# Faster than the speed-of-light

#### Elvin12

Einstein claims that is impossible for any object (with a mass) to reach the speed of light, however I have many questions that I would appreciate if were answered.
To start with, the Largest Hadron Collider can accelerate an electron to a speed equivalent to 99.9999991% of the speed of light and the weight of that electron would be equal to the weight of a train moving at 80 km/h, which is huge (considering the initial weight of the electron), yet not infinitely huge. And it was only a tiny city on the border between Switzerland and France that was able to do it.
Moreover, I once read that it would physically impossible for a space ship moving at the speed of light to reach the end of the universe, because universe is expanding at the speed higher than the speed of light. The only way something could move at the speed of light is if it had an infinite amount of energy, but then that doesn't make sense. For the universe to have an infinite amount of energy, the space it must be contained in must be infinite also, but since the universe is expanding, it can't be infinitely large, because otherwise it wouldn't have a value to its size.
I also read somewhere else that in first second (up to 1^-37 seconds) after big-bang the laws of physics were different, in fact the 4 forces (electromagnetic, gravitational, weak and strong) didn't exist but there was only one force (Unified Force, where gravitational force was as strong as other forces) which later split into two different forces which after split again to become the 4 that we know now. So I thought that maybe in those first second, there was no limit to the speed, and it was that initial push that set us off on the speed faster than the speed of light, and then following the Newton's law of motion, the universe kept increasing at that speed because there were no external forces (assuming that there is nothing outside the universe) that would slow it down.
And also wouldn't the particles during the "inflation" expand at a faster-than-the-speed-of-light velocity?
Any opinion or answer will be appreciated,
Thanks

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#### mgb_phys

Homework Helper
The universe can and does expand faster than light.
Relativity prevents information (which in this case also means mass) going faster than light. There is nothing to stop two points moving apart faster than light - they just can't send information from one to the other.

#### Quinzio

Well, if I'm not wrong in understanding the expanding universe, it's not really that two galaxy at the opposite ends of the universe are moving apart at a pace faster than light.
Rather, there is a creation of space in which the galaxy and all the universe is immersed.
Just like if on a ballon I draw two point, then I inflate the ballon. The point are not exactly moving, it's the baloon surface that is expanding.
Please so. correct me if I'm wrong.

#### Quinzio

If I may add something about travelling faster than light, let's say I make a trip 1 light year long.
I begin the trip, I accelerate fast to 3/5 of c, travel 4/5 of a year, the I stop.
My average speed has been v= s/t where space is 1 light year and time is 4/5 years.
Can't I say I travelled at (5/4)c ?
(Even if relative speed respect my original frame never exceeded c)

#### JesseM

The universe can and does expand faster than light.
Relativity prevents information (which in this case also means mass) going faster than light. There is nothing to stop two points moving apart faster than light - they just can't send information from one to the other.
Are you talking about inertial coordinate systems in SR, or non-inertial coordinate systems in GR? If the former, then it's impossible for two objects with mass to have a relative velocity faster than c, so "there is nothing to stop two points moving apart faster than light" would be wrong (unless you're talking about the 'closing speed' between two objects as seen in the inertial rest frame of a third object, rather than the velocity of one of two object in the other's rest frame, which is how 'relative velocity' is normally defined in SR). If you're talking about non-inertial GR coordinate systems, the speed-of-light limit only applies in inertial frames, so it is quite possible for things (including photons) to move faster than c in a non-inertial coordinate systems and to pass information to each other. As mentioned on p. 4 of http://www.scientificamerican.com/article.cfm?id=misconceptions-about-the-2005-03 [Broken], we actually can observe galaxies that are receding from us "faster than light" in the standard cosmological coordinate system, so clearly we are getting information from these galaxies:
According to the usual Doppler formula, objects whose velocity through space approaches light speed have redshifts that approach infinity. Their wavelengths become too long to observe. If that were true for galaxies, the most distant visible objects in the sky would be receding at velocities just shy of the speed of light. But the cosmological redshift formula leads to a different conclusion. In the current standard model of cosmology, galaxies with a redshift of about 1.5--that is, whose light has a wavelength 150 percent longer than the laboratory reference value--are receding at the speed of light. Astronomers have observed about 1,000 galaxies with redshifts larger than 1.5. That is, they have observed about 1,000 objects receding from us faster than the speed of light. Equivalently, we are receding from those galaxies faster than the speed of light. The radiation of the cosmic microwave background has traveled even farther and has a redshift of about 1,000. When the hot plasma of the early universe emitted the radiation we now see, it was receding from our location at about 50 times the speed of light.

