why is C the speed that it is?

[Iggy]

By my understanding, unity does not imply non-dimensionality.

 

In physics, a dimensionless physical constant (sometimes fundamental physical constant) is a universal physical constant. Because it is a dimensionless quantity, its numerical value is the same under all possible systems of units. Fundamental physical constant may also refer (as in NIST) to

universal but dimensional physical constants such as the speed of light c

, vacuum permittivity ε0, Planck's constant ħ, or the gravitational constant G.

http://en.wikipedia.org/wiki/Dimensionless_physical_constant

[Mr Skeptic]

c=1 is still equal to 1 under any system of units. It is not a dimensionless physical constant, it is a convenience. Think of it this way: whenever you see a unit of time/distance, multiply it by c with those units, and whenever you see a unit of distance/time, divide by c. The result is c=1 and unitless, in this system. This gets your c's out of the equation (c can be everywhere if you are doing certain subjects). You just need to multiply by the appropriate number of c's at the end to get physically significant results.

[Iggy]

c=1 is still equal to 1 under any system of units.

 

I might just disagree with the way you are saying it. I wouldn’t say, for example, that c = 1 in the international system of units. In that system of units c = 299,792,458 m/s. In other unit systems it is normalized to unity. But, I think you mean that c is always equal to 1 in natural units even if it has a different value in other unit systems.

 

It is not a dimensionless physical constant, it is a convenience.

 

Absolutely.

 

Think of it this way: whenever you see a unit of time/distance, multiply it by c with those units, and whenever you see a unit of distance/time, divide by c. The result is c=1 and unitless, in this system.

 

Ok, I think I see where you’re coming from. An example would be that beta of the Lorentz factor is unitless in any system of units:

 

[math]\gamma = \frac{1}{\sqrt{1 - \beta^2}}[/math]

 

And if you had a velocity in m/s (1.5E8 for example) and wanted beta you would:

 

[math]\beta = 1.5 \times 10^{8} \; m/s \left( \frac{1}{3 \times 10^{8} \; m/s} \right ) = 0.5[/math]

 

I agree. I would just avoid saying that c=1 is the same in any unit system which would mean to me, for example, that the equality’s lhs should be unity in SI or any other units:

 

[math]\displaystyle { c } = \frac{1}{\sqrt{\mu_0 \epsilon_0 }} [/math]

[StrontiDog]

In "natural" units c=1 and has no dimensions (along with hbar and G). Generally loved by theorists and reviled by experimentalists.

 

But the 'Natural' unit doesn't always work, now does it? Not in every case, anyway. I've seen C expressed as cm/s, m/s, mph, and I think, km/s. All of these will work just fine in every equation, as long as everything else has been converted. No problem, just arithmetic (as opposed to math. . .)

 

Wouldn't the implied units for C be: light-year/year, or even light-second/second? (no, the units do not cancel, a light-year isn't a year. . .)

 

ie: 1C = the speed of light = 1 light-year/year (Which I'm pretty sure is a true statement.)

 

Which is a little like one mile/h, when you think about it. It's still distance over time.

 

Problem is, that arithmetically (if that's a word) you should be able to pose the problem that:

 

If I make a trip to the Andromeda galaxy (~2 million light-years away) at a constant velocity of C³, how long will it take me to get there?

 

You can do the equation easily in m/s, but it makes no sense in light-years/year. All you have to do is to convert light-year to meters. Easy.

 

By the same token, you can't do the equation for a thousand mile trip if you do it at the cube of one mph, but you can do it if you convert to something other than one (say, 5280 ft/h, which should be the same thing.)

 

So I guess the point is, using C, instead of a number with units, works fine as long as you're not raising it to any exponent, of any kind. If you use it as part of a ratio, it works great. That whole one-to-any-power thing, can be a real pain. But where would we be without it?

 

Bill Wolfe

Edited March 6, 2010 by StrontiDog

[michel123456]

Once upon a time a teacher asked 2 young boys to make a small experiment, measuring experimentally the speed of a snail.

