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Interesting discussion. However, there is a flaw in your example about the car, Robert. You assume there is friction to cause the car to become harder to move as it gets heavier. In empty vacuum there would be no friction. However, space is far from empty. Perhaps the reason something can't reach the speed of light is because as it increases in speed, it begins to collide with more and more "stuff". 
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 Force is a constant, then mass is inversely related to acceleration. 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
You assume there is friction to cause the car to become harder to move as it gets heavier.
Had a second thought and wanted to readdress this: as anything gets heavier it becomes harder to move - no friction necessary. Does this make sense now? Think about things that gain weight (i.e. a baby turning into an adult, or a dump truck with an open and initially empty back in the middle of a rainstorm) - as they get heavier, you need more force to achieve the same acceleration. If you always apply the same force, then as they get heavier your acceleration you put on the object is decreasing.
Hope this demystifies it a bit!
~Robert
Wow, very technical! While I could understand all of that if I wanted to read it (I minored in Mathematics when I got my BS), I skipped it because it is largely irrelevant to the point I was making. And I also already have a headache.
You said:
As your car gets bigger, you start to slow down... By this point you should have slowed to a crawl (if you're even moving at all anymore).
In a frictionless system, this is wholly incorrect. It wouldn't matter how much bigger my car could get, I would NOT slow down. Once I've applied the force to get the car (or any object) to a certain speed, I would maintain that speed until some opposite force (friction or otherwise) caused me to slow down, even if the mass of the object increased 10 million-fold. Yes, it would take MORE force if I wanted to accelerate to a faster speed. But I wouldn't slow down until some force acted upon me. In fact, it would become MORE difficult to slow the car down, because as it gained mass it would need a larger force to decelerate it.
I was just stating, slightly tongue in cheek, that the car analogy is flawed compared to what we are ultimately talking about (accelerating a particle to the speed of light or beyond).
-Jason
There's no better way I suppose to point out that there isn't any flaw with the car analogy than to point out a different analogy that is more visible and easily performed at home to demonstrate this principal.
Okay, force is a constant. If you have a spinny chair (yes, I just did a 180 from wholly technical) and you spun around in the chair, you will notice that if your legs start below you and raise up, you will slow down. Although Aerodynamics could play a role, you'll then notice that by lowering your legs you will speed back up. Overall there is friction in the system at home but you can clearly see that by increasing your "centripetal mass" (it's not politically correct to call it this, but in effect this is what you are doing) or decreasing your "centripetal mass" you will decelerate or accelerate respectively.
To argue that the car example is wrong is to suggest that a bullet train in Japan with 200 cars attached to it and a bullet train with 1 car attached to it accelerate at the exact same rate when you apply the same amount of electricity. Or that I can lift boulders as easily as I can lift pebbles.
I think what you're trying to argue Jason is that the car won't lose velocity? This is 100% true and entirely supported by my example - in my example, once the car (with its godly engine) reaches 100 mph it will never ever see 99 mph again. But as mass increases the argument is that it may get from 100 mph to 200 mph but that it will take longer to reach 300, and longer yet to reach 400. But it WILL reach 100 mph, 200 mph, 300 mph, 400 mph, etc., etc., until we get to the speed of light - it will get pretty darned close but it will never reach that point. Acceleration will decrease as mass increases - it has to, or else we break Newtonian Mechanics and then we're really arguing the last few hundred years of physics are incorrect.
To further illustrate my point, if you did a drag test with a car and then tried the same drag test with a car but added an extra 500 lbs (again we're not worried about friction here), what my "car example" demonstrates is the notion that the two test times will differ, the former being the "faster" time and the latter being the "slower" time. If you can see that, then you can see the point I was trying to convey with the car example. The only difference is in real life if you strapped 100 elephants to your drag car, you probably won't have enough acceleration to overcome friction - but we're not worried about friction.
Perhaps the reason something can't reach the speed of light is because as it increases in speed, it begins to collide with more and more "stuff".
Incidentally, you suggest above that there is friction to concern ourselves with in space. I wouldn't really call it "air friction" but maybe "space friction"?
