Why the speed of light cannot be exceeded according to Special Relativity

[Cartesiantheater]

EDIT- as far as my part in this thread, I don't believe anything I have posted really answers the question until you get to post #20. Before then, I am basically handwaving math, but not explaining where it comes from. Post 20 explains where the math I cite comes from.


They tell you things and then you have to accept them by faith. Well, not anymore. In this thread you will learn WHY the speed of light cannot be exceeded according to Special Relativity , theoretically speaking (well, more like ONE of the reasons the speed of light cannot be exceeded). We will take a small trip through algebra and special relativity, but I promise it will be easy to understand, as long as you read the two semi-hard parts a few times over.

Number 1.)

First, I will introduce you to a Greek letter, gamma: y. This letter represents a algebraic function that shows up in several places in special relativity.

All of the next few things, until you reach "Number 2.)" will simply be me showing that y is used throughout Special Relativity. You don't necessarily have to understand any of them to get the point.

Time dilation (the equation describing the slowing of time between two observers):

Δt = yΔtp

which reads in English as "the change in time measured by the observer moving with respect to the clock measuring time is equal to the change in time measured by the observer at rest with respect to the clock TIMES the value of y." (this means that the observer moving with respect to the clock will experience SLOWER time than the observer at rest with respect to the clock [because y will always be greater than 1, which I'll show in a minute]). The little subscript "p" is just a way to distinguish the two times. Don't worry about it. tp is called "proper time," and is just a way to distinguish between a moving and non-moving perspective.

Length Contraction (similar to time dilation; objects will contract based upon motion):

L = Lp/ y

This means that when you're in relative motion with an object, the measured length of that object will be less than the measured length of that object when you are at rest with respect to it (by a factor of 1/ y). Lp is called "proper length," and will be defined like this: The proper length of an object is the length measured by someone at rest relative to the object.



The Energy equation according to Special Relativity:

E = ymc^2 (the famous "E = mc^2" is REST energy, but TOTAL energy requires the y, because we must include Kinetic energy when we're talking about total energy)

Relativistic Linear Momentum:

p ≡ ymu

where p is momentum, m is mass, and u is velocity. One interesting thing to note is that when u approaches zero, the classical definition of momentum, momentum equals mass times velocity, or p = mu, becomes accurate. (u and p are emboldened because they are vectors, which is irrelevant to this discussion, but just in case you were curious that is why)

Working with Lorentz Velocity Transformations:

If an object has velocity components along the y and z axes, the components as measured by an observer are:

u'y = uy/y(1 - uxv/c^2)

and


u'z = uz/y(1 - uxv/c^2)




The purpose of all this so far? To show that the value y is hugely important in Special Relativity.

And now for the reason that the speed of light can never be exceeded according to the theory:

Number 2.)

What IS y? y is the following:

1/ √(1 - v^2/c^2)

One divided by the square root of one minus velocity squared over the speed of light squared.


It's so simple...

What happens when your velocity is greater than the speed of light? Let's let our velocity be 400 000 meters per second (the speed of light is ≈ 300 000 meters/second), and work out a time dilation problem:

Δt = yΔtp

We'll let Δtp equal 5 hours.

So, what happens when we solve for Δt when our velocity is 400 000 m/s?


Δt = y 5 hours

Now, we plug in y and get the following:

Δt = [1/ √(1 - (400 000 )^2 / (300 000 )^2 ) ] 5 h

Δt = [1/ √(1 - 1.77777778)] 5h

Δt = [1 / √ ( -0.77777778)] 5 h

WHOA! Look! You have a NEGATIVE square root! In real life we cannot work with negative square roots (imaginary numbers).

It turns out that EVERY time your velocity is greater than the speed of light, you will get an imaginary number in your y. This is EXACTLY one of the reasons in Special Relativity that the speed of light can NEVER be exceeded- if velocity is greater than c, you WILL get an imaginary number in your answer.

Now you know- it really is that simple (at least this of the many reasons is).

Enjoy.

[DontBeAfraid]

hmm.... CT, you explained how the math of the equation works but you did not put into words WHY a thing cant possibly go faster than light.... Einstien didnt start with the equation, he started with the observation and wrote the equation to fit it. So, and I really am curious as this is still kinda shady to me, why is it that as matter approaches the speed of light that it takes more and more energy to increase its speed? Why does that energy needed increase exponentially while the speed increases linearly? Or hell, maybe even logarithmically(sp).

I understand it has something to do with "mass increasing"?

[Cartesiantheater]

Actually, and this will be a shocker, "although the Michelson-Norley experiment was preformed before Einstein published his work on relativity, it is not clear whether or not Einstein was aware of the details of the experiment." (Physics for Scientists and Engineers,6th Edition, pg 1251).

To the best of my knowledge, special relativity was a "theory driven theory" and he actually did start with equations- the equations Maxwell used to describe electromagnetism- and he used them in a novel way. "It is important to recognize that Einstein was working on electromagnetism when he developed the special theory of relativity. He was convinced that Maxwell's equations were correct, and in order to reconcile them with one of his own postulates, he was forced into the revolutionary notion that space and time are not absolute." (same text book, pg 1254)

So, special relativity was one of those magical moments where theory was thought up and was shown to make sense before observation (that is, it wasn't designed to describe some observation, other than Maxwell's). The idea came first and only later was confirmed by observation. As far as the Michelson-Morley experiment that inadvertently showed the invariance of the speed of light, Einstein made no mention of trying to explain that- his speed of light postulate only coincidently matched this (this is one of the reasons his theories are considered with such pseudo-holiness).



