Friday, 17 August 2012

The Lightness of Fine-Tuning - Part 2


In Part 1, I talked briefly about the fact that light travels as fast as it possibly can, and c is therefore the fastest anything can travel.  I also introduced the concept of the laser, a law abiding, speedy rider (“laser”) cruising along a freeway which has a speed limit of c.  I pointed out that we we might notice that the speed of a “laser” is c but that this was related to the speed limit, rather than being a characteristic of the laser per se, and that similarly the speed of light isn’t related to light per se.

Before we get into it – a caution for people studying relativity formally: please use the derivation methodologies recommended by your teacher or lecturer.  I don’t believe that what follows is wrong, but it’s not entirely standard.

---------------------------------------

In this part, I want to illuminate an interesting thing you can do with relativity.  It involves some mathematics, but I’ll try to keep it simple.

Relativity sounds scary, but it doesn’t have to be, or at least not until you get into General Relativity!  First of all, Einstein didn’t invent it from scratch, a version of relativity was formulated by Galileo Galilei in the early 1600s.  Today we normally refer to it as “Newtonian relativity”, or “Newtonian mechanics” and most people would be familiar with a version of it from High School:

x' = x - vt

This is the equation you use when working out the answer to the clich├ęd question about two trains that leave different cities and crash a certain amount of time later.  While the question is a bit silly, the equation is excellent and it works fine when used properly.

However, the equation comes with some assumptions which most people aren’t aware of.

Imagine that you are watching a train depart towards Manchester from London (don’t ask me why anyone would want to go to Manchester – it’s just a hypothetical scenario).  Imagine further that the track is completely straight and the Earth (or at least England) is flat and that you have a very powerful telescope.  Finally, imagine that as the rear end of the train passes you at a cruising speed of precisely v = 100 km/hr, you zero two clocks (one that you keep, the other is held by your evil twin on the train).  Precisely one hour later (t = 1hr), there is an explosion.  You happened to be looking through your powerful telescope at the time and know that it was Manchester that blew up.

How far from Manchester was the train at that time?

The picture below illustrates the situation:


Assume that we know that the distance from London to Manchester is x = 300 km.

Therefore when Manchester met its end, the distance between the train and the explosion was:

x' = x – vt = 300km – 100km/hr x 1hr = 200km

Pretty simple, huh?

Now, let’s change the scenario only very slightly.  We’ll let Manchester survive (I know that some of you might be disappointed, but there must be at least one nice person from Manchester …) 

Now you don’t know exactly where the explosion happened but, using smoke and mirrors, we can work out that it was triggered at the same time as you zeroed both clocks.  Now, I didn’t mention it before, but your clock (and that of your evil twin) is extremely accurate.  We could use the clock to work out where the explosion happened by noting down the precise moment when you see the explosion – there’s a bit of a delay because the photons from the explosion don’t reach you instantly.  Let’s say that your evil twin also notes the exact moment that the first photons from the explosion pass.

Now because the clock is so accurate (and you are both so quick-witted), there will be a difference between the time you note down and the time your evil twin notes down – because your twin is moving towards where the explosion happened. During that time, the train will have moved.  This is illustrated below:


Again, not overly difficult, is it?

Although, when you look closely at the picture and think about the situation, you might notice that the time at which the first photon reaches your evil twin, t', has to be less than the time it takes for the first photon to reach you, t.  Let’s say (to make things simple) that we are talking about slow photons where c = 800km/hr, that v = 600km/hr (it’s now a bullet train!) and also say also that you receive the first slow photon an hour after the explosion (t = 1hr) – when did that slow photon pass your evil twin?

We can work it out using a little equation.  If you are stationary, you can calculate that the explosion must have happened 800km away.  You watch our your evil twin moving away at 600km/hr.  Therefore the distance between you and your evil twin is:

vt = 600.t

and the distance between you and the slow photon is


xG = 800-800.t

When these two values are the same, the slow photon is where your evil twin is.  Using some mathemagics, we could calculate that the actual moment at which the slow photon is where your evil twin is is after 34.28571 minutes have passed.  Futhermore, we can work out that at this time your evil twin was 342.8571 km distant from you.

A problem arises, however, when your evil twin’s evil nature asserts itself.  Your evil twin claims to be stationary and that it is you who move!  While this seems like a ludicrous claim on your evil twin’s part, you’re an accommodating type so let’s look at the appropriate equations.

Now we are assuming that you are moving away from the explosion at 600km/hr so that the distance between you and the slow photon is at the start was 200km. (We work this out from the fact that after an hour you have travelled 600km, and the fact that the slow photon has travelled 800km.  It takes an hour for the slow photon to catch up because it is travelling 200km/hr faster than you are.)


xG' = 200-200.t

When this value is the same as the value vt = 600.t, your evil twin and the slow photon are in the same location.  Using mathemagics again, we work out that your evil twin and the slow photon were in the same location after 15 minutes.

Hang on!  There is a problem here!  Your evil twin's claim to be stationary results in an inconsistency in time!  While this seems bizarre, there are situations in which neither of you could know which one is in motion and which is stationary.  Let us stick with trains and see if we can sort this out.

Imagine that both you and your evil twin are in box carriages.  Once you are at a steady cruising speed, thanks to the wizardry of British Rail and the Irish Navigators, you cannot know whether you are in motion or stationary.  The same is the case for your evil twin.

