I Effects of time dilation for near-speed-of-light travel

[PeterDonis]

I'm guessing dx is some kind of differential representing an amount of time or length

All of the terms on the RHS are squares of coordinate differentials. The idea is that you are evaluating arc length along a curve; $d\tau$ is the arc length along a small differential element of the curve. $dt$, $dx$, $dy$, and $dz$ are the coordinate differentials along that small differential element of the curve.

I'm starting to think that a good way to treat velocity is meters per light second

No, that's not a good way, because in those units $c$ is not 1. If you measure distance in meters and $c = 1$, then the unit of time is meters--a meter of time is the time it takes light to travel 1 meter (about 3.3 nanoseconds).

If you measure time in seconds, and $c = 1$, then distance is measured in light seconds (seconds of distance).

I reason c=1 because c = (299792458 meters per second) * (1 second / 299792458 meters)) = (meters per second) * (seconds per light second) = (L/T)*(T/L).

This is not valid. See above.

If we say c is always one, can we still somehow say c = 299792458 meters / sec, as some people like to do?

No. See above.

What happens to the nomenclature for the constant 299792458 meters per second

There is no such constant in units where $c = 1$. That constant is part of a specific system of units, SI units.

 

[Chenkel]

All of the terms on the RHS are squares of coordinate differentials. The idea is that you are evaluating arc length along a curve; $d\tau$ is the arc length along a small differential element of the curve. $dt$, $dx$, $dy$, and $dz$ are the coordinate differentials along that small differential element of the curve.No, that's not a good way, because in those units $c$ is not 1. If you measure distance in meters and $c = 1$, then the unit of time is meters--a meter of time is the time it takes light to travel 1 meter (about 3.3 nanoseconds).

If you measure time in seconds, and $c = 1$, then distance is measured in light seconds (seconds of distance).This is not valid. See above.No. See above.There is no such constant in units where $c = 1$. That constant is part of a specific system of units, SI units.

I am a little confused on the units aspect of relativity.

Why do some people write c = 1?

Why do some people write c = 299792458 meters per second?

I would rather just write c = 1 if I can get away with it, it seems it can potentially simplify calculations and it seems like physicists are fond of doing it, and that is one of the reasons I am curious on how to be effective in this approach.

It's a puzzle to me because I'm wondering what units c = 1 means in the language of the problem that is being solved, if c is defined as the speed of light, then what units is c = 1, and why aren't dimensions specified?

How do people utilize an effective language for describing problems using these units constructs?

I've heard when someone sets the speed of light to 1, they are using "natural units."

Does the 1 in the expression "c = 1" represent one light second per second? And we just say one light second per second instead saying 299792458 meters per second? Is it common to not specify the dimensions of c explicitly, but to just refer to it as the speed of light?

 

[Ibix]

I personally had a bit of trouble with blithely setting $c=1$ (it makes more sense the further you get into the geometric picture of relativity). It helped me to explicitly write down such a unit system, and there's a really easy one: measuring distance in light seconds and time in seconds. The speed of light is, of course, one light second per second.

Light speed is also approximately one foot per nanosecond. Somebody (edit: David Mermin in his book "It's About Time", p22, according to Wikipedia) semi-jokingly proposed the phoot as a name for the light nanosecond...

 

[Chenkel]

I personally had a bit of trouble with blithely setting $c=1$ (it makes more sense the further you get into the geometric picture of relativity). It helped me to explicitly write down such a unit system, and there's a really easy one: measuring distance in light seconds and time in seconds. The speed of light is, of course, one light second per second.

Light speed is also approximately one foot per nanosecond. Somebody (don't recall who now) semi-jokingly proposed the phoot as a name for the light nanosecond...

You had me chuckling with the "phoot."

So when one writes c = 1, you are using "natural units" and are essentially implicitly implying you are measuring distance in light seconds, and time in seconds!

That seems to make some sense to me.

But if I asked someone "how fast was the photon going?" Even if we are using natural units I would want them to say "one light second per second" or "the speed of light" and not just respond with the singular word "one." If they want to write down v = c I will just read "ah yes, the photon is traveling at the speed of light, one light second per second!"

 

[malawi_glenn]

velocity squared units from time squared units?

you mean "time squared units"
and you forgot to square dτ

 

[malawi_glenn]

Why do some people write c = 1?

because algebraically, it does not matter what the value of $c$ is.
you can always insert the missing factors of $c$ in your final expression.
every intro book on relativity explains this.

If they want to write down v = c I will just read "ah yes, the photon is traveling at the speed of light, one light second per second!"

Nope you can not do that, because $v$ is the relative speed of the two interial frames in the lorentz-transformation, which is not valid for $v=c$.

