In fact "c" is just the abbreviation for the speed of light (as far as I know, it is also mostly used for the speed of light in a medium) - but the limit for velocities (or transmittion of information) is the vacuum speed of light.

Thanks Johannes,

I think I said that in my question, but anyway irrespective on nomenclature and conventions (which are of course important), I still have the same doubt (I will try to reformulate it).

Is the speed of light in vacuum "per se" the upper level limit of speed?

or

Is the speed of light in vacuum is already limited by a limit of a superior order?

or as another formulation:

nothing can travel faster than the speed of light in vacuum, but can anything travel AS FAST AS the speed of light in vacuum?

If something can travel AS FAST AS the speed of light in vacuum, it means to me (or I would interpret) that the limit is imposed by a superior order, since I could equally say nothing can travel faster than "this another thing"

If nothing can travel AS FAST AS the speed of light in vacuum, it means to me (or I would interpret) that the limit is the speed of light in vacuum "itself".

Finally, do you see any open non-trivial consequences of either answer to the question?

Thanks,

sam

According to the classic interpretations, the only thing that can travel at the speed of light in vacuum is ... light.

Samuel, objects with rest mass acquire an extra 'dynamic' mass when in motion. Relativity theory provides a formula for that. This mass diverges as the velocity approaches c. Objects without zero mass such as light (or photons, of you like) can travel at c because that limitation does not apply.

In classical electrodynamics you find the value of c to be determined by \epsilon_0 and \mu_0. It is the observation that this value is valid for all inertial frames which ultimately leads to the Lorentz-transformation, including the above mentioned mass-velocity formula. From this perspective, it is not so surprising that the velocity of light is central here.

It is maybe worth mentioning that in the presence of matter, phase velocities above c are possible (the refractive index of all materials is smaller than unity in the hard x-ray regime). The notion of "travel" or "transmission of information" is, however, linked to the group velocity, which does not exceed c.

You can think of the value of the speed of light as a limit statement.

In the Limit , as the mass of an object approaches zero, then the velocity

of the object approaches an upper limiting value asymptotically such the

velocity approaches a constant. That constant is C, the speed of light.

For E = M C^2, then C = square root of ( E / M ) where M approaches

a value of zero, but can never reach zero. M can get quite small, but it

can never reach zero. The energy " E " can get quite large, but it can

never reach infinity. The result is that C is the square root of a large,

but, finite number.

If you think about a Gamma Ray Burst from an exploding star, it is the

absolute fastest, highest energy, lowest mass detectable product which

is received first after it traveled millions of light years across the universe,

and it gets here a very brief time before the star brightens into a Super

Nova. The velocity of light and the velocity of Gamma Rays are statistically

identical, and it could be that the Gamma Rays actually leave the Star

just before the light leaves the Star.

In fact, if you think about it, the difference in the time between the arrival

of the gamma rays and the arrival of the light should be directly

proportional to the elapsed time and the distance that both traveled.

The greater the distance, the longer the time gap between the

arrival of the two frequencies.

It is also possible that the Gamma Rays travel a straighter path

way from the source to us, so we receive them a bit sooner, as light can be

bent around large gravitational objects like galaxies, so the pathway was

just a tiny bit longer for the light than the Gamma Rays.

Food for thought.

In the Limit, all Energy "E", no mater how powerful,

has a maximum possible velocity near C, the velocity of Light.

Michal, what do you consider to be a difference in nature between "gamma rays" and "light"? To me, gamma rays *are* light, i.e. e/m radiation! (just historically, they have a different name, since they are associated with radioactivity)

Just Energy, and Frequency, and the ability to be bent around large

gravitational objects like big stars, and galaxies.

We already know that the visible spectrum can be bent around stars,

so we assume all the other frequencies can also be bent around

very large gravitational objects, but do we really know, or is there an

upper limit where energy and frequency are high enough, and

mass is low enough that bending around gravitational objects

is no longer possible ?

We assume, but ,do we know, and, can we test if the upper end of

Gamma Rays can, and are, OR, can not and are not bent in their

pathway from the source to us.

Interesting speculation.

I can not remember where I saw it, but there was a cross-sectional

view of one of the Astronaut's helmets with a conical crater created

by a Gamma Ray, and the depth of the crater was about half the

thickness of the plexi-glass ( Lexan ? ) face shield.

Any way, back to the original question, is the velocity of

299, 792, 458 meters per second and absolute upper wall,

and the answer is no, but it is really close to that wall.

There is actually a range of values in the 450's to 460's, but like

many other things, if we standardize a value, it makes life a lot

easier.