The boys went to the school yard and drawed parallel lines on the ground at 1 meter of distance. Then they put a snail on one line, and measure time with each one's clock.

After a few ours, the boys came back, arguing to each other for the result, because they both measured different things.

 

The first one had measured an average of 1 minute for each meter.

He had measured a speed of 1m/60sec, or 0,016666..m/s

 

The other boy had measured an average of 1 meter each 1 minute.

He had measured speed as 60sec/m, or 60 s/m.

 

Who was wrong, and who was right?

[KennyC]

Billy Bob was right, the snail won.

[Iggy]

If I make a trip to the Andromeda galaxy (~2 million light-years away) at a constant velocity of C³, how long will it take me to get there?

 

That sounds somewhat like asking how far 1 cubic meter is. The answer is that a cubic meter is not a measure of distance but is rather a measure of volume. c³ would have dimensionality of volume / cubic time which is not a velocity.

 

When quantities (be they like-dimensioned or unlike-dimensioned) are multiplied or divided by each other, their dimensional symbols are likewise multiplied or divided; this corresponds to vector addition or subtraction (on the exponents). When dimensioned quantities are raised to a rational power, the same is done to the dimensional symbols attached to those quantities; this corresponds to scalar multiplication on the exponents.

 

http://en.wikipedia.org/wiki/Dimensional_analysis#Mathematical_properties

 

A physical quantity is expressed as the product of a numerical value and a physical unit, not merely a number. It does not depend on the unit distance (1 km is the same as 1000 m), although the number depends on the unit.

 

http://en.wikipedia.org/wiki/Scalar_%28physics%29

Edited March 6, 2010 by Iggy
changed 'area' to 'volume'

[StrontiDog]

That sounds somewhat like asking how far 1 cubic meter is. The answer is that a cubic meter is not a measure of distance but is rather a measure of volume. c³ would have dimensionality of volume / cubic time which is not a velocity.

 

Not at all, Iggy. I might not be able to do this with a space ship, but I can sure do it with a pencil and a piece of paper. Strange, but true. This is not a: how many pints in a ton, question. (pints of what? feathers. . .mercury?)

 

The numbers work. In my example, how else would you write the square of ten miles per hour than (10)²mph, or 100 mph? I'm not really squaring the units, I'm just manipulating the velocity to an exponent. Hey, I'm not the one agreeing that C is dimensionless. We can leave the dimensions out in some equations, but they are important in others. C is still a velocity and that means it has to have units of distance/time.

 

Take a simple kinetic energy equation: E=1/2mv²

 

If all I ask is: How does KE increase if the speed of a bullet is doubled?

 

Simple answer: 4 times.

 

I haven't used a single unit of mass or velocity or Energy to solve this equation. Does this make all of those 'dimensionless'?

 

My whole point is that C really isn't dimensionless, either. There are only two times when we don't need the dimensions. One is when they are cancelled-out by some other velocity taken to the same exponential level (v²/C²), and when all we’re looking for is a relationship (the bullet equation, above.)

 

Something like G is dimensionless. It is also empirically deduced, not a calculated number, at all. It is a relationship, a modifier constant. And it doesn't matter what the other units for mass and distance are in, it's always the same. All long as you're not mixing miles with kg, that number will never change.

 

Am I missing something, here?

 

Bill Wolfe

[Iggy]

If you want to double or triple a physical quantity then you multiply it by the number 2 or 3. Numbers are dimensionless. You don't square or cube them. The units of velocity are squared in KE=1/2mv^2, like I said, m^2/s^2.

 

The gravitational constant is dimensionful, with dimensions of:

The dimensions assigned to the gravitational constant in the equation above — length cubed, divided by mass and by time squared (in SI units, metres cubed per kilogram per second squared) — are those needed to balance the units of measurements in gravitational equations. However, these dimensions have fundamental significance in terms of Planck units: when expressed in SI units, the gravitational constant is dimensionally and numerically equal to the cube of the Planck length divided by the Planck mass and by the square of Planck time.

http://en.wikipedia.org/wiki/Gravitational_constant#Dimensions.2C_units_and_magnitude

 

Looking at the last few posts again, it looks like I may have missed that Mr Skeptic and Swansont may have been talking about geometric units. In that case I agree that the speed of light is dimensionless since time is measured in the same units as length.

