Wednesday, 29 August 2012

On Flew and Ehrman

While investigating William Lane Craig I came across two people whose names were not previously well known to me – Bart Ehrman and Antony Flew.  I had a bit of a closer look at them and found them to be interesting cases because they both moved on the atheist-theist spectrum as illustrated below, but in different directions.

This spectrum illustrates a range of possible positions between “strong atheism” on the top and “literal fundamentalism” on the bottom.  I might not have accurately captured all of them.  It should be noted that as you get closer to the bottom, you approach a specific God and in the context of this discussion we can assume that this is a Christian God.  As you approach the more fundamentalist position, evidence disappears as a factor.  I don’t mean that evidence itself disappears, but rather that evidence is looked at differently by people with these positions – evidence becomes something that needs to be explained, rather than being an essential element of an explanation.  (See my article on evidence.)

For ease of reference, we can consider this spectrum in light of a nine-point scale:

These are my own descriptive terms.  None of them are intended to be pejorative, with the possible exception of “Agnostic Fence Sitting”.  This category is for those who want to apply Pascal’s wager but, despite trying their best to believe in a God so as to not be punished eternally for failing to do so, are still not able to really truly believe due to the lack of evidence.  In other words: “there’s not really any evidence, but I’ll lean towards the God premise just in case”.

Let’s compare the history of Ehrman and Flew.

These are based on rough estimates.  Bart Ehrman’s crisis of faith that cured him of fundamentalism happened in the mid-1980s and was followed by about 15 years of liberal Christianity.  Flew was tending towards a deistic belief, or a belief in an “Aristotelian god” or a First Cause, in his mid-70s as early as 2001.

In 2007, There is no A God was published by Harper One under Flew’s name, even though the book was largely written by the “co-author” Roy Varghese and “copy edited” by Bob Hostetler.  Flew’s contribution seems to have amounted to granting approval and allowing earlier work to be incorporated in the book.

Note that Flew’s regular publisher was Prometheus Books, whose publishing goal is described as addressing “the educational, scientific, professional, library, popular, and consumer markets”.  Prometheus does have a few books on the topic of religion, but a brief look at the titles indicates that they have a skeptical orientation, in line with the founder’s intent (Paul Kurtz also founded the Committee for Skeptical Inquiry, an organisation of which Flew was a fellow).  Harper One’s area of publishing interest is described as “the full spectrum of religion, spirituality, and personal growth”.  It doesn’t take much skepticism to think that perhaps Harper One was selected as the publisher because Prometheus would not have been able to provide editors who would be as “accommodating”.  

I am perhaps being a bit unfair in labelling Ehrman as a weak atheist.  It is true that Ehrman explicitly denies being an atheist, is antagonistic to New Atheists and claims to be an agnostic.  However, when speaking to Justin Brierly on Premier Radio’s Unbelievable program, Ehrman described his position in terms that correlate with weak atheism.  Ehrman is certainly not a strong atheist but, given that his position is sufficiently similar to mine, I feel justified in asserting that he is a weak atheist.

The question of whether Ehrman is an atheist (of any type) or not is related to another sort of question: where exactly is the line between “atheists” and “others” and the line between “Christians” and “others”?

I would have thought that for a Christian, the line depends on the nature of their belief.  If a Christian is a fundamentalist, then a wishy-washy type who is willing to interpret the Bible (and is not born again) is not a proper believer.  For an atheist, someone who thinks “there is probably something out there” is suffering from precisely the same type of delusion as the fundamentalist; just not to the same extent (we could call this “Deluded Lite”).

The moves that Flew and Ehrman made on the spectrum are, I believe, as illustrated below:

Now, on this diagram, it looks like Ehrman has undergone a significantly greater change, but there is no real scale involved.  To get a better understanding of the major changes of position, we perhaps should look at them differently.

So, in other words, Erhman went from someone who believed it all, to someone who reckoned that there is sufficient evidence to support the assertion that there was a human known as Jesus, but not much else.  Despite this, I’m reluctant to count him as a true opponent to William Lane Craig in a debate because he seems to shy away from the most likely explanation for a story as amazing as that of Jesus in the almost total absence of any supporting evidence outside the Bible.  That it is fiction.

