Why can light travel at the speed c?

[mibaokula]

hey guys, I'm new to this forum so i don't know if this has been already asked before

i saw this equation on special relativity explaining why objects with mass cannot travel at the speed of light

E = mc^2/(1-(v^2/c^2))^0.5
( hard to write on a computer. view the attachment for the equation or link here: http://upload.wikimedia.org/math/2/4/0/24090a520815f2d76b2be996acc6c9e2.png)

this makes sense right; if an object travels at the speed of light, the denominator becomes 0 and if the object has mass (say for example one arbitrary unit) you get a large number divided by zero - in other words an undefined energy to make this happen - impossible
but what confuses me it that if you take a photon which can travel at c, you still get an undefined energy. my question is: how are photons able to travel at c in a vacuum? do they carry undefined amounts of energy? or does the equation not work for light?
thanks

 

[Doc Al]

or does the equation not work for light?

That's it. You need to use this more general equation, which works for all particles, including massless photons:

$$E^2 = mc^4 + p^2c^2$$

(p is the momentum of the particle.)

 

[ghwellsjr]

this makes sense right; if an object travels at the speed of light, the denominator becomes 0 and if the object has mass (say for example one arbitrary unit) you get a large number divided by zero - in other words an undefined energy to make this happen - impossible

Actually, a large number divided by zero is not undefined, it's infinite.

but what confuses me it that if you take a photon which can travel at c, you still get an undefined energy. my question is: how are photons able to travel at c in a vacuum? do they carry undefined amounts of energy? or does the equation not work for light?

In this case you are dividing zero by zero which is indeterminate but can be an exact number, you just have to solve the problem some other way.

 

[Drakkith]

Actually, a large number divided by zero is not undefined, it's infinite.

In this case you are dividing zero by zero which is indeterminate but can be an exact number, you just have to solve the problem some other way.

I thought ALL numbers divided by zero are undefined.

 

[bcrowell]

I thought ALL numbers divided by zero are undefined.

If 0/0=x, then x should be a solution of 0=0x. All values of x are solutions, so any real number is a possible value of 0/0. That's why it's called an indeterminate form.

It's misleading to say that 1/0 and 0/0 are both undefined and just leave it at that, because 1/0 is infinite, but 0/0 is an indeterminate form.

 

[ghwellsjr]

Consider A/B=C.

Then A=B*C, correct?

Now if A=0 and B=0, we have:

0=0*C, correct?

This equation is true for every value of C, correct?

Therefore, 0/0 can be any number you want.

 

[HallsofIvy]

Yes, that is why it is called "indeterminant". But "infinite" is included in "undefined".

 

[bcrowell]

Yes, that is why it is called "indeterminant". But "infinite" is included in "undefined".

I'm not saying it's wrong to call it undefined, just that it's misleading. It hides an important distinction.

 

[Pengwuino]

The past few posts have given me a little confusion because when you're talking about a photon, you're separately taking the limit of $m \to 0$ and $v \to c$ but one is not dependent on the other. If I could decide I simply want to take m to 0, you should find the first term drops out and end of story, it's 0, not infinite.

 

[atyy]

I'm not saying it's wrong to call it undefined, just that it's misleading. It hides an important distinction.

Can you explain more? I have never understood this point. It was always explained to me that the relativistic mass is a good heuristic for showing that only massless particles can travel at the speed of light. I have never found the explanation that the true equation is E=rest mass+momentum² to be any more rigourous, since momentum is not defined in the same way as for massive particles anyway. The only way I know to make it "correct" is to invoke quantum theory and say that mass is ultimately a dispersion relation of a wave. It'd be cool if a beloved 0/0 heuristic has more to it than I learnt.

 

[PAllen]

Can you explain more? I have never understood this point. It was always explained to me that the relativistic mass is a good heuristic for showing that only massless particles can travel at the speed of light. I have never found the explanation that the true equation is E=rest mass+momentum² to be any more rigourous, since momentum is not defined in the same way as for massive particles anyway. The only way I know to make it "correct" is to invoke quantum theory and say that mass is ultimately a dispersion relation of a wave. It'd be cool if a beloved 0/0 heuristic has more to it than I learnt.

I'll reiterate my favorite solution, which I haven't seen any book. I makes no use of division by zero, no funny definition of momentum.