Anyway, to your question.

Unfortunately, THAT question lies mostly in the realm of general relativity, but I'll give it a shot. This is the best I can do at this point.

My text book has an answer, but they put it like this: "It is left as an end-of-chapter problem (Problem 69) to show that under relative conditions, the acceleration a of a particle decreases under the action of a constant force, in which case
a ∝ (1 - u^2/c^2)^(3/2). From this proportionality, we see that as the particle's speed approaches c, the acceleration caused by any finite force approaches zero. Hence, it is impossible to accelerate a particle from rest to a speed u ≥ c. This argument shows that the speed of light is the ultimate speed, as noted in the preceding section." (same text book, pg 1268)

I'll take a look at that later today and see if I can prove that acceleration is proportional to that value, but I have no reason not to trust my text book.. Sorry that once again it is theoretical and mathematical, but look at that equation.

As u (velocity) gets bigger and bigger, closer and closer to the speed of light, you're going to get closer and closer to this: (1 - 1)^(3/2) which equals 0 (because you've got in that equation 1 minus a fraction of the square of your speed over the square of the speed of light. If velocity is EQUAL to the speed of light, you have a value of ONE for that fraction, which would make the value of the proportionality equal zero, which would mean no more acceleration). With a finite force your acceleration will be proportional to a value that approaches zero as your velocity approaches the speed of light. So, using the reasoning of limits, your acceleration will eventually stop, and this will happen BEFORE your velocity ever EQUALS the speed of light, therefore you can never reach that speed.

- unless of course you have an infinite force, I guess (for obvious reasons my text and my professor did not really discuss the possibility and results of having an infinite force, so I've got a blank there, but presumably you'd be able to reach c with an infinite force)

Basically then, you'd need an infinite force to over come the limiting value that you can approach with acceleration, because acceleration is fundamentally limited by the proportionality a ∝ (1 - u^2/c^2)^(3/2).



Again, sorry it is nothing but theory and math, but I've got nothing else at this point.


Now, again, I've not mentioned the mass energy connection, Here's all I got on that:

E^2 = (p^2)(c^2) + (mc^2)^2

where p is the magnitude of momentum and is defined as:

mu/ √(1 - u^2/c^2)

with u as velocity, m as mass, E as energy and c as the speed of light.

One thing to note is that the (p^2)(c^2) represents the kinetic energy involved with the particle- which is important since we're talking about the velocity of the object. Once again I'd have to use the same reasoning as before, you'll get a nonsense answer if you allow velocity to surpass that of light (an imaginary number).


Hopefully the discussion on the proportionality of acceleration will be of some use, even though it's more theory and math.

If a ∝ (1 - u^2/c^2)^(3/2),

the larger the speed you accelerate to the less and less you will be able to accelerate, and therefore you cannot ever actually achieve the speed of light.

The bottom line is this:

As u → c, a → 0.


At this point that's all I got. Does that help any at all?


EDIT- as for WHERE all this math comes from, I presume it comes from Einstein's theory rather than some observation, but to date has been shown to match up with observation. Within the context of the math I can understand why, but as to how and why these particular equations were derived, that is beyond me. I imagine it all starts with Maxwell and electromagnetism.

[DontBeAfraid]

At this point that's all I got. Does that help any at all?

Maybe somebody.... but not me. I already understand how the math works.... What I want to know is ... I guess, by what mechanism or means does equal "spent" force equal different amounts of internal energy on objects equal in everything except relative velocities. I suppose that maybe its starting to look like this might be like asking what makes gravity work. We have lots of examples of the way it works but not really the why or how of it. Like the illustration where planets are rolling around each other on a curved universe.... That has always bothered me in that it uses gravity to explain gravity....

edit: changed a word that didnt fit.

[iulian28ti]

Aren't there some theoretical particles named tachyons that are supposed to travel faster than light?

[Cartesiantheater]

Maybe somebody.... but not me. I already understand how the math works....

Well, if you want a better understanding of why those particular equations work you'd have to start with Maxwell's work on electromagnetism, which is beyond me at this point so you're on your own there.

What I want to know is ... I guess, by what mechanism or means does equal "spent" force equal different amounts of internal energy on objects equal in everything except relative velocities.

What do you mean by "spent" force? I'm not sure what you mean here.

As far as how force and matter are interchangeable, I would say that E = mc^2 and what not are just useful models that happen to explain the observed fact that the decomposition of matter releases energy.

I suppose that maybe its starting to look like this might be like asking what makes gravity work.

That line of discussion takes you right out of special relativity and into general relativity, which of course is quite over my head amigo.

We have lots of examples of the way it works but not really the why or how of it.

Yeah, you're right. I think maybe I worded the title of this thread wrong. It should have been "Why according to the theory of special relativity c cannot be exceeded," implying that we were dealing only with the given mathematical model rather than the reason it works and how it was derived.