Flew, on the other hand, went from someone who didn’t believe a thing to someone who was willing to credit God with creating the universe, but not much else.  Despite this, there are plenty of people willing to claim that Flew’s conversion is a great victory for theism (even though he specifically did not convert to theism).  Why is this?

Well, let’s look at a few interesting facts.

Biola University is not known for being a deist university.


Now I have previously stated that I don’t really count Ehrman as being a true “opponent” to William Lane Craig.  This also means that I don’t count Ehrman as a “convert” to atheism, even though his position is almost completely consistent with my atheism.

A small part of my reasoning relates to the fact that Ehrman remains a New Testament scholar, who believes there actually was a Jesus and who seems ambivalent about the Bible.  However, the vast majority of my reasoning relates to the single fact that it does not matter who Ehrman was before losing God, or indeed after.  My personal lack of belief is neither strengthened by a conversion of a theist to agnosticism nor weakened by the conversion of an atheist to deism.  The pope could convert to atheism, and it’d make no difference.  Richard Dawkins could come out as a closet fundamentalist, and it’d make no difference.  My atheism is an entirely personal lack of belief on my part.

Now that that is out of the way, I want to state that I am appalled by the way some people treated Flew.  It is very sad to watch footage of a man who was clearly showing signs of mental deterioration, evident as early as 1998 during the debate with Craig at Biola University, as he is paraded around by smug theists.  (Remember footage of tired and confused allied soldiers making televised announcements on behalf of Saddam Hussein?  It’s rather similar, except Flew was never returned.)

There is a notorious interview between Lee Strobel and Flew in which Strobel shamelessly, and relentlessly attempts to put words in Flew’s mouth (when the interview was held is unclear but the first reference on the internet that I could find is dated November 2006).  Of particular interest in this interview is that not only does Flew appear to be not at his best, but he also clearly states twice that he is not going to write a book and implies it a third time.  Flew does not mention having just written a book (There is no A God was already mentioned as being circulated in pre-print as early as September 2006).
·         4’:39” onwards

o   Flew: “I haven’t even begun to think about this book that I am certainly not going to attempt to write”

o   Strobel (jocular): “We’re going to get you to write it”

o   Flew: “Oh no you’re not”

·         5’:11”

o   Flew: “I’m not going to write the book”

·         9’:05” onwards

o   Flew: “Well, I think, ah, if we were involved in a discussion for this book, that would have to be considered, yes.  I think it’s a reasonable thing for someone to argue.  It doesn’t mean it’s the right, but it’s certainly right for someone to say that, that”

o   Strobel: (incomprehensible vocalisation but probably) “Yeah”

o   Flew: “Yeah and I think that that is in fact what theists would want to say in some way, wouldn’t it, the whole purpose of this creation was to produce believers.”

I strongly recommend that people listen to at least two debates, one between Flew and William Lane Craig from 1998 and a more recent one between Ehrman and Craig.  Compare the clarity of thought exhibited by Flew and Ehrman and just ponder whether the likes of Craig and Strobel should be fisting the air and claiming victory for the Lord because Flew became a deist.


One last thing … I’ve tried, but I’ve not been able to find any reference to claims by high profile atheists that Ehrman becoming an agnostic should be considered as some sort of victory for atheism.  For anyone trying to prove me wrong, I’d happily accept something said by Dawkins (and any who post articles at the Dawkins Foundation site), Dennett, Harris, Hitchens, Smith, Krauss, Law and Shermer.  The clarity of the statement should approach the title of Gary Habermas’ article (Atheist Becomes Theist – Exclusive Interview with Former Atheist Antony Flew) or that of the book crafted by Roy Varghese (There is no A God: How the World's Most Notorious Atheist Changed His Mind).


This article blends information freely available from Wikipedia, Richard Carrier’s blog, Mark Oppenheimer’s NY Times article, and audio and video of debates involving Strobel, Craig, Flew and Ehrman.  My thanks to the people who made this information available.