For any particle with mass, it is straightforward to derive that:

sqrt(1 - v^2/c^2) = 1 - KE / (KE + E0)

If you want to have a particle with E0=0 and KE > 0, and have it follow as many of the same rules as normal particles as possible, then v=c is immediately implied. No limit, no zerodivide, no alternate momentum definition. THEN, having established this, you use :

E2=mc^4+p^2c^2

to arrive at p=E/c, where we know that for massless particles all E is kinetic, and m=0.

 

[Passionflower]

because 1/0 is infinite

If 1/0 = infinite then 1/infinite - 1/0 = 0 but 1/infinite is undefined so it could not be true.

 

[Matterwave]

If 0/0=x, then x should be a solution of 0=0x. All values of x are solutions, so any real number is a possible value of 0/0. That's why it's called an indeterminate form.

It's misleading to say that 1/0 and 0/0 are both undefined and just leave it at that, because 1/0 is infinite, but 0/0 is an indeterminate form.

The limit as x approaches 0 for 1/x doesn't even exist since the limit approaches plus and minus infinity depending on which way you go, so how can you define 1/0 as infinity? Defining it as negative infinity would also be equally valid.

Defining 1/0 to be infinity causes all sorts of problems. It would define, then, infinity*0=1. But infinity*0 is an indeterminate form.

Also, defining 1/0 to be infinity would first require the definition of infinity to be one of the integers. This definition would ruin the fundamental theorem of arithmetic since infinity could not be uniquely factored into primes.

 

[ghwellsjr]

The OP was asking about an equation for the energy of a moving particle. He stated that a massive particle has an undefined energy when moving at the speed of light and said that was impossible. I've never heard of anyone stating that the energy of a massive particle becomes undefined when traveling at the speed of light. I've always heard that it takes an infinite amount of energy to accelerate a massive particle to the speed of light.

Then he stated that a massless particle (a photon) also has an undefined energy and he wondered if photons therefore have undefined amounts of energy or if the equation doesn't work for light. But the equation doesn't yield an undefined energy for light in the same way that it does for a massive particle, rather 0/0 is indeterminate, it can have any value, you just have to determine that value some other way. It's not that the equation doesn't work for light in the sense that it is no longer true, but rather it just doesn't tell you which of the many possible answers is the correct one.

 

[ghwellsjr]

If 1/0 = infinite then 1/infinite - 1/0 = 0 but 1/infinite is undefined so it could not be true.

I'm not following this at all. If 1/0 is infinite then 1/infinite = 0. Where did you get - 1/0? And why do you say that 1/infinite is undefined?

 

[mibaokula]

That's it. You need to use this more general equation, which works for all particles, including massless photons:

$$E^2 = mc^4 + p^2c^2$$

(p is the momentum of the particle.)

how would this equation work in terms of particles like the tachyon which moves at superluminal speeds (assuming it exists - which it probably doesn't)

 

[Doc Al]

how would this equation work in terms of particles like the tachyon which moves at superluminal speeds (assuming it exists - which it probably doesn't)

It still works for tachyons, but E, p, and m are all imaginary numbers.

 

[mibaokula]

It still works for tachyons, but E, p, and m are all imaginary numbers.

awesome, thanks again for the quick reply. i am somewhat enlightened.
although I'm sure that energy cannot be imaginary.

 

[Doc Al]

although I'm sure that energy cannot be imaginary.

But you're OK with imaginary momentum and mass?

 

[mibaokula]

But you're OK with imaginary momentum and mass?

i guess you're right. its all crazy!
i suppose negative or imaginary mass would sum up to gravity having a repulsive effect or something. i guess that's feasible. if mass is imaginary then using any momentum equation:
p=mv or Planck's constant/lambda; i suppose the momentum also is imaginary.
but the energy works out to be imaginary too (according to your equation) and that surely can't be right?
if you slot the numbers into the original equation i had before, you get a complex number divided by a complex number - a real number. how does that work??
surely tachyons cause the equations to lose their meaning - my main reason for thinking they don't exist

 

[mibaokula]

Actually, a large number divided by zero is not undefined, it's infinite.