Running to Stay Still
The idea of seeing faster-than-light galaxies may sound mystical, but it is made possible by changes in the expansion rate. Imagine a light beam that is farther than the Hubble distance of 14 billion light-years and trying to travel in our direction. It is moving toward us at the speed of light with respect to its local space, but its local space is receding from us faster than the speed of light. Although the light beam is traveling toward us at the maximum speed possible, it cannot keep up with the stretching of space. It is a bit like a child trying to run the wrong way on a moving sidewalk. Photons at the Hubble distance are like the Red Queen and Alice, running as fast as they can just to stay in the same place.

One might conclude that the light beyond the Hubble distance would never reach us and that its source would be forever undetectable. But the Hubble distance is not fixed, because the Hubble constant, on which it depends, changes with time. In particular, the constant is proportional to the rate of increase in the distance between two galaxies, divided by that distance. (Any two galaxies can be used for this calculation.) In models of the universe that fit the observational data, the denominator increases faster than the numerator, so the Hubble constant decreases. In this way, the Hubble distance gets larger. As it does, light that was initially just outside the Hubble distance and receding from us can come within the Hubble distance. The photons then find themselves in a region of space that is receding slower than the speed of light. Thereafter they can approach us.

The galaxy they came from, though, may continue to recede superluminally. Thus, we can observe light from galaxies that have always been and will always be receding faster than the speed of light. Another way to put it is that the Hubble distance is not fixed and does not mark the edge of the observable universe.

