Thursday, 25 October 2012

Do Atheists Believe in Evil?

I recently wrote a little contemplative piece on why theists might think that atheists are evil and got some interesting feedback, some involving me being directed to a place I don’t believe exists and which I suspect the people wishing me bon voyage don’t believe exists.  Oh well.

There was also an intriguing response to an assertion that atheists don’t believe in evil.  Apparently I am wrong in that.

I certainly don’t believe in evil, but perhaps I am being too literal or philosophically rigorous about the term.  Or I am just misusing it.

I seriously don’t think that an agent can be inherently evil, evil (as I understand it) is associated with acts, specifically deliberate acts which are contrary to what is known and understood to be “good” or “right” where there is no overriding justification (such as a higher good).  Conceptually, you could have a person whose acts are preponderantly evil and, for convenience, you could label that person as “evil”, but I don’t think that person is inherently evil since their “evil” derives from their actions.

The question that then arises is “Are there people who commit evil acts under this definition?”

I’d say no.  Not even a violent psychopath is literally evil. For a psychopath, the suffering of others doesn’t register particularly highly or he simply doesn’t care.  Note that most psychopaths aren’t actually violent, despite their poor rap in popular culture.  It’s quite likely that more than a few princes of industry are psychopaths, along with politicians in general.

Remember to commit an evil act (at least in my conception of the term), you must know and understand that there is a good or right way to act, within a broad context, and must act deliberately in a way that is contrary it.  Perhaps examples might help explain.

When it comes to examples of evil, most people like to use events like the Holocaust or perhaps Jonestown.  Not me, so let’s look at “Who Shot Mr Burns?” (Spoiler warning!)

There are two acts that are worth considering as potentially evil: the stealing of candy from a baby and the shooting of a person.

When Mr Burns tries to steal a lollypop from Maggie, it’s not an evil act.  It’s a purely selfish act, he considers that his well-being is more important than the child’s.  Many acts which we might be tempted to brand as evil are, in fact, just very selfish.  The person involved counts the good involved with the act as overweighing the bad, for example if I were intensely selfish then a moment’s peace for me would be worth a lifetime of suffering for you.  This certainly isn’t nice, but it’s not evil.

Burns often seems clueless about human motivations, which would also indicate that his act was not evil.  Particularly ignorant, perhaps, but not evil due to a lack of intent.

When Maggie shoots Mr Burns, it’s not an evil act either.  As an infant, one could say that Maggie could not properly comprehend the consequences of her actions.  That said, there is a hint that she might have deliberately shot Burns.  If she did so, then again, it would have been a selfish act if Maggie calculated that the benefits of having a lollypop outweighed the costs of shooting Burns.

We could take a Sam Harris type of position, arriving at a humanist and atheist conception of evil: “An evil act is one that intentionally increases overall balance of suffering”.

But does anyone actually do that?  Really?

Perhaps there is some twisted person out there who gets a kick out of making people suffer.  A few politicians come to mind …

The thing is that such a person would cause suffering in proportion to the amount of satisfaction she got out of it (I’m thinking of Ann Coulter here).  It returns to selfishness again: my satisfaction is more important than yours and is served by your suffering.

So, is evil really just selfishness?  I don’t think so, I think that when people invoke the term “evil” they mean something more.  And if it does mean “selfishness”, then we’ve already got a perfectly serviceable word for that phenomenon.

Friday, 19 October 2012

The Roadless Travel

I wrote recently about William Lane Craig’s “Taxi-Cab Fallacy” and what I have labelled the “Optimus Prime Fallacy” (the continual transformation between multiple forms of argumentation, especially if it is never made clear what form of argumentation is being used until challenged).

While writing the latter, I had thought about how I could illustrate the problem but I wasn’t able to come up with exactly what I wanted without incurring the risk of devoting inordinate amounts of time to the project.  I had the idea of a relatively simple network of roads (the path that a figurative taxi-cab would take, illustrating the methodology of science) interspersed with rivers, swamps and lava flows (which had to be traversed in quite different vehicles to the standard albeit figurative taxi-cab, illustrating the tendency of an apologist to regularly abandon the taxi-cab in order to get to a predetermined destination).