 

I think we're off topic, in any case.

[StrontiDog]

If you want to double or triple a physical quantity then you multiply it by the number 2 or 3. Numbers are dimensionless. You don't square or cube them. The units of velocity are squared in KE=1/2mv^2, like I said, m^2/s^2.

 

The gravitational constant is dimensionful, with dimensions of:. . .

 

I think we're off topic, in any case.

 

Man, when you're right, your right. I could have sworn G was dimensionless. Though if I'd thought about what units Force are measured in, I would have known there was something wrong with that.

 

I bow to your superior knowledge.

 

And yeah, way off subect. With much of it just being semantics.

 

Thanks for setting me straight.

 

Bill Wolfe

[galen]

Heres an interesting thought

If we define the speed of light as 1 cosmological unit per second

Then,

e=m cosmological energy units

Where does this leave our mass definition?

Edited March 19, 2010 by galen

[michel123456]

Heres an interesting thought

If we define the speed of light as 1 cosmological unit per second

Then,

e=m cosmological energy units

Where does this leave our mass definition?

 

Square?

[Radical Edward]

Heres an interesting thought

If we define the speed of light as 1 cosmological unit per second

Then,

e=m cosmological energy units

Where does this leave our mass definition?

 

exactly where it was before. meters are pretty arbitrary, as are seconds. you can change them as much as you like (but have to multiply your changes through any calculations you do with your new units of length)

[Farsight]

See http://tf.nist.gov/cesium/fountain.htm to read about the NIST caesium fountain clock and http://en.wikipedia.org/wiki/Second for the definition of the second:

 

"Since 1967, the second has been defined to be the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom. This definition refers to a caesium atom at rest at a temperature of 0 K (absolute zero), and with appropriate corrections for gravitational time dilation."

 

In essence lasers are employed to cause hyperfine transitions, which are electron spin flips within caesium atoms. This emits light of a given "frequency", which is measured by a detector.

 

I put the word frequency in italics above because frequency is measured in Hertz, which is defined as cycles per second. What the detectors essentially do, is count incoming microwave peaks. When they get to 9,192,631,770, that's a second. Hence the frequency is 9,192,631,770 Hz by definition.

 

Note the mention of gravitational time dilation in the wiki article. If you were to take this clock and place it in a region of low gravitational potential, it would be like pressing a slow-motion button. All electromagnetic and other processes would then occur at a reduced rate, including the motion of the light towards the detector. However regardless of this, when the detectors get to 9,192,631,770, that's a second.

 

It's important to realise here that in this situation, the light is moving slower and this is why the second is bigger. We then use this second... to measure the speed of light! That's why we always measure the local speed of light in vacuo to be 299,792,458 m/s.

 

 

NB: provided you avoid the radial length contraction of general relativity, the metre is not affected. It's defined as the distance travelled by light in free space in 1⁄299,792,458th of a second, so the slower light and the bigger second cancel each other out.

[Amr Morsi]

It is impossible to have c dimensionless in any system of measurement. It will always be length/time. But sometimes, it is omitted, but, it is known implicitly that any velocity term is divided by c.

 

Actually, I think that the reason the velocity of light has this value is two points:

1. The system we use for measurement.

2. It is a universal constant, exactly such as pi. Actually, it is composed of 2 universal constants (epsilon and mu; permittivity and permeability)where

c=1/sqrt (epsilon*mu)

[Radical Edward]

See http://tf.nist.gov/cesium/fountain.htm to read about the NIST caesium fountain clock and http://en.wikipedia.org/wiki/Second for the definition of the second:

 

"Since 1967, the second has been defined to be the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom. This definition refers to a caesium atom at rest at a temperature of 0 K (absolute zero), and with appropriate corrections for gravitational time dilation."