I don't know though, if we can ever answer the questions of why and how (although string theory makes the claim regarding why certain particles exist).

Regardless, both in practice (up to now) and according to the mathematical model c cannot be passed if you're starting from a slower velocity. If c can be passed, we'd have causal violations, which could be bad news for all of applied logic anyway.

Like the illustration where planets are rolling around each other on a curved universe.... That has always bothered me in that it uses gravity to explain gravity....

I can see how that would. Depending on how I look at it, I feel that way at times as well.

But I think it helps to first make sure that you've got it that there is absolutely no absolute position of rest or absolute frame of reference.

From there, don't think of gravity as anything but a useful word, and instead just envision both acceleration and mass having the same affect on space-time, warping it. The more massive objects or the fastest accelerating objects warp it more, so a much less massive object will tend to move in accordance with the warping of the much more massive object (even though the small object warps space-time as well).

As for why massive or accelerating objects warp space-time, I don't know if there is a satisfactory answer to that question, other than to say that the model that predicts it matches observation really well.

Originally Posted by iulian28ti

Aren't there some theoretical particles named tachyons that are supposed to travel faster than light?

To the best of my knowledge those theoretical particles don't accelerate to the speed of light, but are already moving that fast.

Also, from what I know of these ideas, tachyons, they're not really faster than light particles. And even if they do move faster than light, they cannot transfer information faster than light (or even at all) and as such even in a model in which they exist they can hardly be spoken of to exist anyway. To my understanding (which is exceedingly weak) they are more like ghost particles.

Not only that, but as I understand it these "particles" have a squared mass which is negative. I mean that right there tells ME that the hypothesizing of these particles reveals a fundamental flaw in our models rather than an actual possibility of their existence. How can something have an "imaginary mass,' which is EXACTLY what these particles must have? (tell me, what is the square root of negative 1, √(-1)?

Further, if I understand it correctly, these particles have a minimum amount of potential energy (like a pencil balanced on its tip) and then due to quantum fluctuation it will "tip over" gaining energy. But once it gains a minimum amount of energy it is no longer a tachyon anyway, and instead transforms into a "real" particle.



Honestly, my knowledge of these particles comes straight from Brian Greene books and the like, so don't trust me, but to me these "particles" hardly exist even within the model and are really like ghost particles to me. And since they don't transmit information faster than light, they are really not violating SR or GR anyway. That's important.

These particles find a home much more in string theory. If you're interested in tachyons, I recommend you read some books on string theory.

[DontBeAfraid]

What do you mean by "spent" force? I'm not sure what you mean here.

I just meant force traded for more speed.

[Cartesiantheater]

Originally Posted by DontBeAfraid

I just meant force traded for more speed.

Well, since in relatively low speeds force = mass * acceleration, you must add force to chance speed (which would be acceleration), but other than that, I've got nothing. It's just a relationship.

HOWEVER, maybe you might find some answers in all of this mess?

http://forum.physorg.com/index.php?showtopic=18565&st=0

There are a few guys here qualified to answer the questions that can be answered. Also, maybe you might consider looking into Einstein field equations and their history?

If you want, start a register there and post a thread in the "puzzling questions" section. I'm sure if you're nice enough someone with some decent education will answer some of your questions.

Enjoy.

[TEMELUCHUS]

Wholly crap you folks are like super brains. Now these are the threads that I enjoy reading even though I don't quite get it anymore but sometimes it helps me to remember.
If I could ask a question: An experiment will take place in a few months in Switzerland I believe with the new accelerator. If this succeeds and a minute black hole is formed possibly causing a worm hole, would that be considered faster than light if we are able to move from one existence to another or from one galaxy to another in seconds?

[DontBeAfraid]

Its not the same kind of black hole that most of us are "familiar" with. And nothing will move through it to anywhere for a couple of reasons.... One, it will be incredibly small and two it will be equally incredibly short lived.

[lazserus]

Sorry Cartesiantheater, but that was a HORRIBLE explanation. You explained how it works on a mathematical level and not on a physical level. The issue needs be explored in a light the average person can see, not magic light such as UV or infrared.

Special relativity theory was one that challenged that of ether. Einstein said the speed of light is constant and that should be used as reference, not some imaginary ether that cannot be measured. His conclusion not only stated that the speed of light was constant, but that all observers will measure it the same regardless of relative motion. Hence the name relativity.

The reason nothing can travel at the speed (or beyond) is because of mass. Light, which is constructed of photon, travels at a constant speed and has no mass. In order for a particle with mass to travel at the same speed, it would require an infinite quanity of mass and energy. E=mc^2 means energy equals mass times the speed of light squared. What this simple equations tells us is any object attempting to attain the speed of light must accumulate infinite energy and infinite mass. Mass and energy are directly correlate because mass is but another form of energy.

Experiments have shown in simple forms such as aeronautics that as you attempt to increase velocity (the rate of which an object travels over time) you need more energy. Aeronautics present the best experimental data we have. In order for an aircraft to reach a certain velocity it requires a certain amount of fuel to give it the energy to do so. Using Einstein's equation, electrons have been given extraordinary energy levels in order to reach the speed of light--the expierments always fail. Because the electron has mass, it would take an infinite amount of energy to get it to move at velocities of the photon.