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#### JesseM

Einstein claims that is impossible for any object (with a mass) to reach the speed of light, however I have many questions that I would appreciate if were answered.
To start with, the Largest Hadron Collider can accelerate an electron to a speed equivalent to 99.9999991% of the speed of light and the weight of that electron would be equal to the weight of a train moving at 80 km/h, which is huge (considering the initial weight of the electron), yet not infinitely huge. And it was only a tiny city on the border between Switzerland and France that was able to do it.
But no matter how close you get, you can never actually get a particle with mass to reach the speed of light; that last 0.0000009% would increase the particle's weight by infinity (or to put it another way, it would take an infinite amount of energy to increase the particle's velocity from 99.9999991% to 100%).
Elvin12 said:
Moreover, I once read that it would physically impossible for a space ship moving at the speed of light to reach the end of the universe, because universe is expanding at the speed higher than the speed of light.
It would be physically impossible for a space ship to travel at the speed of light "locally", period. To understand what I mean by "locally" here, I'll quote something I wrote on another thread:
One thing to keep in mind about questions like this is that there is no "objective" notion of speed in relativity, you can only define speed relative to a particular spacetime coordinate system. In special relativity where spacetime is not curved, the idea that the speed of light is always equal to the constant c, and that nothing can move faster than c, is only supposed to hold in inertial coordinate systems (a coordinate system where any object at a fixed spatial coordinate would be moving inertially, i.e. not accelerating). Even in SR things can move faster than c in non-inertial coordinate systems. In general relativity, it isn't possible to have inertial coordinate systems in a large region of curved spacetime, so all coordinate systems covering such a large region would be non-inertial ones, and there'd be no restriction about things not going faster than c. But in GR, the "equivalence principle" (see the nice article http://www.aei.mpg.de/einsteinOnline/en/spotlights/equivalence_principle/index.html [Broken]) says that if you zoom in on a very small region of a curved spacetime, the laws of physics seen by an observer in freefall (not being influenced by any non-gravitational forces) will look almost exactly like those seen by an inertial observer in flat spacetime, with the agreement becoming perfect in the limit as the size of the region becomes arbitrarily close to zero. So in this sense we can say that even in GR, it must be true that at every point in spacetime a freefalling observer looking at his own local region would see that locally nothing can travel faster than c, and this would still be true in any one of those galaxies which in the context of a more global cosmological coordinate system may be moving away from us faster than c.
Elvin12 said:
I also read somewhere else that in first second (up to 1^-37 seconds) after big-bang
A lot less than the first second--do you understand that 10^-37 seconds is a tiny fraction of a second? Also, gravity would only have been united with the other forces for about 10^-44 seconds, i.e. Planck time, after that it'd be separate, then after about 10^-36 seconds the strong force would become separate from the electroweak force, and finally the electromagnetic and weak force would become separate after about 10^-12 seconds (see the timeline here).
Elvin12 said:
the laws of physics were different, in fact the 4 forces (electromagnetic, gravitational, weak and strong) didn't exist but there was only one force (Unified Force, where gravitational force was as strong as other forces) which later split into two different forces which after split again to become the 4 that we know now.
I think all the mainstream theories of the grand unification era, when strong, weak and electromagnetic were united into one, would still be relativistic (Lorentz-symmetric) theories where c would be the top speed. As for the TOE era when gravity is united with the other three forces, there aren't really any complete theories to deal with this, but the attempts I know of like string theory are relativistic too.
Elvin12 said:
And also wouldn't the particles during the "inflation" expand at a faster-than-the-speed-of-light velocity?
Not in the "local" sense I discussed above.

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#### Elvin12

The universe can and does expand faster than light.
Not in the "local" sense I discussed above.
So if the universe expands faster than the speed of light and it is not through inflation, what else would expand the space in the universe? and since vacuum in the universe is not absolute vacuum, would a random generation of space defy one of the laws of physics? Matter can not be created nor destroyed?

#### JesseM

So if the universe expands faster than the speed of light and it is not through inflation, what else would expand the space in the universe?
Do you understand that nothing moves faster than light if you pick a local inertial coordinate system to analyze it? And do you also understand that when physicists say that nothing can move faster than light they are only talking about what is true in inertial coordinate systems, that there is no restriction in the laws of physics that says things can't move faster than c in a non-inertial coordinate system? This need not have anything to do with the "expansion of space", even in a flat SR spacetime you can pick a non-inertial coordinate system where things move faster than c, even though if the same objects were analyzed from the perspective of an inertial coordinate system we would find that none of them were moving faster than c. For example, I could define a coordinate system in my room in which my head was moving away from my computer monitor at 1000c...
Elvin12 said:
and since vacuum in the universe is not absolute vacuum, would a random generation of space defy one of the laws of physics? Matter can not be created nor destroyed?
I don't understand what you mean by "a random generation of space". The expansion of space in general relativity is not random, it follows in a deterministic way from the motion and distribution of matter in the universe.

#### TMFKAN64

let's say I make a trip 1 light year long.
I begin the trip, I accelerate fast to 3/5 of c, travel 4/5 of a year, the I stop.
My average speed has been v= s/t where space is 1 light year and time is 4/5 years.
Can't I say I travelled at (5/4)c ?
I don't know... let's say you make a trip 100 miles long. You accelerate to 60mph, travel for an hour, and stop. Your average speed has been v=s/t where the distance is 100 miles and time is 1 hour. Can't you say you travelled at 100 mph? #### Elvin12

if by v you mean velocity, then shouldn't it be v=d/t (velocity = distance/time) as opposed to v=s/t (velocity = speed/time) as velocity is an s (speed) but with direction.