Saturday, 25 August 2012

Creeping Theism

I’ve observed an interesting phenomenon recently, which I call “creeping theism”.  It might not have started with William Lane Craig, and he doesn’t seem to be the only one exhibiting it, but I noticed it first with his debates.

I have spoken to many Christians about religion in my time: at beach mission, at Sunday school, at High School, at university and in day-to-day interactions.  Until only quite recently, I had never heard any of them describe themselves as a theist.  They never talked about theism.  It was Christianity all the way.

In recent years, however, Christians seem to have introduced this new usage of “theism” wherein theism is cast as being directly opposed to atheism.  While this is consistent with the original usage of the term – that is in opposition to emerging deism in the 1700s – it’s curious for a couple of reasons.

Firstly, the people who use the term “theist” do not consider themselves to be theists per se, they are Christian, or Jewish, or (potentially) Muslim or even some form of Hindu.  Theism is a collective term which includes beliefs that the individual “theist” almost certainly holds to be untrue.  Even a Universalist will hold the exclusivity of the individual faiths to be invalid.  A Christian doesn’t believe “theism”, but rather Christianity.

Secondly, atheism does not have to oppose “theism”, not even strong atheism (or “new atheism”).  Atheists are neither restricted nor inclined to address theism as a whole.

It is this latter point that I think is key.  Creeping theism appears to be an attempt to defend an individual faith by hiding among other similar, but apparently false, faiths.  The “one form of theism among many” argument goes a bit like this:

1.      While you might have some legitimate argument against my form of theism, you can’t possibly prove every single form of theism to be false.

2.      Therefore, it is possible that one form of theism is true.

3.      Insert Plantinga's Ontological Argument here to prove that one form of theism is true (while hoping that no-one notices the logical fallacies involved).

4.      If one form of theism is true, then it must be the best form of theism.

5.      My form of theism is the best form of theism.

6.      Therefore, my theism is true.

On the face of it, this doesn’t seem to make any difference since if you prove the theist’s individual form of theism to be false, then it’s false.  However, this ignores the mental gymnastics of the average Christian.  Say someone (an atheist or some alternate theist), knocks down a vital pillar of a Christian’s world view – for example by convincingly arguing that the errors and inconsistencies in the Bible show that the Bible is in error thus eliminating confidence in the prophesies foretelling the arrival of a messiah and indicating that the Gospels are likely to have been made up.

The Christian may be assailed by doubt, but she will not necessarily be crushed.  She activates the “one form of theism among many” argument, noting that it is unlikely that the specific Biblical errors and inconsistencies raised will affect all forms of theism equally.  Since there is no evidence which successfully addresses all forms of theism, there is therefore refuge in some hypothetical form(s) of theism.  Naturally, the Christian believes that if any form of theism is true then it has to be hers.  Not all forms of theism have been shown to be untrue and, therefore, her "best form of theism" must be true.

I know, it doesn’t make sense but the Greatest Living Christian Apologist regularly deploys equally weak and logically invalid arguments.

If anyone has any other theories regarding creeping theism, why the term "theism" has started to get so much traction in the Christian apologetics world, I’d be interested to hear them. 

Tuesday, 21 August 2012

Mark and the Masked Man

It is generally understood that the first gospel to have been written is that attributed to Mark – here, here, here, here and here (although these are all Christian websites, another Christian website, CARM, says the debate is far from over).

Now the fact that Matthew is normally assumed to be the first gospel is not a big issue, since there may well have just been a few years in it.  The issue I want to raise here is not the order so much as the nature of the story in Mark’s gospel.

According to Mark, the story starts with John the Baptist who was baptising people in the wilderness and proclaiming that someone greater than he was on the way.  That person was Jesus who was baptised by John.  Immediately after the baptism, the heavens opened and a voice from heaven revealed that Jesus was the Son of God, in whom God was well pleased.  Now note that 1) God did not say “This is an element of Me in whom I am well pleased” and 2) if Jesus=God then why is he telling himself who he is?

(It is a bit unclear as to who saw the Heavens open or who heard the voice from heaven, it is possible that apart from Jesus only John was in the vicinity, and the text implies that perhaps only Jesus saw the vision.)

