Different speeds of light w/ Planck Units

[pr0pensity]

OK so my whining for help covers writing a paper about how things would be different if the speed of light was changed to 65 mph.

Right after getting the assignment I started thinking about dilations and all that. Then, I remembered Planck's constant. Light would still have to have the same amount of energy, so the frequency would have to go up. Then I looked up a little more about Planck's theorems and read about Planck units.

so if c is changed in the Planck units, everything is just scaled. space gets longer, time gets longer, charge decreases, etc etc pretty much there's just less energy if the speed of light decreases.

but if everything is just scaled, wouldn't it seem the same? wouldn't it seem like light still goes 300000 km/s?

the only way you would be able to tell any difference would be to compare two sections of the universe with different speeds of light, right?

my main question is about whether or not changing c would actually change the speed of light through space if everything is relatively unchanged.

 

[pervect]

OK so my whining for help covers writing a paper about how things would be different if the speed of light was changed to 65 mph.

It's not really a very well defined question, as I think you are starting to realize.

The first question is whether one is just changing the speed of light (i.e. one is imagining that perhaps the atmosphere has some very high refractive index), or whether one is actually changing 'c'.

Saying that one is going to change 'c' sounds like it should be unambiguous and make sense, but it's actually hard to say what this means outside the context of some particular model.

Probably the most direct interpretation of the question is to assume that the constant, c, used to define meter changes. This keeps the second unchanged, and makes the meter a huge number (in comparison to the standard meter). But this may not be the intent of the quesiton.

One can spend a lot of more or less useless time trying to define exactly what this question could mean.

There is one rather robust (and amusingly unexpected) consequnce that I am aware of (due to another poster, not on this board). If Earth's gravity and/or radius is assumed to not also be changed, i.e. the speed of light change is assumed not to be associated with a simple scale change (as I did above), and one assumes that everything else is the same but "c' is different, then the Earth must be a black hole, because it's escape velocity is greater than 65 miles/hour, i.e greater than c.

Which would mean that we would all die in a very short amount of time, certainly well before you finish writing your paper, as the Earth collapsed to a singularity and every human being on it was ripped apart by tidal forces.

(Well, on second thought, I suppose you could think of the Earth as being made of some sort of exotic matter, and imagine it to basically be a rather exotic entity known as a gravistar, and thus escape this fate.)

 

[JesseM]

This page has a good summary of why it does not really make sense to talk about changes in constants with units like the speed of light, and why it's better to talk about changes in dimensionless constants (constants with no units, like the ratio of a given particle's mass to the Planck mass).

First, the term fundamental constants needs to be explained. Fundamental constants can be considered as natural standards against which everything else can be measured. If something is changing it can be detected by consecutive measurements against natural standards. But if the standards themselves are changing, any detection of this seems to be questionable. In fact, the only chance to see any change comes when a change in the natural standards (fundamental constants) happens in a disproportionate way, so that some dimensionless ratio of constants changes. For example, if absolutely everything in the Universe suddenly increases in size, it cannot be noticed. If, in contrast, the Earth becomes larger but the Sun remains the same, it can be detected by comparing their sizes. This comparison comes in a form of (dimensionless) ratio size of Earth/size of Sun. It tells us that the relative sizes of Earth and Sun have changed. However, there is no way to say whether the Earth has become larger or the Sun smaller.

Now suppose that the speed of light is changing. It can be meaningful only if it can be detected by measurements. To perform measurements we need units. Units do not exist in nature and are invented by people to express quantitative relations in nature which exist. Units to measure length, time, speed, etc. are always expressed in terms of some combination of fundamental constants. Performing measurements means comparing the measured value to a particular combination of fundamental constants. Measuring a fundamental constant means comparing fundamental constants between themselves. Physical laws must not depend on the particular choice of units. Below we illustrate that whatever units are used to measure the speed of light, the claim that the speed of light is changing leads to nonsense.

The speed of light c is most commonly expressed in metres (m) per second (s). Its value is c=299792458 m/s. However, the metre is defined as the distance which light travels in 1/299792458 s [8]. If the speed of light is changing, its value in m/s will still be the same. One may argue that this definition of the metre is not good in a situation where the speed of light is changing. What if we use the old definition instead, 1m = 1/10000000 of the distance from the North Pole to the equator? This just moves the problem into another area: there is no way to distinguish between a change in the speed of light and a change in the size of the Earth (and there is no way to say that one is more likely than the other!).

Let's now take a stick 1m long and say that this is going to be our standard unit for length. And let's measure the speed of light via the time needed for light to travel along the stick. Since we always use the same stick we probably should expect that its length remains the same. One would argue that if consecutive measurements produce different results, the speed of light is changing. However, the measurement of the speed of light using a stick as a standard can be interpreted as the measurement of the length of the stick using the speed of light as a standard. The speed of light is not less fundamental than the length of the stick [9]. Here again we cannot say what is changing, the speed of light or the length of the stick.

The best unit to use to get to a paradox the fastest possible way is the speed of light itself. This is the only single fundamental constant (not a combination of constants) which has the dimension of speed and can be used as a unit to measure any speed. Then c=1 by definition and cannot change!

We see that depending on the units used, the speed of light either remains the same or its change cannot be distinguished from a change in other fundamental constants. Recalling that physical laws must not depend on units, we come to the conclusion that a changing speed of light is nonsense.

 

[rbj]

so if c is changed in the Planck units, everything is just scaled. space gets longer, time gets longer, charge decreases, etc etc pretty much there's just less energy if the speed of light decreases.

but if everything is just scaled, wouldn't it seem the same? wouldn't it seem like light still goes 300000 km/s?

yup. first of all, "fundamental physical constants" are dimensionless constants such as $\alpha$ or $m_p/m_e$ or such. the speed of light, as we measure it with rods and clocks, is a human construct.

if no dimensionless constant had changed, it is meaningless to discuss what if some dimensionful constant (such as c or G) had changed. we could not tell if it had or not. but if a dimensionless constant such as $\alpha = e^2/(\hbar c 4 \pi \epsilon_0)$, we would know the difference, but it is not correct to ascribe that change to c or $\hbar$ or e. it's just a change in $\alpha$ (which has a real effect in the real world) and which of those components that changed is dependent only on how you decide to define your units to measure things. with Planck units, $c = (1 l_P)/(1 t_P)$ so neither c nor $\hbar$ would be measured to have changed. but e would appear to be different. but if you used Stoney units, then e would be constant and $\hbar$ would appear to have changed if $\alpha$ changed.