The reason the speed of light cannot be breeched is due to mass and photons have none. If any of you heard of neutrinos, it is the same concept. Neutrinos travel at the same speeds as photons because they have no intrinsic mass.

Imagine making a spit-wad and using a straw to launch it across a classroom. The wad contains mass based on your saliva content and the paper used. The straw is acts a launching device. The heavier the wad is, the more breath required to launch it to the designated target. I'm sure many of you have created spit-wads in class and adjusted the angle in order for it to reach the person you plan spitting. This is the same concept. You have to provide more energy via breath in order to make a heavier wad travel the same distance. The lighter the wad the less breath required. If you take this same example and apply it to particles you reach a similar conclusion; however, that conclusion results in the requirement of infinite energy to make it travel the speed of light.

Special relativity introduces E=mc^2, but takes no position mass. It was not until general relativity that mass was included in the Einstein relativity equations and theory. Special relativity introduced the simple equation but merely focuses on velocity without mass. THat is neither hear nor there. The reason the speed of light cannot be surpassed is because all other particles contain mass.

[Cartesiantheater]

Originally Posted by Ta'Nahir

Sorry Cartesiantheater, but that was a HORRIBLE explanation. You explained how it works on a mathematical level and not on a physical level. The issue needs be explored in a light the average person can see, not magic light such as UV or infrared.

Maybe, but doing that will only take a math equation and reduce it to a verbal proportionality, as you have done below.

Special relativity theory was one that challenged that of ether. Einstein said the speed of light is constant

Actually, I'm pretty sure that was determined before Einstein by Maxwell, Lorentz, etc. The equations of special relativity come directly from Lorentz.

and that should be used as reference, not some imaginary ether that cannot be measured. His conclusion not only stated that the speed of light was constant, but that all observers will measure it the same regardless of relative motion. Hence the name relativity.

Well, he wanted invariance, but anyway.

The reason nothing can travel at the speed (or beyond) is because of mass. Light, which is constructed of photon, travels at a constant speed and has no mass. In order for a particle with mass to travel at the same speed, it would require an infinite quanity of mass and energy.

Now you have done nothing but represent a mathematical equation in words. This is not any different then what I did, except you've deleted the math. You simply have a proportionality between mass, energy and velocity, and in order to reach the speed of light you must have infinite energy. This is just another math equation.

Starting from the equation (by the way, this is the "real" "E =mc^2" equation, including motion, whereas plain old E=mc^2 is when v = zero in the equation below)

E = mc^2/√(1 - v^2/c^2)

And solving for v you get:

v = √[c^2 (1 - (m^2c^4)/(E^2)]

And you can easily see why energy must be infinite. In order for v to equal c from this equation, the value (m^2c^2)/(E^2) must equal zero.

In order for that value to equal zero, you must let the denominator approach infinity (remember, the lim of 1/x as x →∞ is zero)

So, in order for v to approach c, E must approach infinity.


Here you have what I said and what you said amounting to exactly the same thing, being equal in explanatory power, but one written in words and one in math.

E=mc^2 means energy equals mass times the speed of light squared. What this simple equations tells us is any object attempting to attain the speed of light must accumulate infinite energy and infinite mass. Mass and energy are directly correlate because mass is but another form of energy.

However, please do not forget to mention that E = mc^2 is only an objects "rest" energy.

Experiments have shown in simple forms such as aeronautics that as you attempt to increase velocity (the rate of which an object travels over time) you need more energy. Aeronautics present the best experimental data we have. In order for an aircraft to reach a certain velocity it requires a certain amount of fuel to give it the energy to do so. Using Einstein's equation, electrons have been given extraordinary energy levels in order to reach the speed of light--the expierments always fail. Because the electron has mass, it would take an infinite amount of energy to get it to move at velocities of the photon.

Yes, this is true. But this is experiment, not theory. The explanation is found in the theory, not the experiment. That's why I think the math is the best way to explain why.

The reason the speed of light cannot be breeched is due to mass and photons have none. If any of you heard of neutrinos, it is the same concept. Neutrinos travel at the same speeds as photons because they have no intrinsic mass.

Then why can't massless particles surpass the speed of light?

Imagine making a spit-wad and using a straw to launch it across a classroom. The wad contains mass based on your saliva content and the paper used. The straw is acts a launching device. The heavier the wad is, the more breath required to launch it to the designated target. I'm sure many of you have created spit-wads in class and adjusted the angle in order for it to reach the person you plan spitting. This is the same concept. You have to provide more energy via breath in order to make a heavier wad travel the same distance. The lighter the wad the less breath required. If you take this same example and apply it to particles you reach a similar conclusion; however, that conclusion results in the requirement of infinite energy to make it travel the speed of light.

And this result is explained as well with the math.

Special relativity introduces E=mc^2, but takes no position mass. It was not until general relativity that mass was included in the Einstein relativity equations and theory. Special relativity introduced the simple equation but merely focuses on velocity without mass. THat is neither hear nor there. The reason the speed of light cannot be surpassed is because all other particles contain mass.

But you see, that is really the EXACT same explanation I gave in this very post using pure math- only without the math.