This confusion as to identity never really goes away in Mark:

·         Mark 1:34 – Jesus suffers not the devils to speak, because they knew him

·         Mark 1:44 – having cured a leper, Jesus commands him to say nothing to any man (which the leper failed to do, rather he went out and began to publish it much)

·         Mark 7:36 – Jesus heals a deaf man with a speech impediment, then says that he should tell no man of the miracle (which the man fails to do)

·         Mark 8:26 – Jesus heals a blind man, then tells him to neither go into the town, not tell it to any in the town

·         Mark 8:29-30 – Jesus questions his disciples as to who they think he is, Peter says “Thou art the Christ” and Jesus tells them to tell no man of him

·         Mark 9:9 – after the transfiguration on the mountain, he charges those who witnessed it to speak to none of it (until after the Son of Man had risen from the dead)

·         Mark 11:33 – the chief priests, scribes and elders of the temple had recognised Jesus and asked of him by what authority he did what he did.  Instead of giving an answer he asks them what they think, to which they reply “We cannot tell”.  Jesus then says “Neither do I tell you by what authority I do these things”.

·         Mark 14:44 – Judas Iscariot tells those who come to arrest Jesus that he will identify Jesus to them by kissing him

Now, for a personage sent to Earth to spread the word of God, Jesus put a lot of effort into remaining unknown.  But that effort seems to be totally unsuccessful because people continually recognise him, including the chief priests, scribes and elders of the temple.

The only people who don’t recognise him are those sent to arrest him, necessitating the betrayal by Judas (which would not have been necessary since the high priests could have simply described him to their men).

At the ninth hour on the cross, in Mark 15:34, Jesus says “My God, my God, why hast thou forsaken me?”  This is totally incomprehensible if Jesus=God but makes a lot more sense if he is just a guy who either thinks he is the Son of God, or perhaps is favoured by God.

Interestingly, despite the constant hiding of his identity, one of the last things the Risen Christ says to his disciples is “He that believeth and is baptised shall be saved; but he that believeth not shall be damned.”  Why then did Jesus put so much effort into hiding evidence that would support a belief in what he was preaching?

This "masked man" approach would make sense if Mark's gospel was fiction, like others in which the secret identities of heroes are not generally known (Superman for example, whose cunning disguise is a suit and a pair of glasses).

It should be noted that in Mark 16:15-18 we have the perfect test for the strength of a Christian’s faith in the literal word of the gospel:

·       He said to them, “Go into all the world and preach the gospel to all creation.  Whoever believes and is baptized will be saved, but whoever does not believe will be condemned.  And these signs shall follow them that believe; In my name shall they cast out devils; they shall speak with new tongues; They shall take up serpents; and if they drink any deadly thing, it shall not hurt them; they shall lay hands on the sick, and they shall recover."

Noting that there is no expiry clause on this, it possibly provides a method for proving once and for all whether the Bible is inerrant.  For example, those people who attempt to legislate on the basis of Biblical truth or to change the national syllabus to eradicate science based subjects in favour of creationist accounts should first prove their faith that the Bible is inerrant by taking up a taipan and then drinking a glass of potassium cyanide and cola.  While I don't expect many to take up the challenge, it could cool their enthusiasm for forcing change on others.

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 clichéd 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:


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


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).

Special Relativity: From Galileo to Einstein

The intent of this article (adapted from an original paper I wrote about a decade ago) is to show how Special Relativity can be arrived at by examining Galilean boosts along with the inherent assumptions and then removing invalid assumptions. During the process it becomes clear that there is a potential source of confusion associated with the standard expression of time dilation. Note that this is the “methodical” version; there is also a simplified version which (I hope) will make more sense to those who are less gifted mathematically.

1. Introduction

Galilean Relativity relies on a number of assumptions. A number of these assumptions have been shown to be invalid in the real world and Einstein’s Special Relativity, which dispenses with those assumptions, is a far superior conceptual framework. Einstein arrived at Special Relativity via “mind experiments” and brilliant reasoning. I intend to demonstrate an alternate method of arriving at Special Relativity.