All that either of you have for entertainment is an extremely accurate clock, a pad of paper, a pen and a klaxon.  During your journey (or non-journey, as the case may be), the klaxon goes off twice.  Interestingly, the first klaxon also resets each clock to zero so you don’t bother writing that time down (it’d just say “00:00:00.0000000000”).  The second klaxon doesn’t activate a reset so you both keenly write down the details.

After the journey (or non-journey) ends, you each have a time written down but no idea what it means.  Let me explain what just happened to you.

The klaxon was triggered by a proximity sensor, the first time when your box carriage and that of your evil twin passed each other.  The second was triggered when you were passed by a laser (the Law Abiding Speedy Rider, or “laser”, on a motorcycle, travelling at the legal speed limit, c).  The laser passed the evil twin first, and then, a short time later, passed you.  The situation is illustrated below:


This is basically the same situation as with the explosion at an unknown location, with the slow photons being replaced by a chap on a motorcycle and rather than seeing the explosion you have a proximity klaxon.  There’s just one bit of extra information you need.

The mad scientist running this experiment tells you that the legal speed limit on the Manchester-London Freeway is 800km/hr (for those leaving Manchester only) and that the speed at which you and your evil twin separated was 600km/hr – but refuses to say what your absolute speeds were, muttering something that sounds like “absolute speed relative to what?”

Now, a few notes:

  • I have – once again – made c equal to 800km/hr, this time without defying the laws of physics.  I did this deliberately because I want to prove that the speed of light is nothing special.  I could have said that the laser was riding a magical motorcycle which travelled at 299 792 458m/s but I didn’t.
  • The mad scientist involved is right about absolute speeds.  We don’t know our absolute speed.  Relative to the Earth, I am stationary as I type this.  Sort of.  Some bits deep in the core of the Earth are moving and I am not stationary with respect to them.  But the Earth is not stationary with respect to the Sun, we orbit the Sun at just over 100,000km/hr and the whole solar system is moving at about 1.3 million km/hr relative to the cosmic microwave background.
  • When your evil twin claims to be stationary, this affects all measurements of time and space made in the new “frame” that is thus created.  The claim might seem ridiculously arbitrary, but your claim to being stationary is arbitrary also.  Your claim is that because you are have no speed relative to some elements of a single planet that is tucked away on the fringes of an insignificant galaxy, then you are stationary – despite the fact that you are moving at more than a million km/hr relative to something that pervades the entire universe!  This is somewhat of a ridiculous claim also.
  • Because you cannot know which direction either of you are travelling, the relationship between the way you measure time and the way your evil twin measures time has to be reciprocal – according to each of you, the other utilises different, but consistent units of measurements.  This can be expressed as follows:

or

xE / xG = x’G / x’E

So, where was the laser when you and your evil twin passed each other, according to your evil twin (noting that your evil twin claims to be stationary)?

This is going to take some equations, so I will try to make it as painless as possible (I had a bit of a Freudian slip while typing that and initially wrote “painful” … oops!)

We are going to use the equation just above which, when rearranged, becomes:

x’G = x’E . xE / xG

Now, we have a range of equations from our illustration which I will show again because I put so much time into it:





And from these we can work out the following:

x’E = xE - vtE

xG = x’G + vt’G

Then there are a few equations, but rather than working through them laboriously, I’ll just show them in a little diagram, using colours and arrows which hopefully combine to make it all easier to understand.  Take as much time as you need, or as much as you can bear:




The final equation is, I promise you, one of the standard Special Relativity equations – known as the “Lorentz boost in the x-direction”.  To get the other important equation, the one for time, you can just divide it through by c:


But notice – we did this with a chap on a motorcycle was travelling at 800km/hr and while this is a terrifying speed for a motorcycle it is nevertheless well below the speed of light!  In other words, it’s not the speed of light per se that is what is important, since we can get these fundamental equations using a completely different interpretation of c.

Before we go, let’s make sure that this all works.  Plugging all the figures into the equation (isn't mathematics fun!) we go from:

15 minutes


to 

22.67787 minutes


Interestingly, when we do the same again, to work out what your evil twin will calculate as the time that we would calculate as when the laser passed - if we were stationary, we get: 

34.28571 minutes

---------------------------

Hopefully there will be people who work out that there is something unsettlingly wrong about this.

The fact is that your evil twin on the train will not be in possession of a clock that is limited by the speed limit of the Manchester-London freeway, but if this were the case, then your twin’s clock would read 22.67787 minutes when the laser passed.  Time, remember, is part and parcel of that granularity thing, so it is tied to a real universal speed limit.

In other words, the “speed of time” which is also the speed of light and the “speed of information” is a fundamental characteristic which arises from the very nature of the universe (that is its granularity) and so it is not a separate characteristic that could be fiddled with by a creator.  Therefore, it is not a valid candidate for “Fine Tuning”.

If you remain concerned that my maths is questionable, you can either trust me when I promise that it’s all ok … or you can read the more complex version of the argument (which sadly fails to mention Manchester or the Law Abiding, Speedy Rider).

1 comment:

  1. It seems that an intermediate version was initially posted here! This introduced an error which has been addressed.

    ReplyDelete

Feel free to comment, but play nicely!

Sadly, the unremitting attention of a spambot means you may have to verify your humanity.