Home > Popular Science > Einstein's Relativity  
Roses are red, violets are blue, I'm schizophrenic and so am I. (Oscar Levant)  

Einstein's theory of relativity is not easy to understand because it defies commonsense, and does not apply in obvious ways to our daily life and experiences. For this reason I have decided to use the common "twin goes on a fast space voyage" scenario without the scientific jargon. This is not a proof of the theory, just a description of how it works. If you'd rather skip the details of the journey, you can see a summary of the different effects that are observed as a result of high speed travel near the bottom of this page. There are several images below that show different stages of the journey. I have another page that shows these images as an animated sequence to give an idea of what happens over the entire journey. Relativity will become part of our daily life eventually. For example, NASA designed a craft to travel to Mars at an average speed of 1/100 the speed of light with a maximum speed of around 1/80 the speed of light. Even at this tiny fraction of the speed of light, the effect of relativity needs to be taken into account in communicating with the craft. Firstly I must credit Paul Davies, whose example I have expanded upon. Paul is a talented Professor of Natural Philosophy at the University of Adelaide, Australia. He has written an excellent series of books published by Penguin on the topics of time, space, the cosmos, the start and end of the universe, and so on. The 'thought experiment' here is from his book 'About Time'. I heartily recommend these books to you if you're interested in a far more authoritative account of the universe than you will find here. I would also like to thank Paul Alan Cardinale for his help and patience in improving this example, his tables to clarify what each twin sees and can calculate, and for adding some valuable "Relitavistic Tidbits".
Ann and Bob are twins on Earth, and Bob visits a star 8 lightyears away, stops for an instant, then returns home to Earth. All the time, he travels in a superrocketship at a constant 240,000 kilometres per second (80% of the speed of light). For the sake of the example, we assume that the acceleration is instantaneous, although of course these gforces would crush any mortal. We also ignore any movement of the Earth (which is truly trivial compared to a star 8 light years away). The twins are each equipped with a clock and powerful telescopes so they can view each other's time. For the sake of interest, an obliging alien has placed a clock on the star synchronised to Earth time, which can be viewed both by Ann from Earth, and Bob during his space flight. Please read this important concept!It is actually completely irrelevant to compare Ann's and Bob's times over this journey, as if we could be everywhere in the universe at all times to be able to observe. What Ann and Bob each see at these times are different, and even what they calculate about each other's times (knowing Einstein's formulas, the speed of light, etc) will be different! To put it another way, observers separated by large distances, and moving relative to each other, will calculate different actual times and rates of time for each other. "rate of time" is how fast they see and calculate each other's clock changes compared to their own. Also, they can only guess when calculating what each other's "now" might be like. They will not be able to confirm that their calculations were correct until much later when light has had time to travel from each person's location to the other. The pictures below show how meaningless it is to compare these events when they are far apart and moving. Nevertheless, I have chosen to present it this way, because I think that most readers will want to understand things like: "Where is Bob when Ann sees him arrive at the star?" and "What does Ann see when Bob stops at the star?"
Here is what the twins see just before Bob departs on the 1st of January in the year 2000. The clocks are "year clocks", showing 2000 for January 1, 2000 (so 2000.5 would be July 1 2000, etc). I have shown pictures below at various times through Bob's journey.
Both Ann and Bob are stationary on Earth, and read the clock on the star as 1992.
Remember, this clock is showing Earth time, and it takes light 8 years to travel from this star because it is 8 light years away.
They both see the star as it was in the earth year 1992.
The distance of this star in kilometres is:
At each stage of the journey, I have shown a picture of what each person sees, with some text to describe some of the interesting points. Each section also has an more details link so you can see a table to compare the difference between what each person sees and calculates about the other.
Here is what the twins see at the instant Bob departs on the first of January in the year 2000.
Notice that Bob is now travelling at 240,000 kilometres per second. This is 80% of the speed of light (which is nearly 300,000 kilometres per second). 80% of the speed of light corresponds to a timespace dilation factor of 60%. That means that they each calculate the other's time to be passing at 60% of the rate of their own time. To both Ann and Bob, their own time will pass normally. Bob in this case will not see his watch moving faster or slower than he thinks it should. He is in his "own time" and it will pass normally; it is only his view of Ann's Earth time that will be different. The other difference is that the space between Earth and the star is reduced to 60%. Note that this distance does not just appear to be less, it really is less. Of course, the star does not physically move, only the space beteen Bob and the star has been contracted. Because the star is now closer, it will appear brighter, bigger and will have a stronger gravitational effect (of course, gravity from a star nearly many light years away is negligible at this stage of the journey). Bob's perception of the size of the rocket ship is also normal, because he is travelling with it; it is only the space through which he travels that is contracted. Remember, that Bob perceives his own time passing at a normal rate. At 240,000 km/sec he will expect to arrive at the star in 6 years. Just as this speed reduces the time Ann sees pass on Bob's rocket ship over the entire journey, it also reduces the distance to travel for Bob.