2. Galilean Boost

A Galilean boost is the equation which allows x’ to be calculated in Figure 1 where S is nominally stationary and M is nominally in motion with a velocity v. The equation relies on the following assumptions:

                                            when t = 0 and t’ = 0, S and M are coincident
                                            S and M observe the same event
                                            S and M measure time and space identically
                                            S and the observed event (X) share a preferred frame
                                            information is transmitted instantaneously

The boost is given by:
x’ = x – vt
t’ = t

3. Information is not transmitted instantaneously

We know that it actually takes time for information about the event X to reach each of the observers.

This results in a modification to an assumption, such that event X takes place when S and M are coincident, and t = 0 and t’ = 0.  The extant assumptions are, therefore:

                                            the observed event takes place when S and M are coincident, and t = 0 and t’ = 0
                                            S and M observe the same event
                                            S and M measure time and space identically
                                            S and the observed event (X) share a preferred frame

The modified Galilean boosts are:

x’ = x – vt
ct’ = x’ = x – vt = ct – v.x/c
t’ = t – v.x/c2

4. No preferred frame

We referred to S as stationary, but this merely means that S is stationary relative to a nominated “observer” (conceptually the reader). Relative to M, S has a velocity of -v. There is no valid reason to assume that either of the perspectives (or “frames”) is preferred.

Note that the unprimed frame pertains to S and the primed frame pertains to M.  That is to say:

·         in the unprimed frame, S is stationary and M has a velocity of vs according to S and

·         in the primed frame, M is stationary and S has a velocity of vm according to M.

The magnitude of vs and vm are such that:

vs = -vm = v

The extant assumptions are:

– the observed event takes place when S and M are coincident, and t = 0 and t’ = 0
– S and M observe the same event
– S and M measure time and space identically

The third assumption cannot stand without a preferred frame, since x’m > x’s and xm > xs where vs > 0.

5. S and M do not measure time and space identically

Figure 3 is modified slightly to reflect that vs = -vm = v.

The extant assumptions are:

                                            the observed event takes place when S and M are coincident, and t = 0 and t’ = 0
                                            S and M observe the same event.

According to each observer, the other utilises different, but consistent units of measurements – in the other’s own frame.  The unprimed frame pertains to S, so:

                                                                       xm = .xs        (1)
The primed frame pertains to M, so:

                                                                      x’s = .x’m        (2)

where is yet to be determined.  Taking first the perspective of S:
x’s = xs vt’s 
xs = x’s + vt’s

but since xm = .xs and t’s = x’s / c
                                          xm = .xs = .(x’s + vx’s / c) = . x’s (1 + v / c)         (3)

Then taking the perspective of M:

xm = x’m - (-vtm) = x’m + vtm
                                                                   x’m = xm - vtm         (4)

but since x’s = .x’m and tm = xm / c
                                          x’s = .x’m = .(xm - vxm / c) = . xm (1 - v / c)        (5)

Applying (5) to (3):

xm = . x’s (1 + v / c) = . [. xm (1 - v / c)].(1 + v / c)
xm =  2. xm (1 - v / c).(1 + v / c)
1 =  2. (1 - v2 / c2)
                                                               = 1 / (1 - v2 / c2)½          (6)

6. Lorentz boosts

Understanding a Galilean boost is relatively simple, since there is only one perspective. With a Lorentz boost, which we can derive from the information in the previous section, we have two perspectives to select from – that of S and that of M. Since we have already stated that S is stationary by virtue of being stationary relative to us, we will examine the situation from the perspective of S.
According to S, the distance between the event X and the point at which M observes that event is x’s = xs vt’s.  The Lorentz boost doesn’t seek to tell us this, it seeks to tell us what that distance is in terms of what M observes.  To arrive at the relevant equation, we make use of (2), (4) and (6):

                                                                      x’s = .x’m
                                            x’s = .(xm - vtm) = (xm - vtm) / (1 - v2 / c2)½          (7)

and since x’s = ct’s and xm = ctm so that tm  = xm / c:

                         x’s = ct’s = (xm - vtm) / (1 - v2 / c2)½ = (ctm - vxm / c) / (1 - v2 / c2)½
                                                    t’s = (tm - vxm / c2) / (1 - v2 / c2)½         (8)

It is trivial to show that a similar equation can be derived from the perspective of M, in terms of what S observes, so long as one keeps in mind that vs = -vm = v.