So I guess my point is, at this point NEITHER explanation explains why this law of nature is what it is. Both only describe the law. I probably shouldn't have said "why" in the title, but rather offered a logical backing to the observed physical law.

[lazserus]

My purpose was not to claim that your explanation was wrong, but to provide an explanation the majority of people can comprehend. Displaying equations teaches few--explaining the phenomena behind the equations is the fruit of knowledge which all can enjoy. I say you explanation is horrible because you explained little. You merely stimulated the knowledge others of your mathematic proficiency. Mathematics supports knowledge...it does not truly explain it.

Originally Posted by Cartesiantheater

Actually, I'm pretty sure that was determined before Einstein by Maxwell, Lorentz, etc. The equations of special relativity come directly from Lorentz.

Since we are discussing science, Einstein is accredited with the concept of relativity. In reality, Galileo was the first to introduce relative motion.

Originally Posted by Cartesiantheater

Now you have done nothing but represent a mathematical equation in words.

Ah but I have possibly accomplished more than you have using words other than equations. The art of science is not what you know in its intrinsic form, but how you can share it with the rest of the world. Knowledge is absolutely useless if not understood by the world.

However, please do not forget to mention that E = mc^2 is only an objects "rest" energy

I skipped this on purpose. Complicating the theory does not aid in teaching its basic premise. I mention only what is necessary to understand the concept. This is not a physics class, therefore getting into the nasty fabrics of the theory only strays from teaching.

Then why can't massless particles surpass the speed of light?

Modern thinking considers a photon's rest is the speed of light. Being that particles with no mass can only reach the speed of light, massless particles should in no way be considered for surpassing it.

[DontBeAfraid]

Originally Posted by ct

Here you have what I said and what you said amounting to exactly the same thing, being equal in explanatory power, but one written in words and one in math.

I agree. Ta, you didnt explain it any better or in any more detail than ct.. You just dumbed it down.

[DontBeAfraid]

ok, last night I had somewhat of an epiphany and believe I now understand the mechanism by which an objects mass is seen to increase from the relative perspective of "outside" or "not in motion with" thereby making it need more and more energy for smaller and smaler increases in speed.

It has everything to do with the motion paradox and when I figure out a good way to put it inot words I will.

[lazserus]

In relativity theory there is no paradox when it comes to motion. Cartesiantheater and I explained the same thing in different ways. In his defense, it becomes much more difficult to appreciate and comprehend special relativity without proficiency in the mathematics, which is little more than algebra.

Paradoxical motion comes in when using Newtonian motion to describe the motion of extent celestial bodies, such as the much of the outskirts of our own galaxy. Relativity theory does not change Newton's laws of motion. It changes our conception of gravity, time, and mass's correlation to the previous. Special relativity does cap a limit to the velocity of mass. The speed of light is peak because the photon has no mass while particles with mass are incapable of aquiring the energy to reach relative velocity. Some theories conclude that a photon's velocity is directly related to the fact is is massless. IN essence, a photon's rest is 186,000 miles per second and all massless particles (neutrinos) naturally rest at such velocities.

[DontBeAfraid]

In relativity theory there is no paradox when it comes to motion.

Simply saying it doesnt make it true. If you think it does then you dont understand the paradox.

which is little more than algebra.

When you know me you can talk down to me.

Paradoxical motion comes in when...

The correct ending to this sentence is: "when you consider the motion of ANYTHING".

The speed of light is peak because the photon has no mass while particles with mass are incapable of aquiring the energy to reach relative velocity.

Again you have managed to state what we already know. You have not explained the how or the why of it.

N essence, a photon's rest is 186,000 miles per second and all massless particles (neutrinos) naturally rest at such velocities.

You have still not explained anything with this sentence. You have again rewritten what we have already observed.

So, just to recap what we know; you guys should love this because its about a sentence long:

Objects with mass cannot attain the speed of light because as their relative velocities increase so too does their mass meaning that it takes more energy to propel them.

Now the question I posed was: why does their mass increase with velocity?

The answer to that question is NOT: it takes more energy because their mass increases.

I do believe I have an understanding of the "how" the mass increases and it does pertain to the motion paradox.

edit: Please dont interpret my post as stand offish, its just how I choose to comminucate, continued exchange is the goal.

[lazserus]

Originally Posted by DontBeAfraid

Simply saying it doesnt make it true. If you think it does then you dont understand the paradox.

No, I am saying there is no paradox in relativity theory regarding motion. Special relativity is solid when dealing with motion. Maybe you mean something else? Because paradox is the wrong word in this case. There are not contradictory factors in SR that negate it.

Now the question I posed was: why does their mass increase with velocity?

I see what you are asking and it is difficult to explain. Explaining how is easier than why. The eternal philosophical question is why? The question of why can never truly be answered, but how can.

Say you want to throw a football 30 yards. Ignoring wind and gravity you might simply tell yourself you have to throw it x hard. If you want to throw it 70 yards you know you have to throw it harder, right? Energy and mass are interchangeable here. To lug the football farther you have to apply more energy. If you want to move something heavy you must acquire an amount of energy relatively equal to the mass you wish to push.