Those with an active imagination can conjure up a scene with Craig and a materialist on a road in the wilderness somewhere.  The materialist wants to continue the journey using the taxi-cab, but Craig wants to go across a treacherous stretch of swamp to what he thinks is a road on the other side.  In his head, Craig wishes into existence a philosophical bridge (“This is not a physical argument, this is a philosophical argument”) and bravely sets out across the swamp.  As Craig sinks into the mire, the materialist shakes her head in disbelief, hops into the taxi-cab and heads off towards a proper understanding of the universe.

I wasn’t able to put together a drawing representing that scenario.  I did, however, find a program that allowed me to create a drawing from “road shapes”.  Within a short time, I built up an image meant to represent the journey to truth and understanding by humans over the millennia.

I wanted to illustrate the idea that the god hypothesis, as a simplistic explanation, lies between the understanding of Neolithic humans and what we have developed today.  Initially I thought that this hypothesis should be represented by a blank area in the middle of the map, one that had to be navigated around.

I was going to find an icon to represent the god hypothesis, illustrating the flame to which the theist’s moth is inexorably drawn.  Then I thought, no, the arguments used to defend the god hypothesis are circular in one sense and twisted in another sense.  I should have some road in the blank area, not connected to the rest of the roads (thus requiring a leap of faith to get there from the purely evidential path).

So I inserted something resembling a closed loop race-track circuit with just enough curves to represent the twists and turns of apologetical argument.  In retrospect, I realise that I should have included a couple of switchbacks to illustrate the way these arguments turn back on themselves.

Being a bit of a perfectionist, I still thought it was not quite right.  Craig and his ilk don’t use the same form of argument throughout the entire circuit, they jump from pseudoscience (the eye could not have evolved without design) to pseudophilosophy and pseudologic (Kant argued that reason is faulty which means that reason is faulty, so you can’t use reason) to pseudometaphysics (a transcendent god is not contingent but rather metaphysically necessary) to pseudoethics (you know that rape is bad because of the inherent goodness of god) and so on.  So, I shuffled up the race-track to represent the fact that the arguments for the god hypothesis aren’t coherent as a race-track would imply.

Interestingly, the result is a good illustration of what probably attracts some people to the god hypothesis.  Look at the drawing below.  What do you see?

If I add two roundabouts, the illusion is even more striking:

For those in any doubt, there is not a face in that illustration.  When I put in two circles into the last diagram, these resulted in a stronger activation of the “face detection” zones in your brain.  But even with these circles added, there is no face.  There are only fragmented segments of a race-track with two circles.

We humans are very good at detecting patterns, even if they are not there.

I reckon that it is this ability to detect patterns that leads the religious astray.  We all start off the same sort of way, sifting through our experiences in the search for understanding, but when the religiously inclined see fragments that, if looked at the right way, combine into the image of a god, they leave the path of reason in favour of an argument along the lines of “it must be right because it must be right”.  It looks like it might be a face, so it must be a face.

If you look at the second last diagram and are unable to avoid thinking that the apparent face must have been intentional (that it is a face), despite the fact that I, the person who created it, tell you that I did not intend there to be a face, then you may possibly be the sort of person who can fall prey to religious thinking.

If you think that the hand of god guided me in creating that face, in order to send secret messages to believers out there, then you are definitely the sort of person who can fall prey to religious thinking!

Tuesday, 16 October 2012

Atheists are Evil

I wrote the first draft of this article the day before the news that D’Souza was having an affair broke.  It’s a pity, because D’Souza says so many stupid things and he was shaping up as good material.  As a leading apologist though, his days may now be numbered.  Now back to your regular service …


There are two fundamental reasons why atheists are evil.  First, because of what they think.  Second, because of what they are.

This might sound like a controversial claim, but it’s not an uncommon claim, although the vast majority of people making the claim don’t state it quite so baldly.


Let’s look at the intellectual evil of atheists first, and then we can look at their ontological evil.

For the committed theist, the existence of god is self-evident.  You can see evidence of this in some of their statements and their questions.  There’s an example in the Q&A portion of a debate between Dinesh D’Souza and Peter Singer:

It seems like that when all of the evidence points logically to the conclusion of a God that you tend to come up with a different hypothesis.  So what’s confusing to me, as watching you being the atheist viewpoint, is it seems like, rather than look at the evidence and just see where it leads, you start with the position that God doesn’t exist and see if the evidence can get you there, and when it doesn’t you come up with a different hypothesis and my question is simply is that true what I feel like, you know, or am I wrong?