 

In essence lasers are employed to cause hyperfine transitions, which are electron spin flips within caesium atoms. This emits light of a given "frequency", which is measured by a detector.

 

I put the word frequency in italics above because frequency is measured in Hertz, which is defined as cycles per second. What the detectors essentially do, is count incoming microwave peaks. When they get to 9,192,631,770, that's a second. Hence the frequency is 9,192,631,770 Hz by definition.

 

Note the mention of gravitational time dilation in the wiki article. If you were to take this clock and place it in a region of low gravitational potential, it would be like pressing a slow-motion button. All electromagnetic and other processes would then occur at a reduced rate, including the motion of the light towards the detector. However regardless of this, when the detectors get to 9,192,631,770, that's a second.

 

It's important to realise here that in this situation, the light is moving slower and this is why the second is bigger. We then use this second... to measure the speed of light! That's why we always measure the local speed of light in vacuo to be 299,792,458 m/s.

 

 

NB: provided you avoid the radial length contraction of general relativity, the metre is not affected. It's defined as the distance travelled by light in free space in 1⁄299,792,458th of a second, so the slower light and the bigger second cancel each other out.

 

I'm not sure about that last paragraph. THe speed of light is constant for all observers - that doesn't have anything to do with time dilation in the way you seem to be describing it.

[michel123456]

It's important to realise here that in this situation, the light is moving slower and this is why the second is bigger. We then use this second... to measure the speed of light! That's why we always measure the local speed of light in vacuo to be 299,792,458 m/s.

 

 

NB: provided you avoid the radial length contraction of general relativity, the metre is not affected. It's defined as the distance travelled by light in free space in 1⁄299,792,458th of a second, so the slower light and the bigger second cancel each other out.

 

Right. Very clean explanation.

 

The description counts for the observator A who measures himself, his own time & dimension.

R-Edward' s time dilation counts for another observer B.

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Merged post follows:

Consecutive posts merged

(..)It is a universal constant, exactly such as pi. (...)

 

Geometry....

[Amr Morsi]

O.K. Michel. Let's say as Plank's constant.

 

But, have you ever asked yourself why Pi has this value. It is a property of the universe, although it is a geometric one. I think it is similar to what we are saying here.

[michel123456]

O.K. Michel. Let's say as Plank's constant.

 

But, have you ever asked yourself why Pi has this value.

 

ooooyess.

[swansont]

See http://tf.nist.gov/cesium/fountain.htm to read about the NIST caesium fountain clock and http://en.wikipedia.org/wiki/Second for the definition of the second:

 

"Since 1967, the second has been defined to be the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom. This definition refers to a caesium atom at rest at a temperature of 0 K (absolute zero), and with appropriate corrections for gravitational time dilation."

 

In essence lasers are employed to cause hyperfine transitions, which are electron spin flips within caesium atoms. This emits light of a given "frequency", which is measured by a detector.

 

Not lasers, which by definition are near the optical regime or above. The transition is in the microwave, and everyone I know (or know of) uses a resonant cavity fed by a frequency chain rather than a maser. Hydrogen masers, on the other hand, do use amplified stimulated emission radiation.

 

Also, these are passive devices. They do not run by the detection of emitted hyperfine radiation. Again, unlike a maser.

[Farsight]

Noted Swanson. I rather oversummarised there and left out the microwave cavity. All: see http://tf.nist.gov/cesium/fountain.htm which includes the image below.

 

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Merged post follows:

Consecutive posts merged

I'm not sure about that last paragraph. THe speed of light is constant for all observers - that doesn't have anything to do with time dilation in the way you seem to be describing it.

It's just a different way of looking at it. Imagine you're watching some scene, and can press a time dilation button. Then everything in the scene goes slower. Your button might equally be labelled a slow motion button. But the observer inside the scene doesn't notice slower motion. He uses his NIST fountain clock and counts 9,192,631,770 microwave oscillations to define his second, then he defines his metre as the distance travelled by light in free space in 1⁄299,792,458th of a second, and hey presto, he always measures the local speed of light to be 299,792,458 m/s. Edited April 5, 2010 by Farsight
Consecutive posts merged.

