 | 2.72064781 | 0.36755952 |

94.0000 | 281,804,911 | 630,379,437 | 2.93105191 | 0.34117444 |

95.0000 | 284,802,835 | 637,085,601 | 3.20256308 | 0.31224990 |

96.0000 | 287,800,760 | 643,791,765 | 3.57142857 | 0.28000000 |

97.0000 | 290,798,684 | 650,497,929 | 4.11345035 | 0.24310492 |

98.0000 | 293,796,609 | 657,204,093 | 5.02518908 | 0.19899749 |

99.0000 | 296,794,533 | 663,910,258 | 7.08881205 | 0.14106736 |

99.5000 | 298,293,496 | 667,263,340 | 10.01252349 | 0.09987492 |

99.9000 | 299,492,666 | 669,945,805 | 22.36627204 | 0.04471018 |

99.9900 | 299,762,479 | 670,549,360 | 70.71244595 | 0.01414178 |

99.9990 | 299,789,460 | 670,609,716 | 223.60735677 | 0.00447212 |

99.9999 | 299,792,158 | 670,615,751 | 707.10695797 | 0.00141421 |

%c | m/s | MPH | gamma |
1/gamma |

Wonder no more! (If it isn’t obvious, this is sort of a reference post.)

This table is handy for those who are curious about how much things change at different speeds, but don’t want to tackle the *gamma factor* equation I showed you recently.

Notice that the table isn’t linear — the first and last rows break speed into much smaller increments. That’s because those are the most interesting parts!

The *gamma* isn’t even noticeable until at least *one-hundredth* of *one* percent of the speed of light (which is still a bodacious **67** *thousand* MPH). And at that speed *gamma* amounts to a change of just ten-*billionths*!

At the next notch up, a mere one-*tenth* of one percent of ** c** (a hefty

**670**thousand MPH),

*gamma*is still only half of one-millionth. It’s really not until around

**10%**of

**(a whopping**

*c***67**

*million*MPH) that

*gamma*really becomes noticeable.

If you look at the right column (**1**/*gamma*), that shows the amount of length contraction, so things are still **99%** of their natural length at **10%** of the speed of light! It’s also the amount of time reduction, so Em’s clock at that speed (which is millions of MPH) runs only **99%** slower according to Al.

From Em’s point of view *gamma* says how much Al’s clock is running faster, so according to her his clock is running **1.005** times faster. Still not a lot happening at one-tenth light speed!

Even at *half* the speed of light (as we’ve seen for weeks), the effect isn’t *huge*. Em’s clock and length seem only **86%** reduced to Al. As we’ve seen, that gives a ratio of **10.4** years for her to his **12**.

We need to get to **90%** of the speed of light before *gamma* reaches even *two* (it actually reaches it — not ironically — at **86.6%**). But now things really start to heat up (you’ll notice nearly the entire bottom third of the table involves speeds in the 90% range)!

At **99%** of ** c** the gamma is just over seven, and at

**99.5%**it’s just over

*ten*!

At **99.9%** it’s up to **22+**, and at **99.99%** it skyrockets to **70** (and some change). One more decimal place — **99.999%** — and now gamma is well over *two hundred*. (And notice how extreme length and time contraction have gotten.)

§ § §

At the beginning I said that Em’s **12** LY trip at **99%** of ** c** only takes her

**1.7**years and only

*half a year*at

**99.9%**(and that Al’s time converged on the round-trip time for light:

**12**years).

Let’s work the first one through. We’ll do it from Al’s perspective and from Em’s.