Peter Singer quite happily told the questioner that he was wrong and that he (Peter Singer) didn’t see any of that “evidence” that apparently points logically to a god but rather a wealth of evidence against the Christian conception of god, focussing particularly on suffering.

Then Dinesh D’Souza had the opportunity for a brief response (I’ve taken his stuttering out):

Look, ultimately here … let’s take one more look at this issue of suffering because in a sense it’s demounted as Job did as a complaint against God and the argument would go something like this: “God, why don’t I have both my hands?”

Now the question you have to ask is, if it is the case that there is a God and if it is the case that He created us, all living creatures have been given something to which they are not entitled in any way, namely life itself.

Even the person who has little, or is suffering, is often clinging to life, which is to say life is valuable to us.  We’re still in the plus column because we want to keep living, and so what I am getting at is do we really have a legitimate complaint against God by saying, in a sense, “I’ve only got one hand, that guy has two. Why did you make me a one-hander?  Why did you give me cancer so I only have 47 years of life when I could have had 70?”

It seems to me that these complaints ring hollow when you consider that our entire life and everything we have is in fact a gift from God so, on the premise that God exists, which was his (Singer’s) premise, it looks to me that He has not done anything to any of us that has put us in the negative column.  We’re still in a sense on the positive side of the ledger because we still have something to which we have no claim and have no right against Him, namely the gift of life itself.

Note carefully what D’Souza is implying here.  Singer has argued throughout the debate that the Christian conception of god is inconsistent with the evidence and presents as his primary case the argument that suffering of non-human animals due to non-human causes is inconsistent with a benevolent god.  He follows a chain something like this:

If your god is omni-omni (my term, not Singer’s meaning omnipotent, omniscient, omnipresent, omnibenevolent), why is there suffering?

-       Response, free will is so brilliant that all the suffering in this world is worth the cost, it’s just not possible to implement free will without suffering as a side-effect and, in any case, the longer term goal of salvation makes the suffering inconsequential.

Ok assuming that your arguments are valid; why, if your god is omni-omni, do animals suffer when they don’t have salvation as a pay-off?

-       Singer’s conclusion is that, if he exists, then god is either incompetent or evil, which is not part of the Christian conception of god, therefore god as the Christians conceive him does not exist (another option is that god simply doesn’t care, but that brings god into line with Darwinian evolution – the Christian god apparently does care)

Singer is arguing that the inconsistency in the Christian conception on god is evidence that that particular god does not exist.  Singer is not having a hissy-fit, pouting his little lips and declaring that he’s angry at god because there’s some suffering in the world.  D’Souza however seems to think that if (in Singer’s estimation) God were a bit better at preventing suffering, then Singer would begrudgingly acknowledge Him but until then, Singer’s going to be an atheist just to spite God.

Some theists seem to be unable to get their heads around the idea that atheists simply don’t believe in a god.

My theory is that this is related to the phenomenon by which some theists are locked into their faith, that is by means of a form of wishful thinking – they simply would not wish to live in a universe in which there is no god looking after them.  There’s a beautiful Swedish song, Du Måste Finnas, or “You Must Exist” in which the singer pretty much begs god to exist (this is my translation, not that on the video):

You drove me out, God,
I was ripped from my homeland,
I am refugee and a stranger here
And I find myself in the wilderness*

But you took my child
And now you take me from my husband
I can no longer see a meaning to it all
What is it that you want?
What should I believe …?

My thoughts are dizzying
Before me there is an abyss
I am in turmoil and want to say “no”
[But] the question has been raised.
And now my soul shakes in anticipation of the answer
That you don’t exist
Even though I believed in you.

Who will help me survive this life?
Who will give me the strength that I need?
Who will comfort me, so insignificant in this world?
If you don’t exist,
Yes, then what should I do?

No, you must exist,
You must,
I live my life through you. 
Without you I am no more than a ripple in a stormy sea. 
You must exist,
You must. 

How can you forsake me? 
I would be nothing,
I would be nothing if you didn’t exist.

Never before have I had it in word or thought,
The little word that scares and plagues me so,
The word is “if”, if I prayed all my prayers in vain
If you don’t exist,
Then what should I do?

Who would sense my repentence and later forgive?
Peace in my soul, yes, who would grant me that?
Who would receive me after death?
If you didn’t exist
Then what should I do?