[swansont]

Noted Swanson. I rather oversummarised there and left out the microwave cavity. All: see http://tf.nist.gov/cesium/fountain.htm which includes the image below.

 

I prefer this (2.1 MB pfd)

http://tycho.usno.navy.mil/clockdev/fountain_operation_hires.pdf

 

It's just a different way of looking at it. Imagine you're watching some scene, and can press a time dilation button. Then everything in the scene goes slower. Your button might equally be labelled a slow motion button. But the observer inside the scene doesn't notice slower motion. He uses his NIST fountain clock and counts 9,192,631,770 microwave oscillations to define his second, then he defines his metre as the distance travelled by light in free space in 1⁄299,792,458th of a second, and hey presto, he always measures the local speed of light to be 299,792,458 m/s.

 

One must note that there is no physics that can tell you that you are in this frame, or the one that is not in "slow motion."

[Farsight]

I think there might be a distinction to be found in measuring the fine structure constant [math]\alpha =\ \frac{e^2}{(4 \pi \varepsilon_0)\hbar c}\ [/math], at least when the time dilation is gravitational. See SpaceTime Mission: Clock Test of Relativity at Four Solar Radii but note that I view this as a relativity/electrodynamics matter rather than a string theory test.

 

"SpaceTime is a mission concept developed to test the Equivalence Principle. The mission is based on a clock experiment that will search for a violation of the Equivalence Principle through the observation of a variation of the fine structure constant, α. A spatio-temporal variation of α is expected in some string theories aimed at unifying gravity with other forces in nature. SpaceTime uses a special tri- clock instrument on a spacecraft which approaches the sun to within four solar radii. The instrument consists of three trapped ion clocks based on mercury, cadmium, and ytterbium ions, in the same environment. This configuration allows for a differential measurement of the frequency of the clocks and the cancellation of perturbations common to the three. The observation of any frequency drift between each of the clocks, as the tri-clock instrument approaches the sun, signals the existence of a scalar partner to the tensor gravity. Some relevant details of the mission design are discussed in the paper."

[swansont]

I think there might be a distinction to be found in measuring the fine structure constant [math]\alpha =\ \frac{e^2}{(4 \pi \varepsilon_0)\hbar c}\ [/math], at least when the time dilation is gravitational. See SpaceTime Mission: Clock Test of Relativity at Four Solar Radii but note that I view this as a relativity/electrodynamics matter rather than a string theory test.

 

"SpaceTime is a mission concept developed to test the Equivalence Principle. The mission is based on a clock experiment that will search for a violation of the Equivalence Principle through the observation of a variation of the fine structure constant, α. A spatio-temporal variation of α is expected in some string theories aimed at unifying gravity with other forces in nature. SpaceTime uses a special tri- clock instrument on a spacecraft which approaches the sun to within four solar radii. The instrument consists of three trapped ion clocks based on mercury, cadmium, and ytterbium ions, in the same environment. This configuration allows for a differential measurement of the frequency of the clocks and the cancellation of perturbations common to the three. The observation of any frequency drift between each of the clocks, as the tri-clock instrument approaches the sun, signals the existence of a scalar partner to the tensor gravity. Some relevant details of the mission design are discussed in the paper."

 

I wonder of this is still an active project, that's an old paper. A lot of missions were canceled with the Mars initiative being underfunded, and Lute Maleki is no longer at JPL.

[Farsight]

Damn. I dropped in to http://www.sstd.rl.ac.uk/stereo-soho/announcements.html to look at all the stands then had a chat with the organisers about why I thought this was so crucially important. All by the board I suppose. But it's interesting that Lute Maleki (http://www.oewaves.com/index.php?p=content&mid=2&id=5) seems to be into photonics. For photonics read opticks: Are not gross bodies and light convertible into one another? No, when you "stop a photon" we don't call it a photon any more.