7. Finding x and t in the other frame

Rather confusingly, a different priming convention applies for denoting relativistic effects on space and time.  When considering relativistic effects, a primed value refers to a measurement within a frame in relative motion, as made by an observer from within that frame in relative motion – in terms of units which pertain to the nominally stationary frame.

Consider xs and xm:

                                            xs is a measurement made by observer S within the S frame which, according to observer M, is in motion
                                             what is xs in terms of xm?

This is equivalent to the question what is x’ in terms of x?  We apply (7) to (1):

                                                                       xm = .xs

                                                                        xs = xm / = xm . (1 - v2 / c2)½
                                                            x’ = x . (1 - v2 / c2)½         (9)

 and since xs = cts and xm = ctm:

                                                       xs =  cts = ctm . (1 - v2 / c2)½                                        

                                                              ts = tm . (1 - v2 / c2)½                                              

                                                             → t = t . (1 - v2 / c2)½          (10)

The same logic can be applied to x’s and x’m and the same result achieved.

8. On the result achieved by standard derivations

Standard derivations achieve a different result, viz:

                                                              L' = L . (1 - v2 / c2)½          (11)

                                                               Δt' = Δt / (1 - v2 / c2)½         (12)

This is because, in the standard derivations, Δx’ corresponds with x’ as expressed in the boost while Δt’ refers to something quite different, something that is conceptually inverse to t’ as expressed in the boost.  Conceptually, the term Δt’ refers a period measured in a frame in motion (relative to the observer).  In other words, Δt’ is equivalent to Δtm rather than Δts:

It must, however, be noted that a key assumption in this derivation is that if x = ct then x’ = ct’, and by extension that if Δx = Δct then Δx’ = Δct’, so that the use of the prime notation is consistent.  If the use of the prime notation used in the length contraction and time dilation equations is intended to be consistent, then there is a problem with this assumption as used explicitly to transition from (8) to (9).  Therefore it behoves me to present a derivation which does not rely on this assumption.

9. Finding t in the other frame by other means

Recall that according to each observer, the other utilises different, but consistent units of measurements – in the other’s own frame.  The unprimed frame pertains to S, so:

                                                                       tm = .ts

                                                                       = tm / ts        (13)

The primed frame pertains to M, so:

                                                                      t’s = .t’m

                                                                      = t’s / t’m         (14)

where is yet to be determined.  Multiplying (13) by (14):

                                                            . = (t’s / t’m).(tm / ts)


                                                             2 = (t’s / ts).(tm / t’m)        (15)

According to S, M moves towards the event at velocity v while information about the event moves towards both S and M at c, so:

                                                    ts / t’s = c / (c - v) = 1 / (1 - v / c)       (16)

According to M, s moves away from the event at velocity v while information about the event moves towards both S and M at c, so:

                                                   t’m / tm = c / (c + v) = 1 / (1 + v / c)          (17)

Applying (16) and (17) to (15):

                                    2 = (t’s / ts).(tm / t’m) = [1 / (1 - v / c)].[1 / (1 + v / c)]                    

                                                                2 = 1 / (1 - v2/ c2)

                                                            = 1 / (1 - v2/ c2)½ =         (18)

Applying (18) to (13):

                                                        ts = tm / = tm . (1 - v2/ c2)½                                        

Since ts is a measurement made by observer S within the S frame which, according to observer M, is in motion, then:

                                                              ts = tm . (1 - v2/ c2)½

                                                               t' = t . (1 - v2/ c2)½      (10)

As described in Section 8, this corresponds to the time dilation equation (where tm = Δt' and ts = Δt), so:

Δt = Δt' . (1 - v2/ c2)½
Δt' Δt / (1 - v2/ c2)½      (12)

10. Conclusion (as modified)

The equations for Special Relativity can be derived mathematically by means of systematically removing invalid assumptions associated with Galilean relativity.