No, you must exist,
You must,
I live my life through you. 
Without you I am no more than a ripple in a stormy sea. 
You must exist,
You must. 

How can you forsake me? 
I would be nothing,
I would be nothing if you didn’t exist.

* Note: there’s a play on words here, “ödet” which I’ve translated into “the wilderness” while strictly meaning “desolation” also means “fate”, “the wilderness” works on both a literal and a figurative level, since she’s in pioneer era Minnesota, a long way from home, and she’s also metaphysically adrift as the following verses indicate

Note that the character is rather conflicted.  In part, she is angry at the god she believed in (for forcing her from Sweden, killing her child, putting her in the wilderness and so on).  But she expresses something which I think a lot of recovering theists probably feel at some time or another – a notion that a universe without god, even a god that one is angry with, is undesirable.  Julia Sweeny vividly describes a form of vertigo experienced while she was losing her religion in “Letting Go of God” (or the TED version).

This believing because not believing would be so terrible is a mirror image of what I think D’Souza and people like him believe is going on with atheists.  D’Souza assumes that there are people who, because of ignorant pride or inappropriate priorities, decide that an existent god is not doing a good enough job and who, as a consequence, decide quite irrationally that the universe would be better without god.  These people, according to D’Souza, therefore adopt the intellectual stance of believing in the non-existence of something that obviously exists – thus becoming atheists.

If this were the case, it would be a type of evil.  If god, being the omnipotent and omniscient creator of everything and the source of all good, were to exist (and that existence were self-evident) but insignificant vermin such as humans took it upon themselves to judge god and turn away from god, that would indeed be an evil act on their part.  No questions about that.

(This might be part of the reason that apostates are looked upon so unkindly, although a better explanation is that the policy of treating apostates poorly – up to and including torturing them to death – makes apostasy a somewhat less attractive option for others, which in turn makes it easy to lock people into your in-group.)

However, as far as I know, this isn’t how the vast majority of atheists see things.  Some atheists are particularly disenchanted with the religious, and they do mock theists by saying that even if god did exist (which, they make clear, he doesn’t) then he’d not be worth worshipping because of all the suffering, or because of how much of a megalomaniac butcher he was in the Old Testament, or because gay people are people too, or whatever.

But … a person who is seriously pissed-off with God for His mismanagement is a pissed-off theist, not a pissed-off atheist.

So, that’s the intellectually evil atheist, or rather the straw-man of the intellectually evil atheist.


But atheists are not just evil because of what they think, they also evil because of what they are.  The nature of an atheist is inherently evil.  Fortunately, this inherent evil is not unique to atheists.

I wrote before about a miracle testimonial event that I attended at the urging of a Christian friend.  I was sorely disappointed that I didn’t hear about people being miraculously cured of amputations, or third degree burns, or even cancer.

 (I guess remissions don’t always last – it’s somewhat dark humour, but I have in my head the voice of my friend telling me that I should have come last year, because there was this great speaker, a young woman, who had talked about how she had been miraculously cured of terminal cancer, and that I would have enjoyed that.  Oh, yes, I ask, what happened to her?  The cancer came back and she died, my friend tells me – oblivious to the fact that he is telling me that the miracle that I would have liked to listen to was not, in fact, a miracle at all.)

All the “miracles” that evening were mundane stories about how people in the congregation had been nice to each other.  Now I am all for that, I like people being nice to each other and I really like it when people are nice to me, but that’s far from miraculous … unless you have a rather twisted view of humanity.

In the debate between D’Souza and Singer, D’Souza tries to argue that the Ten Commandments are not made up.  I’ve heard him make the same argument elsewhere, but in this particular instance, he makes a comment to the effect that if he were to be a member of the committee responsible for making up the Ten Commandments, he’d toss three of them away for a start (I don’t know how many he’d finally end up ditching) – because D’Souza doesn’t like to be told not do things.  I don’t know which version of the Bible he’s talking about, but it sounds to me that he’s talking about tossing three of the commandments that appear after honouring your father and mother (in the context of the debate, I can’t imagine that he meant that he’d toss out the first three “I am god” commandments or that he has a burning desire to dishonour his parents).  That leaves him with:

Thou shalt not kill

Thou shalt not commit adultery

Thou shalt not steal

Thou shalt not lie

Thou shalt not covet houses

Thou shalt not covet wives, manservants, maidservants, oxen, asses, etc

To be kind, we could assume that D’Souza is very much into coveting.  He enjoys a good covet.  So, that’s two of his three and he’s got one more commandment to discard.  If he’s got any sense at all, he’d go for “Thou shalt not lie”.  Now that lying is ok, along with the coveting, he can happily lie about all that killing, adultery and stealing he does.

