Saturday, April 10, 2010

Theory of relativity

I went back to K.L. for Chinese New Year celebration a few months ago. Over dinner, I was explaining the theory of relativity to my friends and to my utter dismay, I did a terrible job of it. And in the process of doing so, I realised that I did not understand the theory as well as I had hoped. So here I am again, attempting to explain this theory which I failed to do so miserably months ago and I hope this time, I would do a better job.

Motion is relative
We need to first understand that motion is not absolute, but relative. What do we mean by that?

Let me begin by a simple question: If you are standing still, are you moving? This question may sound absurd and to many, the answer is a quick "no". But what if I repeat this question to someone who is observing you from the Sun? The Earth revolves around the Sun, so if you are standing still on Earth, you must be rotating together with Earth about the Sun. So an observer at the Sun will see you as "moving", while an observer on Earth will see you as "standing still".

Let's look at another example. If you are at the train station when a speeding train whizzes past you. To you, it looks rather obvious that the train is moving - moving FORWARD. But to the passenger on the train, it is you who is moving BACKWARD. For the same motion, two different observers see it differently and there is no one single reference frame which we could base all our observations upon and agree! Therefore motion cannot be absolute. It must be relative, i.e. whether an object is moving forward, backward or not moving at all is completely subjective to the observer or the equipments that are measuring it. A motion or measurement of speed has no meaning if we do not attach a point of reference (or observing point) to it!!

Let's say the train is moving at 50 km/h. If the passenger in the train throws an apple vertically into the air, the apple would have landed back onto his hand. But to the person at the train station, it would seem like both the apple and passenger had moved forward in the same speed and thus the apple landed onto the hand of the passenger again when it drops. Now imagine that the passenger now throws the apple in the direction of the moving train at a speed of 10km/h, what would the person at the train station see? He/She would have seen the apple travelling faster than the train! The apple has to be travelling faster than the train in order for the passenger in the train to see the apple going forward. The speed of the apple would SEEM to be travelling at 50+10=60 km/h. In other words, the velocity of the apple is the "sum of the velocity of the train and the velocity of the apple being thrown":

v_sum=v_train + v_apple

v_sum is the velocity of the apple as seen by the person at the train station
v_apple is the velocity of the apple as seen by the passenger on board the train
v_train is the velocity of the train as seen by the person at the train station

If, however, the passenger is now throwing the apple in the backward direction, i.e. in the opposite direction of the travelling train. Then, the person at the train station would see the speed of the apple as "the speed of the train minus the speed of the thrown apple", i.e.

v_sum=v_train - v_apple

Speed of light is constant in vacuum
From the above, it would seem that 'speed' can be added linearly. The above is true for Newtonian physics - physics before theory of relativity. But the observant reader would have noticed that I emphasized the word "seem" to suggest that Newtonian physics is not entirely true, especially when we approach the speed of light. Motion is still relative, but the assumption that "v_sum=v_train + v_apple" no longer holds true! Let us see why.

Center to the theory of relativity is the discovery that the speed of light is constant in vacuum. In the late 19th century, carefully conducted Michelson-Morley experiment had proved this so. But how can this be true if speed can be added linearly, like the apple thrown in the train?

Imagine now that instead of throwing apple, the passenger turns on his/her flashlight and point it in the direction of the moving train. If the observer/person at the train station carries equipment that could measure the speed of light, what would the measurement be? Based on what we have discussed earlier, we would have expect that the speed of light measured by the person at the station is the speed of train plus the speed of light. Just like,

v_sum=v_train + v_apple

with v_apple now representing the speed of light as seen by the passenger.

but if v_apple is the speed of light, then v_sum would have to be greater than the speed of light. But this is NEVER the case! One may argue that perhaps the speed of the train is so small compared to the speed of light that the equipment is not sensitive enough to pick this up. But very careful measurements have been made and every time the speed of light remains constantly the same, whether it is measured by the person at the train station or the passenger in the moving train. This is very peculiar indeed and the assumption that speed can be added linearly must surely be wrong!

The physicists who came upon this experimental result were dumbfounded and they needed to re-visit their fundamental understanding of what 'speed' really means and how measurements are made (did they measure the speed of light wrongly?).

How do we measure distance and time?
There are many ways to measure distance but the basic principles are the same. We measure distance, for example, by using many rods of the same length and lining them up together. If we find the 3 rods lined up exactly along the side of the table, we say that this table is 3-rod long. The common 'rod' that we used is called a meter, so we can claim that the table is 3-meter long. But what if the table is just a little short of the 3 rods? How do we measure the length then? We could, of course, then use shorter rods. That is why we have centimeters, milimeters and inches. They are just 'rods' with different lengths.

Measuring time is a bit more difficult, but again the basic principles are the same. Time can be defined as 'how long it takes for an event to happen'. Now if we can find an event that always take the same amount of time to happen, we can use that as our reference to measure the time for everything else - just like the rod for measuring the distance. The first clock, invented in China, was based on the event of a dripping water. Today, most mechanical clocks measure time based on the event of the ticking of a second.

