Speed of light and Euler's constant

[hassan23]

TL;DR

Does f = nce² have any significance where c is the speed of light and e is Euler's constant and f and n are variables?

I am a newbie and the last time I did physics was in secondary school. I was researching the topic of pi and came across the euler's constant and was wondering about the relationship between euler's constant and the speed of light specifically ce². I think that works out to be one third the speed of light.

 

[Nugatory]

TL;DR Summary: Does f = nce² have any significance where c is the speed of light and e is Euler's constant and f and n are variables?

No. The easiest way of seeing this is to consider that we can make the speed of light come out to be any arbitrary number just by changing our units.

For example a century ago we would all agree that the speed of light was about 186000 miles/second. As a result of several historical accidents a mile is defined to be 5280 feet. Now we ask what a foot is, and we learn that it is more or less the the length of the foot of some long-deceased English king. So.... If the king had had a different shoe size light would still be moving at the the same speed but we wouldn't call that speed 186000 miles/second. That suggests that relationships between the true constants like pi and e and quantities like the speed of light or sound whose values change with our choice of units must be accidents with no real significance.

On the other hand, there are physically measurable quantities with values that are meaningful because they do not depend on our choice of units. For example, if I am running from a hungry bear, both my speed $S_N$ and the bear's speed $S_B$ will depend on the units we're using. However, the ratio $S_B/S_N$ is the same no matter what units we use because the units cancel - and the outcome depends on whether that ratio is greater than or less than unity without considering the units at all.

A less frivolous example is the fine structure constant (Google for that) which is also a ratio in which all the units cancel. We routinely see questions about "what would happen if the speed of light were something else?"; these questions always turn out to be asking what would happen if the fine structure constant had a different value.

 

[hassan23]

Correction e=eulers number which is different from eulers constant (γ).

 

[Nugatory]

Correction e=eulers number which is different from eulers constant (γ).

We figured that's what you meant :)
But still good to correct

 

[Vanadium 50]

specifically ce².I think that works out to be one third the speed of light.

You think incorrectly. This implies $e = 1/\sqrt{3}$. This is true neither for Euler's number nor Euler's constant.

 

[hassan23]

Thanks everyone for the replies. I have now moved on to learning about the fine structure constant and I see a formula for it that has the speed of light and pi in it. This brought me back to my question here. I wonder if there is a link between Euler's constant and the fine structure constant.

 

[berkeman]

Thread closed temporarily for Moderation...

 

[Mark44]

his brought me back to my question here. I wonder if there is a link between Euler's constant and the fine structure constant.

No and again it's Euler's number, or e, which is different from the e that appears in the fine structure constant.

Per this wikipedia article, https://en.wikipedia.org/wiki/Elementary_charge
$e = \sqrt{4\pi\alpha}\sqrt{\epsilon_0\hbar c}$
In this definition e is the unit of electric charge, not the Euler number.

Thread will remain closed, as the question has been answered.