Here is what the twins see in Ann's Earth year 2005, and Bob's rocket year of 2003.
There are 3 things that affect how you see time on distant objects:
The combination of these three effects determines how we calculate the actual time and "aging rate" of other objects. Notice that they see each other having "time rates" slowed to 33%, and each person's view of the other is delayed by the time it takes light to travel between them. This delay means that at Ann's 2005, she will see the rocket in the position it was when light left it. Ann sees Bob's rocket 2.2 light years away, showing the time as 2001.7. At Bob's 2003, he sees his position halfway between the Earth and star. He can now see that the clock on the star actually is synchronised with Earth time, because they both show the same time: 2001. At the halfway point, it takes light the same time to reach his rocket from both Earth and the star (regardless of whether he is moving or not). During his 3 years travel so far, he has seen the Earth clock running at 33% of his own rate, and the Star clock racing ahead at 3 times his own rate, from 1992 to 2001.
Now Bob has actually reached the Star, and is still travelling at 80% of the speed of light towards the it. It took Bob only 6 of his years to get to the there. In Ann's time 2010 she sees Bob 4.4 light years away from Earth, with the rocket time showing 2003.3.
For the sake of this example, Bob only stops for an instant to check all the times and distances before he returns home again.
Now that he has stopped, Bob sees both the Earth clock and the Star clock advancing at the same rate as his own. He sees the Star time as 2010, and the Earth (being 8 light years away when he is stopped) showing 2002. Even though the diagram shows that Bob now sees the Earth 8 light years away, the Earth or Star have not moved, it is the space between them that has increased (that is, it is not contracted). Ann still sees Bob on his way to the Star, 4.4 light years away with the rocket clock showing 2003.3.
Bob is still at the star, but travelling back towards Earth at 240,000 km/second.
Now that Bob is travelling away from the Star, and towards the Earth, he sees the Star clock advancing at 33% of his own rate, and the Earth clock advancing at 3 times his own rate. Also, because he is moving at 80% of the speed of light, the Earth is now 4.8 light years away instead of 8.
Now Bob is actually half way back, but at Ann's time of 2015, she still sees Bob on his journey to the star, 6.7 light years away. She will not see Bob at the half way point on the return journey until Earth year 2019. On the journey to the star, Bob has seen the Earth clock running at a third of his own time rate (due to the time it takes light to travel this distance, and the Doppler effect of him moving away from the Earth). On the way back, he sees Earth time running 3 times faster than his own. He knows from the relativity formula that Earth time is running at 60% of his own rate during the entire trip (ignoring his brief stop at the Star).
Bob is now halfway between the Earth and Star, and can again confirm that the Earth and Star clocks show the same time at the same time: 2011.
After 18 of Ann's years, she finally sees Bob arrive at the star. Ann sees both the rocket and the Star 8 light years away, with the rocket clock showing 2006, and the Star time showing 2010. Remember this is exactly how Bob saw the rocket and Star times when he arrived. Ann is correctly seeing this as is was, and it has taken 8 of Ann's years for this image to reach Earth.
This is not a special time for Bob, who is by now well on his way home.
Ann also sees Bob stop at the Star for an instant. The only thing Ann sees change is the rocket's time rate is now the same as her own, and no longer advancing at 33% of her own rate, because the rocket is now stopped and not moving away from her.
This time, the only change Ann sees is the rocket is now heading towards her at 80% of the speed of light, and its time is racing forward at 3 times her own rate.
After 19 of Ann's years, she sees Bob halfway home. Of course, Bob will actually arrive home in 1 more of Ann's years. Ann sees the rocket time showing 2009, and still advancing 3 times faster than her own.
And finally, Bob arrives back home! In the picture, Bob is still moving at 240,000 km/sec towards Earth, so they still see each other's time rates running 3 times their own.
When Bob stops moving, and the twins are together again in the same time and place, they will see that on average Bob's rocket clock has been running 60% slower than Ann's Earth clock, and that Bob has actually only aged 12 years compared to Ann's 20. Bob or Ann have not been "shortchanged" in their life; Ann has lived 20 years while Bob was living just 12. Notice that they both measure the same average speed of Bob's journey. Ann on Earth sees that Bob has taken 20 years to travel 16 light years, while Bob sees himself taking 12 years to travel 9.6 light years. Both give the same answer of 240,000 km/sec, because Ann sees Bob's time run slower by the same factor that Bob sees distances reduced.
Now that they are finally in the same place at the same time (and not moving in relation to each other), they can agree again on what each other sees and calculates.