Special Theory of Relativity: Time and space is relative
From the discussion so far, we know that when we approach the speed of light, things are not always as they seem. For one, speed can no longer be added linearly and that the speed of light is always constant, no matter how you measured it. What does this tell us about time and space?

If the speed of light is always constant, something else must be changing. Speed is defined as the distance travelled divided by the time taken. If speed remains constant, then the distance travelled or the time taken or both must have changed in such a way that the speed of light is always constant. For example, if v = a/b, then v remains constant if a and b increase or decrease by the same amount. It turns out, after some mathematical considerations (called Lorentz transformation), that both the time and distance must have changed. This suggests that not only motion is relative, but time and space must also be relative (note that space is just length in 3-dimension)!

Even without the mathematics, but just logic considerations, we would have came to a similar conclusion. Let's now go back to the example of measuring light speed on the train. The passenger on board the train, travelling at the speed of the train, shines his/her torch light in the direction of the moving train. The person at the station makes a measurement of the speed of the light. According to Newtonian physics, he/she should obtain a speed faster than light in vacuum, but he/she does not. To correct for a smaller value for speed (v = a/b), either the numerator have to be smaller or the denominator have to be bigger or both. The numerator turns out to be the distance (smaller) and the denominator is the time (bigger), and so our measurements for distance and time must have changed accordingly. It must have because space has contracted, and time has dilated (moved slower).

Hence, the theory of relativity predicts that as we approach the light of speed, space contracts and time dilates (move slower), i.e. the measured distance is shorter and the time taken for the measurement is longer (since 'distance' or 'length' in 3-dimension is space).

But the adept reader would have argued that if space contracts for an object that is travelling at the speed of light, would not the rods - which we make measurements with - contract as well. Thus, we would have counted the same amount of rods for the contracted distance. The fallacy of the argument lies in the fact that this reader had failed to see where the measurement is being made. The measurement of the object travelling at the speed of light, is made by the person standing 'still' at the train station, using the rods provided. These rods, are not travelling at the speed of light. They are WITH the person, standing 'still' and therefore do not experience space contraction. So the measurement was made by using rods at still, to measure the distance covered by an object travelling close to the speed of light, which is experiencing space contraction. And the same explanation goes for the time dilation.

We have so far covered the "special" case of theory of relativity. So what this theory say is that not only motion is relative, time and space are also relative. There is no way we can say for absolute, 1 second is 1 second and everyone else agrees. 1 second is different for everyone. 1m is different for everyone. But at speeds much less than light, these differences are OK. There is, however, a bit more to this theory ...

The Twin Paradox
We can claim that A is moving away from B at light speed. But since motion is relative, this is analogous to saying that B is moving away from A at light speed. Remember, the relativity of motion states that there is no way we could know whether A or B is really moving, it only depends on where your reference is.

Consider this: Ali and Wong are twins. One day, Ali sat on an alien spaceship and whizzed off to outer space at light speed. After a certain time Ali came back to Earth, also at light speed, and met his twin brother Wong again. When they compared their watches, they realised that Wong had aged faster than Ali.

Let's consider the problem in detail and see why this is such a tricky problem.

What does Wong see?
1. Ali travelled away at light speed.
2. Since Ali was moving at light speed in relation to Wong, Wong observed that Ali's time ran slower (dilated). (For argument's sake, let's just assume that for every 1 hours that Ali had travelled, Wong observed that it only took 2 hour.)
3. Ali stopped his space ship when Wong's clock recorded 10 hours had passed. Due to time dilation, Ali's clock recorded only 5 hours.
4. Ali now return to Earth and Wong recorded that it took another 10 hours, but again, Ali's clock recorded only 5 hours.
5. In total, Wong waited on Earth for 20 hours, but observed Ali's clock to have only ran for 10 hours.

What does Ali see?
1. Ali travelled away at light speed.
2. Since motion is relative, Ali observed that he was stationary but it was Wong who was travelling away at light speed.
3. Ali stopped his space ship when his clock recorded 5 hours had passed. Due to time dilation, Wong's clock only recorded 2.5 hours. (because to Ali, it was Wong who was travelling at light speed and thus experiencing time dilation)
4. Ali travels back to Earth and recorded that it took him another 5 hours, but again, Wong's clock only recorded 2.5 hours.
5. In total, Wong waited on Earth for 5 hours and Ali travelled for 10 hours.

Herein lies the paradox: Upon returning from his travel, Ali thought Wong was younger but Wong thought Ali was younger. How can they both see each other as being younger than themselves? One of them had to be wrong! Carefully conducted experiments suggest that Ali (who travelled away in light speed and returned) is the one who should be younger. Then, where did our understanding of theory of relativity gone wrong? Surely a theory must be able to explain the results of an experiment!?

The main issue here is that because motion is relative, i.e. we cannot distinguish who is moving or who is at stationary. Ali sees Wong travelling away at light speed, but Wong sees the same of Ali too, just like in the train example above. Therefore, it is perfectly logical (if motion is relative) to claim that either of them had experienced time dilation.

But since experimental results suggest otherwise, there must be something different between Ali and Wong. What is the difference? How can physics distinguish them? Can we distinguish who is the one who is really 'moving away'?