Funny discussion. Maybe it's worth to remember that the discussions related to c and c to represent the "upper limit" (at least in Einstein's papers) is always related to an information exchange with an observer, i.e. the identification of light/photons as an information transfer between the observerd system and the observer. As such, c emerges as the asymptotic limit (i.e. infinity) of velocities. Things became messed up by the usual euclidean and, moreover, by Weyl's affine Interpretation of coordinates, but strictly identifying c from the viewpoint of an hyperbolic geometry and understandinc Lorentz transformations as hyperbolic transformations, everything works fine. And if Information exchange is related to c (in local coordinates), how do you want to observe velocities larger than c? ;-)

A very interesting read is the 1927 write up of the 1925, 1926 Michelson

Experiment, and the write up of the US Coast and Geodetic Survey that had to

determine a first order survey between Mount Wilson Station and Antionio

Station atop San Antonio Peak.

The station to station distance was 35,385.53 meters, and

the instrument to instrument distance was 35,425.1 meters.

Just to be clear, the speed of light is precisely c = 299,792,458 m/s as a matter of definition because the meter is defined to be 1/299,792,458 of the distance that light travels through vacuum in one second. This is the first resolution of the 17th General Conference on Weights and Measures from 1983 (http://www.bipm.org/jsp/en/ListCGPMResolution.jsp?CGPM=17). The present consensus on how use this definition to obtain a standard meter in practice was decided in 2002.

Also, building on Michael and Kai and Rolf's points, the Lorentz transformations suggest that the energy required to accelerate a massive object from rest to a given velocity relative to a specified frame of reference diverges as the speed approaches the speed of light. There is not enough energy in the universe, therefore, to accelerate the object to the speed of light, let alone to surpass the speed of light.

Interestingly enough, the same Lorentz transformation equations suggest that a massive object initially traveling faster than the speed of light could never decelerate to sub-luminal speeds. This is the starting point for the theory of tachyons. Sadly, there is no experimental evidence for the existence of tachyons.

@David: I'm not sure whether we are talking about the same "Lorentz transformations"... first of all there are no "suggestions", and secondly how do you represent a "massive object initially travelling faster than the speed of light"...? maybe you know of another Version of Lorentz transformations than me...

@Rolf: The Lorentz transformation equations, together with the work-energy theorem, yield the relation

E^2 = p^2c^2 + m_0^2 c^4

for relativistic energy of an object of rest mass m_0 moving with constant momentum p through an inertial frame of reference. The Lorentz-invariant momentum is related to the particle's speed by

p = m_0 \gamma v

where

\gamma = c/\sqrt{c^2 - v^2}.

We can rewrite the energy-momentum relation in terms of the speed to obtain the

relativistic energy

E = m_0 \gamma c^2

Some people define the relativisitic mass to be m = \gamma m_0 and interpret this result to mean that the object's mass diverges as its speed v approaches c. Others say that the energy required to accelerate the rest mass m_0 diverges. The two interpretations yield equivalent results for measurable quantities.

These results all rely on the properties of the Lorentz transformation equations. Different transformation equations (such as the Galilean transformation equations) yield different results for the kinetic energy.

A tachyon in special relativity would have a real-valued energy. Its mass, therefore, would have to be imaginary. Although this might sound outlandish, having a complex-valued coupling to the gravitational field is just unfamiliar. A community of theorists is actively working on the possibility that dark matter might consist of tachyons interacting with conventional matter gravitationally through this unfamiliar coupling.

Among the most highly cited papers in this field is Padmanabhan, Phys. Rev. D 66, 021301 (2002).

Even though this is unfamiliar and for not leaving anything out, I point out that in the Theory of Reference Frames the relativistic mass is given by m=mo(1-v^2/2c^2) with all consequences that it implies.

I see - we are going to discuss on various interpretations of certain people and communities... maybe even some eminences...?

I went through the insanity of studying Michelson's 1925, 1926 Mount Wilson to

San Antonio Peak experiment of the speed of light in smoke filled air, and

found that he made a 6 cm error in his distance calculation which gave him

his value of 299, 796 km per second.

Had he not made that error by rounding from 35,424.44 meters to

35,424.5 meters his value (using his correction ) would have been

299, 792 +/- 4 km/sec, rather than 299, 796 +/- 4 km / sec.

The secret is in the number of mirror faces exposed to the light,

which can be calculated by revolutions per second times the number

of faces per revolution. ( R.P.S. time facets )

(528 x 8) = (352 x 12) = (264 x 16) = 4224 for N round trips of light.

One round trip is actually 2 x 35,424.44 meters = 70,848.88 meter and not

2 x 35,424.5 meters = 70, 849 meters as Rounded by Michelson.

The air speed of light in smoky conditions was:

70,848.88 x 4224 = 299,265,669.1 meters per second = 0.998242821 c ( vac)

rather than

70,849.00 x 4224 = 299,266,176 meters per second = 0.998244512 c (vac).

In actuality he rounded both the distance up to 35,424.5

and he rounded the speed of

light in air up to 299, 270 km per second.