What we need to be mindful of (this is me being politically correct about my statement above with velocity not being lost) is that as we increase mass we maintain the same momentum which means the car does lose velocity - however I consider this to be irrelevant since we already state that the car/object/etc. is accelerating which makes up for any loss in velocity from an increase in mass. Plus as we examine our limits, we know that the change in mass cannot completely oppose the speed gains from acceleration.
Momentum is given by I in the equation I=mv and we know that changes in a system without introducing force (i.e. changes to mass) are met with an opposite change in velocity. We can further state that mv1=mv2 should mass change and also that mv1=mv2+mv3 should a mass split into two parts or should two masses come together to make a whole.
To reiterate since we've carried on this far (for others now catching up), I'm playing devil's advocate explaining how Einstein theorized the world works when my feelings are the complete polar opposite on his theories.
And I'll add my humorous note of the day:
"Every time a particle gains mass by accelerating, Newton kills a kitten." 
~Robert
Jason, it occurred to me as an afterthought on the way in to school today that I should try explaining it from a different angle - one that better illustrates what I'm trying to convey with the car example.
I've been saying Force is a constant and if we're talking about what happens in a vacuum then it couldn't be more true. Here is exactly what my car example is based on:
Let's say Force is 100 kilogram-meters-per-second-squared. We can drop units for now since I think we understand that if we always use mass in kilograms and acceleration in meters-per-second-squared that we don't need to keep up with the convention of using units.
Okay so Force = 100. We also know F=ma. So here's the following chart that expresses what I based my example on:
You'll notice that this is a simplified version of the limit performed in the right column in my proof above. If you keep increasing mass, acceleration will keep decreasing until eventually it starts getting so close to zero you have to go out several decimal places to check for sure if it's not zero.
Hopefully this clears it up? This basic premise is the source of how I came up with my "car example" and you'll notice there is no friction involved.
~Robert
I understand your point about acceleration decreasing, however you are (at times, or at least how I am reading it) using speed synonymously with acceleration, which is incorrect. Yes, when I say speed I mean velocity. Speed is the measure of velocity without a direction (ie, velocity is the speed of an object in a given direction).
Thus, when you originally stated that as you add mass you lose speed, that was an incorrect statement, unless there is some other force (eg friction) that is counteracting against the mass in the opposite direction. What you will lose is acceleration, assuming the applied force is a constant. But you will never lose speed/velocity until some opposite force starts to work against the mass.
Also, I like the idea of space friction. 
Thus, when you originally stated that as you add mass you lose speed, that was an incorrect statement, unless there is some other force (eg friction) that is counteracting against the mass in the opposite direction. What you will lose is acceleration, assuming the applied force is a constant. But you will never lose speed/velocity until some opposite force starts to work against the mass.
Ah I see where the confusion is. In the car example, the car is NOT accelerating, it is moving at a constant rate. As mass is added, the car's speed reduces proportional to the mass gain. This is an over-simplified version of what we are talking about.
How it is different:
There is no constant rate of acceleration/deceleration (this helps simplify it)
There is no constant force acting on the car, other than the incurred deceleration from the change in mass (again, this helps simplify it)
The car's velocity approaches zero instead of the speed of light (this is where you will notice the difference and where I think the source of the confusion is)
I made this example not wholly parallel to the notion of a particle approaching the speed of light for a few reasons:
There isn't really a good, tangible example of particles approaching the speed of light that everyone can relate to...otherwise, of course, these theories would have been invented a while before Einstein***
It nicely illustrates the notion that Force is constant (your engine's capabilities) while relating mass approaching infinity to velocity approaching a fixed number (in this case, zero)
It's discrete math so those without an understanding of Calculus can grasp the notion
***Of course, nobody's car grows in size, but people can imagine how a vehicle accelerates while nearly empty and light versus full and heavy.
In particular, the basis that needed to be demonstrated was the effects on velocity as mass approaches infinity while force is a constant. The car example is meant to help connect the dots from one to the other. Considering you understood what I meant by everything BUT the car example, clearly you did not need the car example to grasp the basics behind objects approaching the speed of light (besides which you mentioned being able to get through the calc ).
Glad we sorted THAT one out!