No. We cannot distinguish who is really 'moving away' but they could distinguish who is 'returning'. Now, the adept reader should realised that so far all our attempt to explain theory of relativity has been made on constant speed. No acceleration (change of speed) was involved. If the train moved at constant speed, then neither the passenger nor the people at the station would be able to tell who is moving. But what if the train suddenly brakes? The passenger on board the train would see the people at the station going backward but suddenly losing this speed. The people at the station would see the passenger going forward but suddenly losing this speed too. But the people at the station will not feel the 'force' of the brake, the passenger will!! The passenger will lunge forward as a result of the sudden brake and will have to hold on to something to prevent injury. The people on the station feel nothing at all!

So while we cannot distinguish the two based on speed alone, we can distinguish them based on a change in speed, i.e. acceleration. Acceleration - that was the difference, that was what broke the symmetry between Ali and Wong. As Ali turned his ship around and returned to Earth, he experienced significant acceleration. And it would seem from this case (since Wong aged faster) that the effect of acceleration was to 'distort' Ali's observation on Wong. During the turnaround, Ali's time stood still while Wong's time ran for another 15 hours such that when Ali returned to Earth he recorded 10 hours of travelled time while Wong recorded a total of 20 hours (5 hours for Ali's outgoing and incoming flight + 15 hours of the turnaround). The acceleration had obviously no effect on Wong.

Now, after this 'adjustment', both of them would have agreed that Ali aged slower.

The above example is a bit extreme, of course. No one could travel at light speed and turnaround at light speed IMMEDIATELY. If the turnaround was not that extreme (but still in the order of light speed) then Ali's time would not have stood still while Wong's time ran for another 15 hours. Instead, Ali's time would just run much slower compared to Wong's time e.g. 1 hour of Ali's time to 15 hours of Wong's time. Therefore, acceleration seems to give similar effect to time (and length) as travelling near the speed of light would. Acceleration would make time run slower and length contracts too. This is the basis for the general theory of relativity!

General Theory of Relativity: Gravity changes time and space

The main difference between the Special Theory of relativity and the General Theory of relativity is the inclusion of the effect of gravity for the latter.

While explaining the twin paradox, I explained that acceleration had a similar effect on time and space (remember that length contraction is equivalent to space contraction). In science, if there is ABSOLUTELY no way to distinguish between two things, then these two things are said to be same. Is there a way to distinguish between the effect of acceleration and a gravity field?

The answer is no. In the twin paradox, Ali made an instantaneous turnaround to return to Earth. During this turnaround, he experience acceleration. The effect of this acceleration is analogous to the effect of gravity. During this turnaround, Ali would be pushed back to his seat just like a gravity field is pulling him. Therefore, if acceleration could cause time dilation and space contraction, so can gravity. They are equivalent.

We can now rephrase what we know about gravity based on our understanding of the twin paradox. The effect of gravity is that time dilates and space contracts for observers close to the gravity field. For the twin paradox, we could have well said that during the turnaround, Ali was close to the gravity field and experience significant time dilation while Wong was sufficiently far enough that he experienced none.

Let's assume that time and space are the two axis of a big fish net. If we place a big object (like a planet) at the center of this fish net, this will create a huge gravity field. If gravity causes time dilation and length contraction, you could imagine that the net has a higher "density" near where the planet is located. The net will look as though it is bent around that region. That is why general theory of relativity says that space-time is warped (bent) by gravity field. Now, if we roll a ball across the net, you could imagine that the ball will initially move in a straight line but then curve inwards as it approaches the bent. This is how general theory of relativity explains the gravity pull!
(source: nrumiano)

And that's all to the theory of relativity! The rest are just maths and more maths. I hope this have helped you to understand the theory better. There is another excellent website on this here.


kimtiam said...

Wah! Great job!

Snowpiano^ ^ said...

Lazy to read~~Explain again when u come back yah! :D

joonyi said...

so, theory of relativity is used to explain something that is absolute (speed of light)? interesting..

DeluSion said...

You wrote that the person in the train would "feel" the "force" (or rather, acceleration) when the train changes direction, whereas the one at the platform won't, and apparently this provides an intuitive understanding about how the symmetry is broken....

I'm really really intrigued...

Where does this "feel" come from?

I try to picture myself being freely suspended in "space" and moving in one direction away from a certain object. If the object is to change its speed, by intuition, I wouldn't "feel" the "force"....

Why is this so?

Can you direct me to any web resources or suggest any reading material to me?

(My level of physics: STPM)

Thanks a lot for writing these articles.

shinliang said...


Thanks for reading. There a lot of web resources that are very interesting, and suitable for different levels. You can check this out:

and also the article written by Einstein himself in 1916 "Relativity: The special and general theory", which is very very well written in a logical manner. It really depends on the reader, which usually have different style preferences.

For Quantum physics, you can read Feynmann. No one explains better than him!

As for why we feel the "force"... hmmmm.. that's a good question. I'm not sure if we already have an answer for that. But I think for a simple answer, it's because of Newton's 1st Law on inertia.

DeluSion said...

Thank you Shinliang!