Al sees Em make a **12** LY journey at **0.99 c**. The fundamental motion equation is

**=**

*d***(where**

*vt***is time,**

*d***is velocity, and**

*v***is time — 60 MPH,**

*t***, times 2 hours,**

*v***, equals 120 miles,**

*t***).**

*d*Here I have ** v** (0.99 c) and

**(12 LY); I need**

*d***!**

*t*I get that with ** d**/

**=**

*v***(120 miles ÷ 60 MPH = 2 hours). So Al sees Em taking (**

*t***12**÷

**0.99**=)

**12.12**years to make the trip. He knows her clock runs slow; the chart tells him it runs

**0.141**times slower. What seems

**12.12**years to Al will seem like

**1.7**years to Em.

From Em’s point of view, *space* (and Al) is moving at **0.99 c**, so it suffers extreme foreshortening —the same

**0.141**factor. So that

**12**LY now looks like

**1.69**LY!

We have ** d** (1.69) and

**(same 0.99) again, so**

*v***1.69**÷

**0.99**=

**1.7**years. Em sees her time interval ending up the same length Al does — a bit under two years.

I’ll leave the math for what happens at **99.9%** of light speed as an exercise for the interested reader.

§ § §

Tomorrow is Friday, and in this series I like to discuss topics closely related to light on Fridays. It’s the International Year of Light, for one thing, and obviously light has everything to do with Special Relativity.

In the next post I’ll pick up a thread I started two Fridays ago and talk about light clocks (yes, I know, a heavy subject). They’re the reason time dilation *has* to occur.

Length contraction can be hard to wrap your head around, but there turns out to be a simple basis for time dilation, a simple reason Al *has* to see Em’s clock running slower.

Funny thing is, it was one of the first SR things I read about back in high school. It took me *years* for the light bulb to finally go on about it, but I can be as slow as anyone on the uptake.

Hopefully I can explain it in a way where it won’t take years for your light bulbs to light!

We’re almost done! (Which is to say, I’m almost out of diagrams.) Light clocks to end the week, and some final thoughts next week, and I’ll call it a series.

*(Besides, it’s getting nice out here, and baseball is in full swing now, and I don’t know about you, but I’m really tired of thinking — let alone writing — about Special Rela-friggin’-tivity!)*

April 16th, 2015 at 6:53 pm

My light bulb may be dimmer than most, W.S. Smiles, but I enjoyed this about time and space which always makes me want to add ‘continuum’s.’ Why is that?

April 17th, 2015 at 1:02 pm

Well, the “space-time continuum” is a common phrase when talking about this stuff. Some just call it “space-time” others “the space-time continuum.” Depends on how many words you wanna use, I guess. Both phrases say the same thing. (I suppose, if quantum theory turns out to be completely correct, and space and time are quantized, then “continuum” may no longer really be a correct word, but (A) we’re a looooong way from being able know if space-time

isquantized, and (B) some of us secretly hope it isn’t (I might talk about that in this series next week).As for your light bulb, I wouldn’t worry about it. We all have our own “wattage” and there is a lot more to life than being a bright bulb in the marquee. (I’m a good example that being capable with science and stuff and being capable in life are two completely different things.)

April 17th, 2015 at 6:00 pm

It’s a shame we can’t get length contraction by itself. It’d make a respectable FTL mechanism.

April 17th, 2015 at 7:47 pm

How do you figure?

April 18th, 2015 at 9:15 am

Just being whimsical. If we could contract the distance to Proxima Centauri from 4.24 light years to 4.24 light hours, that would be a cool thing. I know it’s incoherent, that you can’t get length contraction without time dilation and mass increase. But it would be sweet if you could.

April 18th, 2015 at 2:13 pm

Oh, I gotcha. (Ha! Despite having written about it several times in the series, I apparently have a blind spot to thinking about relativistic travel as “time travel” — must be the lack of a return ticket.)

From the universe’s point of view, the ship would be contracted, but so what. From your point of view, the distance would allow you to make the trip in a short time. Cool!

But wait. What about the times? The universe has to see the ship take somewhat longer than light, so there’s no FTL advantage there. From your point of view, you’d reach your destination in short time, but you’d arrive there years into the future… Meh! It just goes to show how interconnected contraction and dilation are. Even when you try to ignore one, it sneaks back in as the inverse of the other. The geometry is impossible otherwise.

Just one more nail in FTL’s coffin! 🙂