Thus, when you originally stated that as you add mass you lose speed, that was an incorrect statement, unless there is some other force (eg friction) that is counteracting against the mass in the opposite direction.
Actually to say "when you add mass you lose speed" is an entirely true statement as per the conservation of momentum. No acceleration involved, no energy added or deducted from the system; +mass = -velocity.
The conservation of momentum is mainly what applies to the car example, but again, it's not a perfect parallel to the notion of particles approaching the speed of light BUT there isn't really anything that parallels particles approaching the speed of light. As explained above, I used the conservation of momentum to illustrate the effects an increase in mass play on velocity - it was a convenient and oversimplified example.
Also, I like the idea of space friction.
Right, well technically it exists - air-friction is exactly the same thing, but generally speaking you encounter less "stuff" in space than you do in Earth's atmosphere (unless you flew into a celestial body i.e. a planet or a star - then it's not flying, it's falling with style...and painful, disgusting death). Effectively there are likely some stray gas, liquid, and solid particles drifting about in space from various things (debris from Space Travel projects included) and they would behave in the same way thin air in the outer atmosphere...unless it's more than a few particles and then we could be talking projectile motion instead of friction (since friction is really meant to illustrate a sort of "casual" force dealing with small and immeasurably many collisions).
~Robert
So we have obviously gone way off topic here, so this is my last post on the subject, especially since you finally made me have to look something up... that's where I draw the line!
I can understand how mathematically, if you were to increase the numerical portion of mass that the velocity would decrease assuming the momentum is constant (M=mv - momentum being "M"), so if M = 1 then you get 1/m = v, and thus any increase in mass becomes a decrease in velocity.
However, in this example when adding mass to a car, I don't think you can assume that the momentum is constant because you have to consider the mass added is also going at the same velocity when it is added, thus you get additional momentum since: (m1+v) + (m2+v) = M1+M2.
Another way to think about it as two objects in a closed system (no external forces acting on either), both masses going the same speed and coming together (ignoring the fact they would normally continue in a straight line and are somehow able to "curve" together to prevent a collision and potential lose of velocity due to the vector change). Both objects have their own momentum which is added together the moment they become "one". In more scientific terms, a perfect inelastic collision ... m1v1+m2v2 = (m1+m2)v
I guess it comes down to how exactly the mass is increased in the system, since you really can't have spontaneously generated mass. If the added mass has no initial velocity, you will of course lose velocity as the two masses are combined together. However, if you assume the added mass has the same velocity as the original mass, you will obviously not lose any velocity at all.
Also, my comment about saying I like the idea of space friction was more about the term "space friction", since I don't think I've ever heard it coined like that before, not that I didn't think it actually existed.
-Jason
Agreed, we have gotten far off topic, so I will close with a formalized proof that by the conservation of momentum velocity always changes when mass changes.
(BTW for those who know what it means to say "Lemma" in a formalized proof, it is very fitting here since the theorem that velocity changes when mass changes is a lemma for the above proofs regarding bodies approaching the speed of light)
Edit: the above proof demonstrates that there can never be an instance when a non-zero change in mass results in no change in velocity. It is true, however, that if a body is at rest then a change in mass will not necessarily trigger a change in velocity - explosions being an exclusion to this, however bodies that begin at rest and explosions do not have any place in the above discussion, hence the reason such instances were intentionally excluded from the proof. We don't need to show that a body with zero inertia remains at zero inertia with mass changes - notice how even here the conservation of momentum holds true.
Furthermore, if you were correct in your theory Jason, then you would have single-handedly disproved Einstein's theory of relativity and the notion that the speed of light is the maximum speed, both of which I'm still convinced are false. Sorry to say you didn't disprove Einstein's theories.
~Robert
Last edited by RLongtin (2011-10-24 02:19:46)
I'm pretty sure that perfect inelastic collisions have been proven time and again, yet they don't disprove Einstein's theory of relativity. Without spending too much time pouring over your proof, I think the flaw in it is perhaps initial I being equivalent to final I. If that were the case, that would require mi to be equivalent to mf, which means you add no mass to the system at all.
Ok, so I broke my own rule.
Quick, distract everyone with Lemmings!
*is distracted*
Ooooh! Lemmings!