We've established the concept. The special theory of relativity's focus is on relativity: relative motion.- motion relative to the observer. If you got on a track an ran as fast as you could, reaching a speed close to that of light, you would not notice yourself getting heavier. The person recording your time would be the observer noticing. The point of the theory is partially philosophical in that only observers can measure, not the object itself. With energy and mass being interchangeable, the more mass that increases the more energy and the more energy used the more mass accumulated.

You have to remember that Einstein put SR down on paper and it was not tested for over a decade. It was all upstairs. Maybe a simpler way to think of the concept is through vehicle collisions. If you put a truck in neutral and pushed it, you could only get it so fast due to a limited supply of kinetic energy. If you add 4 friends, the truck will move much faster. The truck weighs the same amount, but once it impacts another object at the velocity you and your friends pushed it, it will impact that object as if it were x times heavier.

I probably didn't answer your question, but I may have answered others'. If you really want to dig deep, check out these sites for better answers. I warn you, they are mathematically heavy.

http://math.ucr.edu/home/baez/physic...y/SR/mass.html
http://galileoandeinstein.physics.vi..._increase.html

The second one has some real-world applications to flesh things out, but I can't safely say they will help.

[DontBeAfraid]

Say you want to throw a football 30 yards. Ignoring wind and gravity you might simply tell yourself you have to throw it x hard. If you want to throw it 70 yards you know you have to throw it harder, right? Energy and mass are interchangeable here. To lug the football farther you have to apply more energy. If you want to move something heavy you must acquire an amount of energy relatively equal to the mass you wish to push.

Ignoring wind AND gravity I would not need any more force to throw the ball 1 million miles than I would need to throw it 1 foot. If you understanding of the motion paradox is this profoundly wrong then Im not sure you should be posting "answers" in here, aristotle.

If you got on a track an ran as fast as you could, reaching a speed close to that of light, you would not notice yourself getting heavier. The person recording your time would be the observer noticing. The point of the theory is partially philosophical in that only observers can measure, not the object itself.

You like to state what we have already covered...

With energy and mass being interchangeable, the more mass that increases the more energy and the more energy used the more mass accumulated.

Now Im almost certain you do not understand the concept. Or maybe you do and you are having as hard a time as I am putting it into owrds.

If you put a truck in neutral and pushed it, you could only get it so fast due to a limited supply of kinetic energy. If you add 4 friends, the truck will move much faster. The truck weighs the same amount, but once it impacts another object at the velocity you and your friends pushed it, it will impact that object as if it were x times heavier.

Rolling friction has nothing to do with this conversation. Here, on earth, there is no practical example which can be used to describe physics of an object becoming more massive as its velocity increases.. The only thing on earth that even approaches velocities this high would be a particle zipping around a super collider. It has to be taken into account a little bit when dealing with satellites but not much.

No, me throwing a football or pushing a car, simply has nothing to do with why no object can surpass he speed of light.

I probably didn't answer your question, but I may have answered others'. If you really want to dig deep, check out these sites for better answers. I warn you, they are mathematically heavy.

I appreciate your enthusiasm.

[Cartesiantheater]

Okay, this discussion was tactically flawed from the beginning. I had thought that the math itself would be sufficient, however, if I cannot show you how the math is derived from reality, then it is meaningless. But if I CAN show you how the math is derived from reality, then the math can be used as a proof. After spending a bunch of time trying to learn how it was derived (they don't teach you that part in a survey of physics class; had to figure it out on my own time, thus it took forever), I am ready to show where the math comes from, which in turn will show why the speed of light is a necessary speed limit.


HOWEVER, that isn't necessary at all! The only thing necessary is to follow one of the two fundamental assumptions of special relativity to its logical conclusion (however, I really don't believe this first part is satisfying. Read the entire post, especially the part about the math, because I think it is vital).

The two major postulates of special relativity are that:

1- The laws of physics are the same for every inertial frame

2- The speed of light is constant for every inertial (non-accelerating) frame

Number 2 says that the speed of light is constant for every inertial frame, but it does NOT say that it is the FASTEST speed. Experiments (Michelson-Morley, for example) have confirmed that the speed of light is constant for inertial reference frames.

Now, let us follow that to its logical conclusion.


Consider two observers at rest with respect to each other in the same location (ingore the impossibility of two people being at the same point in space). Call them Alice and Bob. Let's say Alice emits a circle of light that propagates out radially at the speed of light. Both Alice and Bob would forever be at the center of the circle of light as it propagates.

However, what if, at the instant that Alice sends out the light signal, Bob begins to move to the right at half the speed of light, but without accelerating. That is, Bob moves at a constant speed with respect to Alice. According to the principle of the constancy of the speed of light (postulate 2), the speed of light must be the same for both Alice and Bob. What this means, if you think about it deeply enough, is that BOTH Alice AND Bob must be in the center of the circle of light as it propagates, even though Bob is moving away from Alice at a constant speed. The only possible way for both Alice and Bob to both be in the center of the circle of light is if space and time distort as they move in just the right way so that both see themselves as being in the center of the circle. (This is where time dilation and length contraction come from- in the thread "The Universe" I gave a decent explanation of how that happens)

But there is one other thing we have yet to consider: If Bob remains in the center of the light circle (from his perspective) despite moving away from Alice at half the speed of light, then with respect to the edge of the circle Bob never "gains any ground." That is, the light is STILL moving away from him at the speed of light, because he remains in the center of the circle.

