Can you go so fast that after say one second, light has traveled less than a planck length further than you did (with respect to an outside observer of course)? Is c the actual speed limit, or is the speed limit slightly less than c? Maybe I'm not stating this properly and forgive me if not, but I think you know what I mean.
As your speed increases your inertia also increases and it becomes harder and harder to accelerate you further. [tex] m=m_0\gamma[/tex] [tex]\gamma=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}[/tex] [tex]F=ma=am_0\gamma[/tex]
A massive object can never achieve c. Assume that the total energy at rest is [tex]E_0=m_0c^2[/tex] The energy when the object reached speed [tex]v[/tex] is [tex]E_1=\gamma m_0c^2[/tex] The total work expended is [tex] \Delta W =E_1-E_0=(\gamma-1)m_0c^2[/tex] For [tex]v->c[/tex] [tex]\Delta W[/tex] goes to infinity.
I think that's an interesting question actually. The Planck length and related quantities aren't present in the theory of special relativity, so the answer within the framework of SR is clearly that the speed limit is exactly c. Light travels 299792458 meters in one second. You're asking if it's possible to travel more than 299792458-l_{P} in one second, in the universe we live in (as opposed to the one described by SR, where it certainly is possible since there's no Planck length). There's nothing special about a second, so we should be able to replace "one second" with any other unit of time in your question and still get the same answer. Let's choose "one Planck time". Since the speed of light is one Planck length in one Planck time, your question becomes "is it possible to travel more than zero Planck lengths in one Planck time"? It's funny that when you break it down like that, it appears that 0 and c are the only possible speeds, but we know that's not the case, so there's definitely something strange going on here. Maybe speed in a quantum theory of space-time is the probability that we will "jump" a Planck length in a Planck time. So I don't think anyone really knows the answer to your question, since there's no complete quantum theory of gravity. (A quantum theory of gravity would almost certainly also be a quantum theory of space-time). I wonder if the candidate theories like strings and loop quantum gravity have a clear answer to this question. Perhaps someone will tell us that in this thread. (Wink wink, nudge nudge).
Just imagine you are inside a spaceship travelling at the fastest possible speed less than c. You stand up and try to walk forward. Would you find some mysterious force preventing you from moving and thus breaking the "speed limit"? Of course not. So there can't be such a fastest speed. I'm no expert on quantum theory, but I don't think it is right to think of the Planck length as being "the smallest possible distance". It's more like "the smallest distance you can measure" (and even that's probably an over-simplification). Also, in quantum theory, it is usual to measure momentum rather than speed. There is no theoretical momentum limit. You might get a better answer by asking this question in the Quantum Physics forum. In the real Universe, there is a practical upper limit. The faster you go, the more energy you need, so eventually you would run out. So, to give a ludicrous example, your kinetic energy could never exceed the total energy of the whole Universe!
That's pretty much what I was getting at, but you expressed it much more eloquently. It seemed that something odd was going on with this, but I didn't know how to express it. Nice answer. Perhaps 0 and c ARE the only speeds and it only appears that they aren't. Formulated another way, is it possible to move 1/2 a planck length from your current position?
How can mass of an object be = to zero? How can the mass of an object be = to zero ? If mass is zero would it still exist? How can nothing be something? Does this mean that light can not be a particle ?
An object with zero mass may only exist if traveling at the speed of light. In this case, the object would show a nonzero relativistic mass equal to its kinetic energy. Example: photons.
It has a mass equivalent to it's kinetic energy, per Einstein's famous E=mc^2. If the kinetic energy is E then the relativistic mass of a massless object is m=E/c^2. Additional info here: http://en.wikipedia.org/wiki/Mass_in_special_relativity
Relativistic mass is just another name for the energy? according to Wikipedia. So we know that mass is an expression of energy from e=mc^2? So if an object has mass of 0 then 0 = E/c^2 = ? Im trying to make sense of this.
[tex]m_{0}^{2}c^{4}\gamma^{2}=E^2=p^{2}c^{2}+m_{0}^{2}c^{4}[/tex] If mass=0 then energy equals momentum times the speed of light.
Unfortunately questions like that are only well-defined within the framework of a theory, and we still don't have the theory we would need to even ask that question in a way that makes sense mathematically. (The concept of "position" is well-defined e.g. when we're talking about classical point particles moving in a space-time that can be represented mathematically by a smooth manifold, but there's no reason to believe that space and time in the actual universe is anything like a smooth manifold on small scales).