That, plus they forgot to take into account the fact the Earth rotates.
That, plus they forgot to take into account the fact the Earth rotates.
What are we talking about now? O.o
Without spending too much time pouring over your proof, I think the flaw in it is perhaps initial I being equivalent to final I. If that were the case, that would require mi to be equivalent to mf, which means you add no mass to the system at all.
I think you'll find that if you spent a little more time on the proof, it's obvious that mass is said to change in the proof - I said so in the initial conditions before I even started! All I did was demonstrate conclusively that a change in mass has to result in a change in velocity, excluding 0-velocity systems which have an inertia of zero in the first place. It's a definitive and inarguable proof - putting the words proof and inarguable in the same sentence is extremely redundant since you can't argue against a proof, to do is...illogical. Proofs don't have flaws if they're written right.
I do have a degree that pretty much only says "I write proofs good".
You only have to agree with the initial conditions, and then check to make sure that I didn't divide by zeros. And on a side note, if you look at line 6, you'll see an inelastic collision (a very specific one that can be used to prove that inelastic collisions with 0-mass particles results in no change of velocity).
But we were never arguing inertia has anything to do with disproving Einstein's Theories - if you remember this all spun off of an example that was just meant to show the effects on particles as they acquire infinite mass. It's the notion that particles with mass can move faster than light that threatens (not disproves) Einstein's Theories. If CERN's results (it should be noted these are CERN and OPERA's results) are correct - and the more I read, the better and better those results look - then it puts a gigantic hole in Einstein's Theories as it disproves a pivotal Math equation used by Einstein in the development of his Theory of Relativity and then like dominos it also disproves countless other concepts such as time dilation and black holes - things that we can actually observe but now the rationale behind them is gone so we'd have to redefine everything from Einstein forward. If it can be shown that Neutrinos move faster than light, the last several decades of Physics are...well...subject to criticism.
~Robert
Last edited by RLongtin (2011-10-28 20:48:35)
FML, okay yet another person that didn't read OPERA's report in detail. OPERA didn't miss relativity, they didn't forget Time Dilation, the GPS had nothing to do with their final results.
...Mischa should I just post the whole thing here again??
Nevermind, I'll just go ahead and post what I posted on the link Euan provided (minus a comment about the writer being full of fail).
It's clear, firstly, that Elburg doesn't know if CERN and OPERA accounted for time dilation as a result of velocity - he says it appears that they did but it's not "explicitly" stated. Furthermore, Elburg failed to realize that the GPS device was not a part of the final results. He expected OPERA to show calculations of this yet they don't, and he too failed to realize how OPERA accounted for time dilation - they used PDF curves to compare particle wavelengths to get a precise measurement of both DISTANCE and TIME. Those results don't lie and don't require GPS to do.
Not to mention, Elburg failed to realize that OPERA and CERN utilized two things with the GPS devices that even if they were the basis for the experiment would eliminate the error Elburg claims existed - firstly, the GPS device used by CERN and OPERA doesn't call on one satellite as Elburg suggests, but rather as many as 16 simultaneously, and since Earth's spin is relative to each, the GPS device runs comparisons between all of the satellites to remove time dilation as a result of Earth spin. Secondly, the GPS devices were running in something called "common-view mode" which was developed in conjunction with Physicists from the UK National Physical Laboratory to reduce uncertainty as a result of Earth movement and its effects on GPS signals.
Really, don't take one physicist's article as proof when he is merely suggesting a possibility and admits he doesn't have all the facts.
Have we gone off topic? :-P
Ash
FML, okay yet another person that didn't read OPERA's report in detail. OPERA didn't miss relativity, they didn't forget Time Dilation, the GPS had nothing to do with their final results.
...Mischa should I just post the whole thing here again??
Well, it certainly helped me understand things quite a bit more!
So.... what you're saying is that it would take an FTL neutrino to allow the members of Firefly to go back into the past and keep the movie Serenity from happening and therefore effect the aftermath of what we know did happen?
|
...yes...yes, exactly my point.
~Robert
Huh what? What were we talking about again? This was the thread about a Serenity sequel, right?
Well, more about the events that would have taken place in the Firefly universe after the movie Serenity.
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