Ultimately, the assumption that the speed of light is constant for all reference frames caries with it the logical consequence that no observer moving slower than the speed of light can ever reach the speed of light, because in all frames the speed of light is constant, including ones of very high speeds.





Now onto the math. Before you panic, this is juts the Pythagorean theorem, and it will show you what is going on with all of this (including the increase in mass as you approach the speed of light).

What we are dealing with is coordinate transformations. For example, if you are standing on a bus going 40 miles per hour with respect to some observer, and you throw the ball 40 miles per hour from your perspective, then from the perspective of the person at rest, the ball travels 80 miles per hour. We've all experienced something like that. This is a coordinate transformation. In your coordinate system, the ball moves away at 40 miles per hour. However, because your coordinate system is moving with respect to the observer at rest, in his coordinate system the ball is moving away at 80 miles per hour. This is the fundamental issue dealt with in special relativity. Transformation of coordinates from "at rest" systems to "in motion" systems.

The problem, however is that the transformation discussed above (the Galilean Transformation) is inconsistent with the postulate of the constancy of the speed of light (I also discussed this in the Universe post mentioned earlier). If you follow this example it will be easy to see why the transformation isn't identical to the Galilean transformation, and what it implies.


Imagine two observers, each with clocks calibrated exactly. The two clocks work as follows: a beam of light shoots up from one mirror, hits the other mirror, then reflects back down to the original point. Light beam goes up = "tick," light beam goes down = "tock." We will assume that the speed of light is constant for all inertial frames, as per the the postulate of the constancy of the speed of light (and as per experimentation).

Here are the two clocks at rest with respect to each other.

clocks at rest.gif

Now, imagine that one of the clocks moves away at some speed less than the speed of light. Remember, assume that light travels the same velocity in both frames, since the two are in the same inertial frame (there is no acceleration going on between them- all velocities are constant). You will see that, because the light beam must travel at velocity c in both instances, the light beam travels a longer distance in the moving clock than the stationary clock. Here is an animation of path the light travels in both clocks, one at rest, and one in motion with respect to the other (traveling at some velocity, v).

clocks in motion.gif

Now, it is easy to see that the path the light follows in the moving clock is a triangle, while the path the light follows in the stationary clock is a straight line. It is also easy to see that in the moving clock, if you draw a perpendicular line from the top of the triangle to the base, that line is the same distance the light travels in the stationary clock. Keep that in mind, because it will be important shortly.

Remember, the speed of light is the same for both observers. And remember that "Distance = rate times time," D = rt. Consider the path the light beams travel. The distance for the clock at rest is shorter than the distance for the clock in motion. Thus, we will define the distance for the clock at rest to be D1 = r1t1. We will then define the distance the beam travels in the clock in motion to be D2 = r2t2.

Again, remember that the speed of light is the same for both clocks. That means that r1 = r2, so now we will rewrite the two distance formulas as D1 = ct1, and D2 = ct2. The rate the beam travels is the same for both clocks- the speed of light.

Now, because the two distances are different, and because the rate the beam travels is the same in both, the times must be different as well. Keep that in mind.

Now, consider the triangle formed by the moving clock, and remember what I said about "the line drawn from the top of the triangle perpendicular to the base is equal to the distance traveled by the beam from the clock at rest." If you draw that line, you split the triangle in half. We will consider this half triangle. The hypotenuse of the triangle is equal to D2, the leg going up is equal to D1, and the leg pointing sideways is equal to another distance, D3. This third distance is the distance the clock travels in time t2. It depends upon how fast the clock is moving, so we will label this third distance D3 = vt2.



This is what that "half" triangle looks like.




The hypotenuse is D2, which is the speed of light times t2, the vertical leg is D1, which is the speed of light times t1, and the horizontal leg is D3, which is the velocity of the mirrors, v, times t2.


Now, using algebra, we will divide all three sides by ct1, thereby making the vertical leg equal to 1, the hypotenuse equal to t2/t1, and the horizontal leg equal to t2v/t1c. This gives us the following triangle:



End of part 1.

[Cartesiantheater]

Part 2.

One more algebra step. For simplicity, we will let the value t2/t1 = y. The triangle now looks like this (click on it to make it larger- the site isn't letting me host another image):

light triangle 3.JPG


This is the triangle we will work with. We will now use the Pythagorean theorem to solve for y. The Pythagorean theorem states that the sum of the squares of two legs of a right triangle equals the square of the hypotenuse of that triangle. Therefore:

y^2 = 1^2 + y^2 v^2/c^2

Step by step algebra to solve for y:

y^2 - y^2 v^2/c^2 = 1

y^2 ( 1 - v^2/c^2 ) = 1

y^2 = 1/( 1 - v^2/c^2 )

y = 1 / √( 1 - v^2/c^2 )


There you have it. We have just derived the Lorentz factor. This equation, which is a mathematical description of a real physical relation (the triangle formed by the beam of light in the moving clock), states that it is impossible for the velocity of the moving clock to be equal to or greater than the speed of light. EDIT- The reason the Lorentz factor precludes faster than light travel is, as has been mentioned earlier in this thread, if v >c, then you would have a negative square root, which is nonsense, in a physical sense. Or, you can look at it as a natural consequence of the constancy of the speed of light for all inertial frames, as discussed earlier.