#### TMFKAN64

I just copied what Quinzo used in #4 above. That post refered to "space" which is why s was used for distance.

I suppose instead of just trying to make a joke, I should have pointed out the importance of making all of the measurements in the same frame of reference when you calculate velocity, but...

#### pervect

Staff Emeritus
If I may add something about travelling faster than light, let's say I make a trip 1 light year long.
I begin the trip, I accelerate fast to 3/5 of c, travel 4/5 of a year, the I stop.
My average speed has been v= s/t where space is 1 light year and time is 4/5 years.
Can't I say I travelled at (5/4)c ?
(Even if relative speed respect my original frame never exceeded c)
What you are describing is not a velocity - but it's a useful concept, it just needs another name. Some authors, see for instance http://arxiv.org/abs/physics/0608040 , call it a celerity.

I'll take the liberty of quoting from the paper, since I suspect people won't read it otherwise. (the numbers are footnotes which are references in the original paper).

Sometimes, one single quantity in a theory splits into several distinct quantities in the
generalised theory. One example is the velocity in classical mechanics. In the widest sense, velocity or speed means the covered distance divided by the time needed to cover it. This is uncritical in classical physics, where time and distance are well defined operational concepts that are independent of the frame of reference, in which they are measured. In Special Relativity, these concepts depend on the frame of reference in which they are defined, if the frames are not at rest with respect to each other. This makes it necessary to distinguish between the different possibilities regarding the frame of reference in which the spatial and the temporal intervals are measured. This is best illustrated with a common situation of measuring the velocity of a rolling car. Firstly, the velocity of the car can be measured by driving past kilometer posts and reading the time at the moment of passing the post on synchronized watches mounted on the posts. Secondly, the driver can also measure the velocity by reading the corresponding times on a clock which is travelling with the car. Thirdly, a person with clock standing beside the street can measure the times on his clock at the moments, when the front and the rear ends of the car are passing him. The travelled distance is then taken from a measurement of the length of the car in the frame of reference of the car. A fourth possibility measures the velocity of the car up to an arbitrary constant by measuring its acceleration using an accelerometer travelling with the car, e.g. by measuring a force and using Newtons Second Law, and integrates the measured acceleration over the time measured with a clock, also travelling with the car.

In classical mechanics, all four measurements are equivalent and give the same value for
the velocity. In Special Relativity, the first possiblility gives the coordinate velocity, which
is often referred to as the genuine velocity. The second and third possibilities are equivalent, but are hybrid definitions of the speed. The temporal and spatial intervals are measured in different frames of reference. This speed is sometimes called celerity [1, 2], or proper velocity [3, 4]. In addition, proper velocity is the spatial part of the vector of four-velocity . The fourth definition of a speed, sometimes called rapidity [1, 2], is somewhat distinct from the other concepts of speed in so far as it can only be determined as a change of speed. The need to measure an acceleration in the moving frame by means of measuring a force qualifies rapidity to bridge kinematics and dynamics. This seems to be not critical in classical mechanics, if the concept of force is accepted as an operational quantity. However, it can also be used to determine the relativistic version of Newtons Second Law if viewed from the accelerated frame of reference.

#### cfrogue

Do you understand that nothing moves faster than light if you pick a local inertial coordinate system to analyze it? And do you also understand that when physicists say that nothing can move faster than light they are only talking about what is true in inertial coordinate systems, that there is no restriction in the laws of physics that says things can't move faster than c in a non-inertial coordinate system? This need not have anything to do with the "expansion of space", even in a flat SR spacetime you can pick a non-inertial coordinate system where things move faster than c, even though if the same objects were analyzed from the perspective of an inertial coordinate system we would find that none of them were moving faster than c. For example, I could define a coordinate system in my room in which my head was moving away from my computer monitor at 1000c...

I don't understand what you mean by "a random generation of space". The expansion of space in general relativity is not random, it follows in a deterministic way from the motion and distribution of matter in the universe.
Please show me the math for this.