 

[Chenkel]

because algebraically, it does not matter what the value of $c$ is.
you can always insert the missing factors of $c$ in your final expression.
every intro book on relativity explains this.Nope you can not do that, because $v$ is the relative speed of the two interial frames in the lorentz-transformation, which is not valid for $v=c$.

Doesn't the clock for a photon basically never tick for anyone observing the photons clock because the Lorentz factor increases without bound as velocity approaches c? What's wrong with plugging in v=c in the Lorentz factor and just treating the Lorentz factor as a infinite hyperreal?

 

[Chenkel]

because algebraically, it does not matter what the value of $c$ is.
you can always insert the missing factors of $c$ in your final expression.
every intro book on relativity explains this.Nope you can not do that, because $v$ is the relative speed of the two interial frames in the lorentz-transformation, which is not valid for $v=c$.

My last reply just talked about treating Lorentz factor as an infinite hyperreal, not sure if this can be done, but I'm sure my analysis is not on point considering my lack of knowledge in special relativity.

 

[Ibix]

So when one writes c = 1, you are using "natural units" and are essentially implicitly implying you are measuring distance in light seconds, and time in seconds!

Or years and light years (useful because $g$ is close to 1 light year/year² - actually 1.032), months and light months, whatever.

But if I asked someone "how fast was the photon going?" Even if we are using natural units I would want them to say "one light second per second" or "the speed of light" and not just respond with the singular word "one." If they want to write down v = c I will just read "ah yes, the photon is traveling at the speed of light, one light second per second!"

As you get more into the geometry it makes more sense that you want to use the actual same units for distance and time. The Lorentz transforms are the Minkowski geometry equivalent of rotating a Cartesian coordinate system in Euclidean geometry. In Euclidean geometry there's nothing wrong with measuring vertical distances in fathoms and horizontal ones in nautical miles, but it's easier if you use the same units for everything. Thus some textbooks will explicitly start measuring time in meters (one three hundred millionth of a second) or something.

Personally I set $c=1$ and don't worry about whether it's "actually" 1 or "actually" 1ls/s. The maths is the same and I can always insert the $c$s again by dimensional analysis if necessary.

 

[malawi_glenn]

I'm sure my analysis is not on point considering my lack of knowledge in special relativity.

Why not just pick up a book about relativity and study it? I can recommend the book by Morin "Special Relativity: For the Enthusiastic Beginner" which is quite cheap.

It is hard (impossible) to learn a new subject on a forum.

 

[Chenkel]

Why not just pick up a book about relativity and study it? I can recommend the book by Morin "Special Relativity: For the Enthusiastic Beginner" which is quite cheap.

Thanks for the reference, I'll check it out!

 

[jbriggs444]

Light speed is also approximately one foot per nanosecond.

Grace Hopper used to hand out nanoseconds at her lectures. (one foot wire segments). The one time I heard her speak, wire was too expensive and I picked up a salt packet of picoseconds instead. (Google says pepper. I remember salt. Go figure).

 

[robphy]

Grace Hopper used to hand out nanoseconds at her lectures. (one foot wire segments). The one time I heard her speak, wire was too expensive and I picked up a salt packet of picoseconds instead. (Google says pepper. I remember salt. Go figure).

Go to t=46m00s [for milliseconds... start at t=45m07s]



About Adm Hopper


https://en.wikipedia.org/wiki/Grace_Hopper

 

[PeterDonis]

Why do some people write c = 1?

Because they're using units called "natural units" in which $c = 1$.

Why do some people write c = 299792458 meters per second?

Because they're using SI units, in which ##c# is defined to have that value.

I'm not sure what the issue is with this. Surely the existence of multiple different systems of units is no mystery.

Does the 1 in the expression "c = 1" represent one light second per second?

It can. Or it can mean one meter per light-meter (meter of light travel time). Or it can mean one light year per year. The point is that the unit of distance and the unit of time are related by the time it takes light to travel the distance. That makes things much simpler mathematically (you don't have stray factors of $c$ all over the place and have to worry about whether you've gotten them all right) and makes spacetime diagrams easier (because the worldlines of light rays are 45 degree lines and the units on both the space and time axes are the same).

 

[PeterDonis]

My last reply just talked about treating Lorentz factor as an infinite hyperreal, not sure if this can be done

No, it can't. There is no such thing as an inertial frame in which a photon is at rest.

 

[Chenkel]

Why not just pick up a book about relativity and study it? I can recommend the book by Morin "Special Relativity: For the Enthusiastic Beginner" which is quite cheap.

It is hard (impossible) to learn a new subject on a forum.

I got the book recently and I've been studying it, it's interesting so far and I think I'll learn a lot from it.

Thank you for the suggestion