There are several observable effects of this high speed journey to a distant star: TimeProbably the most difficult comprehend is the different time flow rates. Bob and the rocket have aged only 12 years while Ann and the Earth have aged 20 years. Both have identical start and end times, however, each person observes a different rate of time flow in the other. MassA similar effect applies to mass: Bob in his fastmoving spaceship would measure the Earth having a different mass than Ann would measure on Earth. Likewise, Ann would measure a different mass of Bob's spacecraft than Bob would measure. Both Ann and Bob feel their normal weight, however, they would measure different results in each other. DistanceOther effects of the journey are simply related to Bob's contracted distance to the star (from 8 light years to 4.8 light years at the start of his journey) due to his speed in the spacecraft. There is no argument that the distance really is less (it doesn't just seem or look less), because what he sees as a straight line to the star is actually a shorter curved geodesic path than the geodesic path he sees at rest. So on this journey Bob has actually travelled 9.6 light years (4.8 to the star plus 4.8 return) while Ann has observed him travel 16 light years. GravityThe inverse square law shows that Bob and the spacecraft would sense about 2.8 times more gravitational attraction from the star due to being 4.8 light years away from the star instead of 8. Of course, due to this rule, Bob would not actually be able to feel any gravitational effect from the star at all until he gets quite close. BrightnessStars radiate energy in all directions, however, observers see only the portion of visible light that enters their eyes. Brightness is a human perception, however, a technical measure of light intensity is luminance which is measured in "candela per square metre". The inverse square law also applies to luminance which will be about 2.8 times higher at Bob's contracted distance of 4.8 light years compared to 8 light years at rest. But for this exercise, we are actually interested in the change of perceived brightness due to Bob being closer to the star at the start of his journey. Humans perceive brightness in a logarithmic relationship to luminance (sensitivity of the eye decreases rapidly as the luminance of the source increases). For example, a star that delivers twice as much measured luminance is not perceived as "twice as bright". There are other complicating factors such as the way our eyes adapt to changing luminance levels over time and to all visible objects; not just the object of interest. Also, human perceptions of brightness differences are highly subjective estimations. We know that due to Bob's increased speed towards the star, the distance to the star decreases and therefore Bob might be able see an increase in the star's brightness. SizeHumans tend to ignore the changing size of fixed objects as we move closer or further because it is such a common daily occurence. It's usually only when we're surprised by the size of an object that we notice, because our mental compensation for an estimated size turned out to be wrong. So for the sake of Bob, who is suddenly 4.8 light years away from a star instead of 8 light years, let's assume he can objectively view the difference in size. There is a tangential effect as we move closer to an object because it occupies a larger angle of our view. Look up the trigonometric definition of tangent on the web or in WIKI for more information. Like gravity, this is trivial at a distance of nearly 5 light years. The star would actually look like a point light source until Bob has travelled close enough to the star for it to be visible with area as a flat (2 dimensional) disc. And More?The information above on gravity, brightness and size is based on Euclidian (straightline) geometry and not Riemannian (elliptic) geometry that applies so critically in this scenario. I don't expect that these effects to behave differently over curved lines instead of straight, however, I would be interested if others more knowledgeable than me can shed any light (pun intended) on how this might further affect perceptions.
Just after he left, he saw Ann's clock was advancing at ^{1}/_{3} normal. Six years later he saw Ann's clock at 2002. When he turned around he saw Ann's clock advancing at 3× normal. Six years later, he saw Ann's clock at 2020. Half the time, he saw Ann's clock running at ^{1}/_{3} speed, and the other half he saw it running at 3× speed; that averages out to ^{5}/_{3} normal. (^{5}/_{3}) × (12 years on Bob's clock) = (20 years on Ann's clock). What Bob CalculatesJust after he left, Bob knows his speed relative to the Earth, and calculates that Ann's clock advances at 60% of his own rate. Six years later he calculates that Ann's clock shows 2003.6. When he turns around, he calculates that Ann's clock jumps ahead by 12.8 years to 2016.4, and as he returns, he knows it is again advancing at 60% of his own rate. Another six years later, he calculates Ann's clock shows 2020. So ... (60% of 12 years on Bob's clock) + (12.8 year jump at turnaround) = (20 years on Ann's clock).
Ann sees Bob's clock running at ^{1}/_{3} speed for 18 years, then running at 3× speed for 2 years. This gives an average of (^{1}/_{3} × 18+3 × 2) / 20 = 0.6 (i.e. 60%). (60% of 20 years on Ann's clock) = (12 years on Bob's clock). What Ann CalculatesAnn calculates Bob's clock running at 60% of her time rate for 20 years. (60% of 20 years on Ann's clock) = (12 years on Bob's clock).
"How can people age different amounts of time, then meet together later in the same place and time?" Even if you've understood none of the above, the simplest answer I can give is that travelling at high speed changes how fast you see time pass on distant objects. For this example, Ann who stays on Earth sees Bob's rocket time progress at ^{1}/_{3} of her own rate for 18 years while she sees it flying away from Earth. She then sees the rocket's time run 3 times faster for 2 years while watching it return to Earth. In the rocket, Bob sees Earth time progress at ^{1}/_{3} of his own rate for 6 years while flying away. He then sees Earth time progress 3 times faster than his own for another 6 years while returning to Earth. The twins' times are different when Bob returns because Ann has seen Bob's time running slower than her own on average, while Bob in the rocket has seen Earth time running faster than his own on average.