Thank you.

#### Al68

Please show me the math for this.

Thank you.
A simple example is the reference in which I am currently at rest (earth's surface). The star Sirius is 8.6 light years away, and therefore moving at almost 20,000 times c relative to earth's surface.

(2)(3.14)(8.6 ly)(365 rev/yr) = ~20,000 c = coordinate velocity of Sirius relative to earth's surface.

#### Mentallic

Homework Helper
A simple example is the reference in which I am currently at rest (earth's surface). The star Sirius is 8.6 light years away, and therefore moving at almost 20,000 times c relative to earth's surface.

(2)(3.14)(8.6 ly)(365 rev/yr) = ~20,000 c = coordinate velocity of Sirius relative to earth's surface.
wait... what?
I'm not understanding your logic in multiplying by the amount of days Earth has in a year. Firstly you have the circumference of the circle $2\pi r$ and then...?

#### cfrogue

A simple example is the reference in which I am currently at rest (earth's surface). The star Sirius is 8.6 light years away, and therefore moving at almost 20,000 times c relative to earth's surface.

(2)(3.14)(8.6 ly)(365 rev/yr) = ~20,000 c = coordinate velocity of Sirius relative to earth's surface.
Use SR's multiple velocity equation.

No,

I want to see this in non-inertial frames.

#### cfrogue

wait... what?
I'm not understanding your logic in multiplying by the amount of days Earth has in a year. Firstly you have the circumference of the circle $2\pi r$ and then...?

#### Al68

wait... what?
I'm not understanding your logic in multiplying by the amount of days Earth has in a year. Firstly you have the circumference of the circle $2\pi r$ and then...?
Just to make the units work out. Alternatively, we could say that Sirius is (8.6)(365) light-days away. Either way, Sirius' velocity would be about 20,000 light-years/year or 20,000 light-days/day, or 20,000 c.

#### Al68

Use SR's multiple velocity equation.

No,

I want to see this in non-inertial frames.
Earth's surface is non-inertial. I was just giving a simple example of an object's coordinate velocity exceeding c in a non-inertial frame.

#### JesseM

Use SR's multiple velocity equation.
Only applies in inertial frames.
cfrogue said:
No,

I want to see this in non-inertial frames.
A rotating frame in which the surface of the Earth is at rest is a non-inertial one. Since points on the surface of the Earth aren't moving inertially, there is no inertial frame where they are permanently at rest.

Anyway, if you want something more mathematical, let x,t be the coordinates in some inertial frame. Then define the following coordinate transformation to get a non-inertial frame:

x' = x + t*100c
t' = t

If an object is at rest in the inertial frame, then it is moving at 100c in this non-inertial frame. For example, suppose one point on the object's worldline has coordinates x=0 light-seconds, t=0 seconds in the inertial frame, and another point has coordinates x=0 l.s,t=10 s in the inertial frame. Then in the non-inertial frame, the first point has coordinates x'=0 l.s.,t'=0 s and the second has coordinates x'=1000 l.s., t'=10 s. So, in this non-inertial system, in 10 seconds the object has moved 1000 light-seconds, a speed of 100c.

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#### yuiop

The universe can and does expand faster than light.
Relativity prevents information (which in this case also means mass) going faster than light. There is nothing to stop two points moving apart faster than light - they just can't send information from one to the other.
Let us say we have a distant galaxy that appears to be moving away from us at 5c. We can send information to that galaxy and obviously the galaxy can send information to us, or we would not be able to see it. I suspect mgb meant to say "they just can't send information from one to the other" faster than the speed of light.

The statement "There is nothing to stop two points moving apart faster than light" also needs clarification. mgb was careful to use the word "points" rather than particles so I assume he is referring to two locations in space moving away from each other. Now the trouble is that it is difficult to visualise how to measure the velocity of one piece of vacuum relative to another piece of vacuum, so it might be helpful in these discussions to think of the vacuum as a fluid with special properties, the presence of which can be inferred from those properties. It is a bit like air. We may not be able to directly see air, but few of us have any doubts about the existence of air, as we can readily see the effects of movement of air in our every day lives.