What all this means is that if we assume that the speed of light is constant for all inertial reference frames, then it is impossible for any object to travel faster than the speed of light. Because this assumption (that the speed of light is constant for all inertial frames) is backed up by experimental reality (as discussed earlier), the mathematics are valid for real objects.


Now, what the Lorentz factor, that we have just derived is:

Using those triangles we have derived a scaler factor to be used whenever a reference system is in motion with respect to another. For example, if a ball is being thrown from on a train, in order to find out what that ball's coordinates are with respect to the coordinate system at rest, we would take the Galilean transformation (discussed earlier) and multiply it by the Lorentz scaler factor. The reason we do this is because in our world the speed of light is constant for all inertial frames. Therefore, whenever a frame is in motion with respect to another, we must multiply its coordinates by the Lorentz factor in order to determine where its position is with respect to the stationary coordinate system in a universe in which the speed of light is constant for all inertial frames (a universe like ours).

Another example is that of a moving object's mass. If a massive object is moving with respect to an "at rest" coordinate system, then in order to calculate that objects mass from the perspective of the stationary system, we must use the Lorentz factor, because we live in a world in which the speed of light is constant for all inertial frames- which causes distortions in space and time, which then distorts everything within space and time, including mass. Therefore, to determine the relativistic mass of an object in a moving reference frame, you must multiply that value by the Lorentz factor, which then implies, via the E = mc^2 relation, that an object's relativistic mass will approach infinity as the object's velocity approaches the speed of light (this was discussed earlier in this thread).



The simple matter is that our intuitive notion about coordinate transformations was inconsistent with a universe in which the speed of light is constant for all inertial frames. We do not live in a universe in which the speed of light changes. That was an assumption made by the Galilean transformation. If we are in a universe in which the speed of light is constant for all inertial frames (which we are), then all coordinate transformations must by logical necessity (as derived in this post) use the Lorentz transformation, and NOT the Galilean one alone. Because the Lorentz transformation necessarily implies that we cannot move faster than light, and because we must use this transformation (because we live in a universe in which the speed of light is constant for all inertial frames), it is a natural consequence that faster than light travel is impossible.



I don't know if this is any better than my first post, but at least you now know where the Lorentz transformation comes from. So now, anything this formula suggests to you should no longer seem like magic, but rather a natural geometric consequence of a universe in which the speed of light is constant for all inertial frames.

Is this one any better at all?

[Lightmind]

Not sure on all this math... but to save all our sanity's and CT's time, I'm not gonna contest, and I'm not an expert either.

DBA, you asked why something gains mass as it approaches the speed of light. I'll try;-

All equations MUST balance. Both numerically and units.

AT the speed of light a particle's energy has E=MC^2
C... The speed of light, is a constant, so cannot change. Therefore the only factor that can change is mass.
Disregarding potential energy (gravity) and Kinetic energy (relative motion). As a particle(m) accelerates (v^2) it gains energy E, as in E=mv^2, v here is a variable and so can change. If we substitute E we get mv^2 trying to become MC^2
If v increases, only an increase in M can balance the equation. But because mass is common to both sides m=M, and the energy of the particle, at any one point in time, is constant. Increasing m means reducing v. ie the bigger it gets the more energy required to maintain acceleration.

I'm not sure that there is anything in this equation that points to Infinite energy requirement, but it would certainly take a lot. Like I said, I'm not an expert. Whether it's a "final barrier" or simply a believed one, due to current knowledge I think only time will tell. Similar science was postulated for the speed of sound, and even at the advent of passenger travel in locomotives, i.e. traveling at faster than 30mph, the reduced air pressure would suck the air from our lungs!

There have been postulated other methods of at least data transferal at hyper-light speeds. Specifically Quarks created from the same atom. There is also a thread on here on AO, (though I can't remember it's name), regarding changes in DNA occurring instantaneously, without regard to distance.

One of these days.... who knows, I'm no expert

[serge_regulator]

After reading this thread I have to say....Wow.... Some interesting descriptions of some complex theories.

I wont go into too much detail but there is a legacy of physicists in my family. I had it explained to me this way:

Energy has mass all to itself. Example: A coil spring at rest weighs less than the same spring when compressed and under tension. The only difference is the potential energy. Velocity is also energy. The more velocity an object has, the more it weighs. The more it weighs, the more energy it takes to accelerate it. You need an infinite supply of energy in order to accelerate an infinite mass. It cant be done.

Simple.

[Lightmind]

LOL.... I've been thinking of a way of explaining it in less words....as in "I'll define that principle in two."

Maybe someone can explain why it is proven/assumed that any mass at C must be infinite energy? As C is a quantifiable value.

Secondly, hasn't general (and therefore special) relativity been disproven with the advent of Quantum physics? In that photons, can have the characteristics of particle OR wave, but not both?

[serge_regulator]

Funny you should ask that.... This was the answer I got:

If you are ten feet from a wall and you move exactly half way there on every turn, how many turns will it take you to reach the wall?