Uh...where are you getting the friction from Jason? O.o I never assumed there was friction in the system, I assume no friction as in a vacuum. Are you assuming acceleration increases? Force is a constant in my example, so naturally as mass increases, acceleration decreases - this is not a result of friction, if we were factoring that in we would never reach the speed of light and that would be the end of discussion.

In that example we're only talking about mass increasing. As mass increases, acceleration decreases because the amount of force needed to achieve the same acceleration has to increase proportional to the change in mass.

Maybe it would be clearer if I posted pictures of equations?

You can see from the equation that if **F**orce is a constant, then **m**ass is inversely related to **a**cceleration. Meaning if mass increases, then acceleration must decrease. This is why Einstein argued mass increases as you accelerate, and specifically this is why Einstein argues your mass approaches infinity - because then acceleration approaches 0.

In light of this (no pun intended) I've written my own proof that mass approaches infinity as velocity approaches the speed of light, however it is based on the assumption that light is an upper limit on velocity of a particle (in other words - based on Einstein's Theory of Relativity).

Furthermore, sorry for some of you - I wasn't going to escalate it to proofs or math, but in many ways it is just easier to make myself understood clearly when using math, assuming the audience understands the math I'm doing. I've shared some resources to help you comprehend the math so hopefully you follow my work.

Real quick, just so I don't lose you here, I'm using a form that you might better recognize as something else, so here is the explanation:

The expression on the left is the same as the expression on the right but for the purpose of doing derivatives I keep everything in the form on the left. This is just so you can remind yourself at various steps that what the left expression means is just change in *x* (whatever *x* may be) over change in *t* or *time*.

Also throughout this, you will see a great many variables. *v* represents velocity, or speed with direction, *a* represents acceleration with direction, *t* represents time as I already said, *F* is force with direction, *m* is mass, *c* is a constant representing the speed of light (appx. 300,000 kilometers per second).

Sorry for those who have math anxiety or do not understand limits. Limits aren't difficult to wrap your mind around, it's just what a function approaches as another value approaches a constant (we'll be looking at what happens as velocity approaches the speed of light and as time approaches infinity - another way of saying "it will never happen"). Here's a resource for coming to terms with limits if you need it: http://www.calculus-help.com/tutorials/. Here's my proof:

Left Column

We take a known equation and demonstrate first that to reach the speed of light there is some acceleration and some time frame in which to accelerate that will bring an object/particle to the speed of light (line 3). What is worth noting here is that we cannot limit acceleration, but if we say that it happens when time reaches infinity then that's equivalent to saying "when pigs fly" i.e. "it will never happen". So we know from this step that we have to examine the limit as time approaches infinity (this is 1 of 2 assumptions that proves Einstein's theory with regards to mass approaching infinity).

Next we implicitly derive the equation for when velocity is the speed of light (lines 4, 5) and then define our "change in a over change in t" (lines 6-8). Next we examine the limit as velocity approaches the speed of light (line 9).

Middle column

Now we need to examine the acceleration as velocity approaches the speed of light and as time approaches infinity (line 1).

Substituting (line 2) we can reduce the limit of a to a simpler form (line 3). We conclude that as velocity approaches the speed of light and as time approaches infinity that acceleration approaches (but technically never reaches) 0 (line 5).

Right column

Okay now we know that if you approach the speed of light that acceleration slows to 0. Now we can limit the equation for Force (line 1). We isolate mass (line 2) and then apply the limit as acceleration approaches 0 (line 3) which we proved only a moment ago. Applying the limit, we can see that mass approaches positive infinity (line 5).

So restating some of the above, here are the things to note. Einstein assumes the following:

there exists a maximal speed which particles cannot achieve, and

the upper limit of speed is the speed of light.

The former means that as velocity approaches that speed, time approaches infinity (i.e. it will never happen). The latter means that as velocity approaches the speed of light, then there must be something that causes the particle's acceleration to approach 0. From the work above, we see that mass approaches infinity which explains how acceleration approaches 0.

On a side note, when we say "assumes" in Math (or Physics), we don't mean that it makes a you-know-what out of *u* and *me*, assumptions are acceptable if the basis is valid and can be proven (which is why it remained a Theory, because we cannot prove the speed of light is an upper limit, otherwise it would be a Law and we wouldn't be having this discussion).

So Einstein's Theory is plausible if we can either prove light is an upper limit or if we can prove that particles actually gain mass as they increase in velocity. I left out the Lagrangian, but if you take the limit as velocity approaches the speed of light, you draw the same conclusions. I just figured the above would be more familiar since it's based on the simpler equations you are likely to see in, say, a High School Physics class.

The work at CERN however would disprove both the notion that light is an upper limit and then (like dominos) the notion that mass approaches infinity as you approach the speed of light. Which brings us to where we are in the discussion today.

~Robert