Now let us say we observe a particle (A) going to the left at 0.9c and another particle (B) going to the right at 0.9c then in a naive sense we might say they are going apart from each other at 1.8c. However, if we use the equation for relativistic addition of velocities, the result is less than c. This means that from the point of view of each of the particles, the other particle is receding at less than c and this is always the case locally. So why is that when A and B are many light years apart, they can measure each other to be moving at velocities greater than c? The answer is that SR can not account for this because it assumes a flat spacetime everywhere. In Cosmology over vast distances the spacetime is like an expanding fluid and objects are restricted to moving at less than c relative to that spacetime fluid. The result is that the velocity of the particle combined with the velocity of the fluid can add up to a value greater than c. On the other hand, two nearby objects that are moving in the same direction, are moving in the same local patch of fluid and because they are constrained to moving at less than c relative to the local fluid, then they are constrained to moving at less than c relative to each other locally. You could for example imagine the objects as little powered boats that have a maximum velocity of cb relative to calm water. In a river two such boats could have a relative velocity greater than cb is one is slow water and the other is in fast flowing water. Now if we had a rapid whirlpool, a small un-powered boat on the perimeter of the whirlpool could have a tangential velocity much great than the cb relative to a non rotating island in the centre but the important thing is that the boat always has a velocity of less than cb relative to the water local to it. This is some ways similar to the case of Sirius star appearing to have a tangential velocity relative to the Earth of 2000c. The important thing is that the star always has a velocity less than c relative to the spacetime fluid that is swirling around the Earth (in the frame where the Earth is regarded as non rotating.) So in some ways, spacetime can be visualised as a dynamic fluid that can expand, contract, curve, warp, rotate or flow and all motion is less than c relative to the spacetime fluid in the immediate vicinity of the particle, giving a informal "picture" of the meaning of local.

#### Ich

There is no "spacetime fluid" relative to which things are moving or not. You know, "The idea of motion may not be applied to it." (A.E.)
Further, spacetime curvature is not necessary for this kind of superluminal speed. Superluminal speeds easily occur when you measure distance vs. proper time, as it is done in cosmology.

#### JesseM

wait... what?
I'm not understanding your logic in multiplying by the amount of days Earth has in a year. Firstly you have the circumference of the circle $2\pi r$ and then...?
I think he was taking the fact that Sirius travels (2)(3.14)(8.6 ly) per day in this rotating coordinate system (since in this system Earth is fixed while the stars rotate around it), and then dividing it by c=(1/365 ly/day), in order to figure out the factor by which Sirius' velocity exceeds c in this coordinate system. Analogously, if I know an object travels 150,000 km per second, I can divide by c=300,000 km/s to conclude the object is traveling at 0.5c.

#### Elvin12

Now if we had a rapid whirlpool, a small un-powered boat on the perimeter of the whirlpool could have a tangential velocity much great than the cb relative to a non rotating island in the centre but the important thing is that the boat always has a velocity of less than cb relative to the water local to it. This is some ways similar to the case of Sirius star appearing to have a tangential velocity relative to the Earth of 2000c. The important thing is that the star always has a velocity less than c relative to the spacetime fluid that is swirling around the Earth (in the frame where the Earth is regarded as non rotating.) So in some ways, spacetime can be visualised as a dynamic fluid that can expand, contract, curve, warp, rotate or flow and all motion is less than c relative to the spacetime fluid in the immediate vicinity of the particle, giving a informal "picture" of the meaning of local.
I spent the last three days trying to figure this out, and at last i get it, thanks so much kev.
But one thing that still puzzles me is the theory of inflation. The speed at which our universe was increasing during that miserable time (the period of inflation) was thousands times the speed of light on a non-inertial (in my understanding) coordinate plate?