(News reports out today, the day of my editing of what I wrote yesterday, indicate that D’Souza could have wanted to ditch “Thou shalt not commit adultery” and “Thou shalt not lie”, I guess that leaves him scope to take out “Thou shalt not covet wives, manservants, maidservants, oxen, asses, etc” – I’m not currently aware of the marital status of D’Souza’s squeeze.)

The point is that D’Souza was basically implying that he would keep the commandments, as silly as half of them are and as bleeding obvious as the other half are, because he’s Christian and he believes that god wants him to obey them.  Without them, he would be like the caricature of an atheist that he depicts: uncaring, totally amoral and intensely covetous.  (Edit – actually, even with them he’s a bit like that.)

It seems to me that people like D’Souza should be theists and even atheists should want people like D’Souza to be theists, because they are psychopaths who are only kept in line (edit – to a certain extent) by a comforting fairy tale.

This viewpoint, that without theism, humans are like wild beasts, is a recurring theme.  D’Souza even went as far to claim that, before the advent of Christianity, there was no compassion.  So, what he’s specifically claiming is that if you are not a Christian, you are highly likely to be an uncaring, dishonest, vicious, baby-discarding, cannibalistic, light-fingered, indiscriminately copulating and covetous beast.

I bet they don’t put that in the travel brochures.


Please note, I am not really saying that atheists are evil.  Other than Ayn Rand, of course.


D’Souza really made a mess of the last section of this – it seems that even if you are a Christian you can still be a bit of a beast.  It just means that you are a hypocritical beast.

Sneaky sneaky

I put a new gadget on the blog and, like magic, advertising appeared.   Not happy.

I just wish I had noticed earlier, but it's gone now.

Monday, 15 October 2012

Something of gravity

I don't usually write short blog articles, or fire from the hip, but here goes.

I recently experienced a modest amount of interest in one of my physics related posts.  I'm quite happy with that, but one comment made has been bothering me a little.  I was driving home in my car today and it struck me what the issue was.

Here someone called supracedent complained that my article A Little Expansion on the Lightness of Fine Tuning was bollocks.  That's ok, it may well be bollocks.  Science is mostly bollocks, waiting for a better version of bollocks to be discovered.  The thing that was niggling at me was that when he clarified his complaint, he made this comment:

Spacetime curves according to gravity and acceleration. We know this because of the position of stars during an eclipse, modern experiments like gravity probe b, even GPS verifies it.

Hang on a minute.  Those are all General Relativity issues (because they are all gravity related, and hence acceleration related, effects).  In my equations, in all the articles about relativity type stuff, I've only been dealing with Special Relativity effects (purely uniform rectilinear relative motion, nothing fancy).  That's really unfair, supracedent!

Oh well.  I guess logical errors aren't the exclusive domain of apologists.

Friday, 12 October 2012

On Time

This might get a tiny bit complicated.  I’ve tried to keep the maths simple but towards the end, where I specifically address potential concerns that might be raised by physics students, it does get a bit tricky.

Again, a warning for people studying relativity in a formal setting, 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.


We are quite used to the idea of movement with respect to time, although we normally express it in terms of time passing by us, rather than thinking in terms of us passing through time (minutes pass, the years pass by and so on).  When we raise the idea of movement, there is a linked idea of the rate of that movement.  In other words we could ponder the question “at what speed do we move with respect to time?”

There is actually an answer to this question.  In a recent post I talked about how the expansion of the universe can be considered to be time as opposed to the universe expanding with time or over time.  Another way of expressing this is to say that if you were to be at rest (in spatial terms), then you’d still have a speed “through” time – the rate at which you travel from one temporal location to another.  That speed would be c.

But what happens when you are not at rest?  (Note that for the sake of simplicity, I am only talking about uniform rectilinear motion here.)

Here’s an illustration with some simple mathematics (note that we have kept the “good twin” and “evil twin” notation, where the “good twin” is notionally at rest, see the Lightness of Fine-Tuning articles in which this concept was introduced - Part 1 and Part 2):

What this illustration is showing is the idea that our speed through space-time is conserved (or invariant).  We can change the direction of our travel and thereby experience some transition through space at the expense of transition through time.

