# How close to light speed can you theoretically get?

• Meatbot
In summary, questions were asked about the possibility of traveling faster than the speed of light and the role of the Planck length in this scenario. It was discussed how the concept of speed in a quantum theory of space-time may differ from classical theories and the limitations of our current understanding due to the lack of a complete quantum theory of gravity. The practical upper limit of speed was also mentioned, as well as the concept of zero mass and its implications. The connection between mass and existence was also brought up, as well as the question of whether light can be considered a particle due to its massless nature.
Meatbot
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.

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As your speed increases your inertia also increases and it becomes harder and harder to accelerate you further.

$$m=m_0\gamma$$
$$\gamma=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$$
$$F=ma=am_0\gamma$$

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Meatbot said:
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.

A massive object can never achieve c.
Assume that the total energy at rest is $$E_0=m_0c^2$$
The energy when the object reached speed $$v$$ is $$E_1=\gamma m_0c^2$$

The total work expended is $$\Delta W =E_1-E_0=(\gamma-1)m_0c^2$$
For $$v->c$$ $$\Delta W$$ goes to infinity.

There is no known limit to gamma.
The Planck length is not a limit on anything.

Meatbot said:
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)?

Yes.

Meatbot said:
Is c the actual speed limit, or is the speed limit slightly less than c?

'c' is an unattainable limit for objects whose mass is not zero.

Meatbot said:
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.
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-lP 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).

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Frederik,
trenchant analysis. A new Zeno paradox maybe ?

Meatbot said:
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.
Just imagine you are inside a spaceship traveling 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!

Fredrik said:
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.

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?

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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 ?

jlorda said:
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.

nanobug said:
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.

So are you saying that it does have a relative mass? I'm not sure what you are saying.

jlorda said:
So are you saying that it does have a relative mass? I'm not sure what you are saying.

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.

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 = ? I am trying to make sense of this.

jlorda said:
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 = ? I am trying to make sense of this.

$$m_{0}^{2}c^{4}\gamma^{2}=E^2=p^{2}c^{2}+m_{0}^{2}c^{4}$$

If mass=0 then energy equals momentum times the speed of light.

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Meatbot said:
Formulated another way, is it possible to move 1/2 a Planck length from your current position?
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).

## 1. How is the speed of light measured and what is its value?

The speed of light is measured using a variety of experiments, including the Michelson-Morley experiment and the Fizeau experiment. The current accepted value of the speed of light is approximately 299,792,458 meters per second in a vacuum.

## 2. Is it possible to travel at the speed of light?

According to Einstein's theory of relativity, it is not possible for any object with mass to reach the speed of light. As an object approaches the speed of light, its mass and energy increase infinitely, making it impossible to reach the speed of light.

## 3. What is the fastest speed that an object with mass can theoretically reach?

The fastest speed that an object with mass can theoretically reach is 99.99999999999999999999951% the speed of light, also known as the Lorentz factor. This is because as an object approaches the speed of light, it experiences infinite mass and energy.

## 4. Can anything travel faster than the speed of light?

According to our current understanding of physics, nothing can travel faster than the speed of light. The speed of light is considered to be the universal speed limit, as it is the maximum speed at which information and energy can travel.

## 5. How close to the speed of light have we been able to travel with current technology?

With current technology, the closest we have been able to travel to the speed of light is approximately 99.9999991% the speed of light. This was achieved by the Large Hadron Collider, which accelerates particles to high speeds for scientific experiments.

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