If he had rounded the distance down to 0.4 , and the air speed up,

he would have gotten 299,792 +/- 4 rather than 299,796 +/- 4 km per second.

We however must remember that he used slide rules and log tables to

do the calculations where we have 10 digit calculators that cost less than

$10.00 , and he did the experiment with a $ 10,000.00 grant, and it

took a Survey of the distance by the US Coast and Geodetic Survey

to the highest accuracy every achieved by any survey in any country

to get the station to station distance of 35,385.53 meters .

Michelson muffed it in the home run of the station to instrument

distance by using 38.97 meters rather than 38.91 meters.

Sloppy hand writing ?, or was it just a rounding procedural error.

OK,... I COULD admit I have no life, OR, I could say it was raining on Sunday,

and I had nothing else to do.

I WILL use the raining excuse !

Dear Michael,

the result of the Michelson experiment (absence of interference) is independent of the real value concerning the physical speed of light, because a prospective error in the calculation of that speed would have effect in all directions and would give always the same result: absence of interference.

The Michelson-Morley Interferometer experiment was testing the "ether"

for a difference in the speed of light in various directions with a null result.

The Michelson 1925, 1926 Experiment was to determine the speed of

light between Mount Wilson, and San Antonio Peak where the distance

between the two stations was determined by repeated measurements

for a first order survey by the US Coast and Geodetic Survey using

eight - 50 meter long INVAR tapes, four base lines, a calibrated 15 lb weight

to stretch the tapes, and numerous tower supports to support the tapes

so they would sag exactly like the tapes had sagged when they were

calibrated by the US Bureau of Standard at a very specific temperature.

The exact distance between the beginning and ending of the light

path is extremely important in determining the speed of light.

The error of only 1 mm in the distance would have changed the speed by

2 x 4224 / 1000 = 8.448 meters per second per round trip of light.

Since the light made eight round trips , the error of one millimeter

would be 8 x 8.448 = 67.584 meters / second per millimeter of

distance error.

For a 1 cm error it would be 10 x 67.584 = 675.84 meters / second per

centimeter of distance error.

Since his error is 6 cm, the total error is 6 x 675.84 = 4055.04 meters per second,

more or less. If we use the measuring limitations he used,

his error was 4 km / second.

His value obtained was the average of eight sets where he obtained a weight

mean of 299, 796 +/- 4. subtracting the 4 ( 6 cm distance error ),

he should have gotten 299, 792 +/- 4 km per second.

The key to how this works is the number of spinning mirror flats, and the number

of round trips per second of light travel that creates the interferometer.

For all of the spinning mirrors to give the same result, the Glass8, the Steel 8,

the glass 12, the steel 12, and the Glass 16 need to have the same number

of total flats past the same point in the same amount of time.

That number is 4224. That number multiplied by the exact distance

gives the value of c ( in smoky air ) which is lower than c ( Vacuum ) .

In his second season, he changed the mirror to 16 flats, and reduced the

speed to 264 revolutions per second ( 15, 840 RPM ), He still had one

mirror explode at this spin rate.

His glass 16 flats mirror were made to be spun by air in either direction

so he could average the errors.

Since he already had Foucault's number of 299, 796, he had to have

known almost exactly how long the base line needed to be in order

to get an even number of round trips to make the system work.

35,424.44 x 2 x ( 528 x 8 ) = 299,265,669.1

35,424.44 x 2 x ( 352 x 12 ) = 299,265,669.1

35,424.44 x 2 x ( 264 x 16 ) = 299,265,669.1

He deliberately made his base line a little less, so

he could put his sending and receiving location behind the

station to station location so he would only need to add

a relatively small distance to his set up to be able to

make only additive adjustments to the distance.

The station to station distance is 35,385.53 , so the

total added one way light path is

35,424.44 - 35,385.53 = 38.91 meters ( one way light trip )

which is 77.82 meters ( two way light trips ).

The light travels 299,265,699.1 / ( 35,424.44 ) = 8448

one way trips, or 4224 two way trips each second.

His mistake was rounding:

35,424.50 - 35,385.53 = 38.97 meters.

Anyway, a Modern application would be to accurately determine the

density of air. As the density goes up, the speed goes down, and the

distance to the 4224 multiple would reduce.

As the density of air decreases, the speed of light in air goes up,

and the distance to the 4224 multiple would increase.

You could choose two peaks that are far enough apart that the spin

would be much reduced to some multiple of 1750 RPM, like 4 times

1750 = 7000 RPM, so the distance would be 80,160 meters apart,

or roughly 50 miles between mirrors.

One mirror would move in or out along the base line, and when the

resonance is reached, the distance, the velocity, and the air density

would be known.

I am sure they could do this more reliably with lasers, and atomic clocks.