This result should not be a surprise.  As Einstein said; “Energy cannot be created or destroyed, it can only be changed from one form to another.”  This is a rephrasing of and expansion on the first law of thermodynamics, per Rudolf Clausius in 1850; “In all cases in which work is produced by the agency of heat, a quantity of heat is consumed which is proportional to the work done; and conversely, by the expenditure of an equal quantity of work an equal quantity of heat is produced.”

Both statements express the conservation of energy principle and, from the most celebrated of all equations formulated in the 20th Century, we know that E = mc2.  This is, strictly speaking, saying that the energy of a body at rest is the mass times the square of the speed of light.

What I am suggesting is that a body can be said to have an invariant space-time type of "kinetic" energy, related to an invariant space-time “speed” of that body.  In other words:

E = m.vspace-time2

Hopefully, people reading this now understand that we have an invariant "space-time speed" (at least to the extent that the maths is accessible to them).

Please note that this is not a standard way of dealing with these concepts.  I don’t think it’s wrong, but if you might not want to spring this on your physics teacher without prior warning, certainly not if he or she is teaching you a different way of looking at relativity and the consequences thereof.


Neither m.vspace-time2 nor m.vspace2 is intended to represent classic Newtonian kinetic energy which is given by Ek ≈ ½mv2.


For completeness (and for the sake of pesky physics students who will probably be utterly convinced that the above must be wrong), I’ll now show how we can arrive at this kinetic energy equation.

If you don’t like maths, or physics, and for some reason you have nevertheless made it thus far … turn back now!

The equation E = mc2 tells us that mass and energy are equivalent and for this reason we should be careful about talking about the mass within a system or the energy within a system, since it is “mass-energy” that is conserved, rather than mass or energy separately.

Extending this concept slightly further (and again going slightly beyond the standard conceptionalisation), what we know as mass can be considered as equivalent to the concentration of mass-energy in a body – the more energy in a body of a fixed size, the more mass it has and, perhaps less intuitively, a body with the same amount of energy in a size reduced by the effects of relativity will also have more mass.  This will be difficult for a lot of people to stomach.  Surely if you have more mass in a smaller space that just means it’s denser mass, but the same amount of mass?  Well, yes and no.  If there is more energy in a given volume at rest, then that equates to more mass, and the same energy in a smaller volume at rest would be the same mass, but denser. However, if you have length contraction acting on a volume, then as the volume decreases due to relativity the relativistic mass increases.

So, (where V is volume)
Mrel 1 / Vrel
(As an aside, another way to look at this is to say that as the concentration of mass-energy increases, the more difficult it becomes to accelerate that mass-energy and change its inertia.  This is the fundamental understanding of mass.)

According to an observer at rest, the volume of a concentration of mass in motion is foreshortened in the direction of the motion (due to length contraction).  Since the volume has gone down (according to the observer at rest), the relativistic mass goes up in inverse proportion.  I've returned to the good twin/evil twin notation, sub-G and sub-E.

VG = LG . A

VE = LE . A

LE = LG . (1 – v2/c2)


VE = VG . (1 – v2/c2)

Therefore (since mG is inversely proportional to  VG)
mE = mG / (1 – v2/c2) = mG . (1 – v2/c2)

This is the standard equation for relativistic mass, where mG is the rest mass (the good twin is notionally at rest) and mE is the mass in motion.

(For those keeping track, I should mention that we are now completely back inside the scope of standard method, it’s only that I have slightly different notation.  While making an aside, may I add that if you follow the link above, note the objection that Einstein had to the concept of “relativistic mass”.)

If the total energy of the body is considered to be the sum of the kinetic energy of that body and the inherent energy of that body (that is the rest energy), then:

Etotal = Ek + Eo


EE = Ek + EG


Ek = EE – EG

Ek = mE.c2 – mG.c2

Ek = mG / (1 – v2/c2).c2 – mG.c2

However, where v is sufficiently small (feel free to test out this approximation using a spreadsheet, or if you are old school, a pen and paper)

1 / (1 – v2/c2) ≈ (1 + ½v2/c2)


Ek ≈ mG.(1 + ½v2/c2).c2 – mG.c2

Ek ≈ mG.c2 + ½mG.v2 – mG.c2

Ek ½mG.v2