In this case you are dividing zero by zero which is indeterminate but can be an exact number, you just have to solve the problem some other way.

try typing 1/0 into a calculator - its undefined. if you think about it this way, even if you had an infinite number of "zeros" you cannot get 1 - hence its not infinity but undefined.

likewise with 0/0, you might think that the answer is infinity as the other guys said there are many answers - infinite answers not just infinity. 1 x 0 is 0. 2 x 0 is 0. 3 x 0 is 0 and so on using this same logic.

so back to my original question, although its been answered, using that particular equation, photons must have an undefined amount of energy

 

[mibaokula]

The past few posts have given me a little confusion because when you're talking about a photon, you're separately taking the limit of $m \to 0$ and $v \to c$ but one is not dependent on the other. If I could decide I simply want to take m to 0, you should find the first term drops out and end of story, it's 0, not infinite.

i think (if einstein's special relativity is right), they are dependent on each other. for any object to travel at c m/s, it must have no mass. so if you plug the numbers in the equation, you get 0/0 - which sure is confusing!

 

[atyy]

i think (if einstein's special relativity is right), they are dependent on each other. for any object to travel at c m/s, it must have no mass. so if you plug the numbers in the equation, you get 0/0 - which sure is confusing!

In my understanding, the idea that massless particles can travel at the speed of light isn't a purely classical idea. In classical physics, light is a wave, not a particle. So the 0/0 thing is a "heuristic".

With quantum physics, everything becomes a wave, and the "mass" of a particle is seen to be a property of a wave. In this framework, since some waves have "mass", then light can be massless. However, there is no 0/0 - what this means is that all frequencies of light travel at the same speed. A massive particle is a wave in which different frequencies (energies) travel at different speeds.

 

[mibaokula]

In my understanding, the idea that massless particles can travel at the speed of light isn't a purely classical idea. In classical physics, light is a wave, not a particle. So the 0/0 thing is a "heuristic".

With quantum physics, everything becomes a wave, and the "mass" of a particle is seen to be a property of a wave. In this framework, since some waves have "mass", then light can be massless. However, there is no 0/0 - what this means is that all frequencies of light travel at the same speed. A massive particle is a wave in which different frequencies (energies) travel at different speeds.

dunno about that but in the standard model, the "photon" has a mass of zero. so i kinda put that in the equation. i do not believe any electromagnetic wave has a mass although they can have momentum depending on their wavelength

 

[atyy]

dunno about that but in the standard model, the "photon" has a mass of zero. so i kinda put that in the equation. i do not believe any electromagnetic wave has a mass although they can have momentum depending on their wavelength

A massless photon simply means that all frequencies of light travel at the same speed. The equation is not strictly valid for massless particles, although it is suggestive that such particles can exist.

 

[mibaokula]

A massless photon simply means that all frequencies of light travel at the same speed. The equation is not strictly valid for massless particles, although it is suggestive that such particles can exist.

yeah, but the second equation explains massless particles, right?

 

[atyy]

yeah, but the second equation explains massless particles, right?

The equation for massless particles is 1/wavelength=constantXfrequency.

 

[mibaokula]

That's it. You need to use this more general equation, which works for all particles, including massless photons:

$$E^2 = mc^4 + p^2c^2$$

(p is the momentum of the particle.)

i meant the equation which i have now attached in this post

 

[ghwellsjr]

try typing 1/0 into a calculator - its undefined. if you think about it this way, even if you had an infinite number of "zeros" you cannot get 1 - hence its not infinity but undefined.

likewise with 0/0, you might think that the answer is infinity as the other guys said there are many answers - infinite answers not just infinity. 1 x 0 is 0. 2 x 0 is 0. 3 x 0 is 0 and so on using this same logic.

so back to my original question, although its been answered, using that particular equation, photons must have an undefined amount of energy

What does your calculator say if you type in 0/0? What does it say if you take the square root of -1?

0/0 is not infinite, it is indeterminate because it can be anything.

 

[mibaokula]

i just realized i misunderstood your point. i think we meant the same thing. 0/0 is indeterminate because it has an infinite number of answers - i.e. any real number. what i meant to say is that it is not enough to say the answer is 1, 0, or infinity

 

[ghwellsjr]

i just realized i misunderstood your point. i think we meant the same thing. 0/0 is indeterminate because it has an infinite number of answers - i.e. any real number. what i meant to say is that it is not enough to say the answer is 1, 0, or infinity

If you just ask the question, what is 0/0, there are an infinite number of answers, but that doesn't mean that your equation for the energy of a massless particle has an infinite number of answers, it just means that you need to determine the answer some other way.

 

[mibaokula]

in other words, that equation doesn't fully describe light (which has no mass)
but would it describe tachyons with negative/imaginary mass (assuming they exist - which i really don't believe at the moment)?