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Useful metric for high-relativistic velocities


0111narwhalz

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I am trying to develop a good metric for velocities extremely close to c. However, to achieve this, I first need to understand what makes a "good metric." It is quite clear that the simple decimal representation (xc) is deficient, as it makes values such as .999c seem very similar to .99999c while making values such as .001c seem very far from .25c (though divergence from Newtonian kinetic energy is vanishingly small). I've started using a metric which is based on counting the nines after the decimal point (e.g. .999c => "3 nines" and .99999c => "5 nines"). This helps with the issue of distinguishing values very close to c, but it doesn't make much sense in general (How do you define .05c? Or .9993c?).

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This should be a job for logarithms, but expect a pesky minus sign.  Also you want to use the largely forgotten "log" (a favorite of sliderules) instead of "ln" (which is taught in calculus and modern students typical exposure to the logarithm).  So log10(1-v/c)

.999-> -3
.99999-> -5
.9993-> -3.154 (essentially "3.154 nines")
.25-> -.12
.01-> -.004


 

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Is velocity informative here at all?
You anyway move with almost the same speed, no matter how long you accelerate.

In fact you just have a deal not with a growing speed, but with almost constant speed and growing Lorentz factor. (1/sqrt(1-(v/c)2)).
As if you have paid all your debts and now are depositing incoming surpluses.

So, maybe you would just talk about "close-to-light speed with Lorentz factor 25.5, 25.6, 25.7..." ?
(When this factor gets big, too, move to logarithmic Lorentz factor).

Edited by kerbiloid
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There is already a recognized metric for working with speeds near the speed of light.

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

Since many of the physical effects (time dilation, length contraction, relativistic mass, etc.) are linear w.r.t. the Lorentz factor, it's probably what you really want to use.

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Hmm.  I read the title, and thought "rapidity".  And no one else has mentioned it.

Rapidity is similar to Lorentz factor, except that instead of being linear with all the relativistic dilations, it's linear with applied impulse.  It's essentially "internally perceived velocity" in that, if you have a "torch" drive or other long-long-term constant boost system, and keep accelerating to the point where Newtonian physics would put your velocity near or above that of light, your rapidity will reflect the expected Newtonian value -- and can be used to calculate perceived trip times in the same way.  Even better, rapidity is additive (at least in one dimension, which is usually what you care about for things like ship travel times).  The rapidity of light is infinite.

The math involved (which I don't really understand; my calculus is about like my Spanish -- it was never good, and that was a long time ago) is similar in complexity to getting Lorentz factor, but the result is a lot more intuitive.  The units aren't those of velocity, but when your Lorentz factor is small, rapidity is proportional to velocity, diverging only when dilations start to become significant.

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3 hours ago, Zeiss Ikon said:

Hmm.  I read the title, and thought "rapidity".  And no one else has mentioned it.

Rapidity is similar to Lorentz factor, except that instead of being linear with all the relativistic dilations, it's linear with applied impulse.  It's essentially "internally perceived velocity" in that, if you have a "torch" drive or other long-long-term constant boost system, and keep accelerating to the point where Newtonian physics would put your velocity near or above that of light, your rapidity will reflect the expected Newtonian value -- and can be used to calculate perceived trip times in the same way.  Even better, rapidity is additive (at least in one dimension, which is usually what you care about for things like ship travel times).  The rapidity of light is infinite.

The math involved (which I don't really understand; my calculus is about like my Spanish -- it was never good, and that was a long time ago) is similar in complexity to getting Lorentz factor, but the result is a lot more intuitive.  The units aren't those of velocity, but when your Lorentz factor is small, rapidity is proportional to velocity, diverging only when dilations start to become significant.

That's the thing about metrics -- they are tools. Which one you use (or even invent) depends on what you are trying to do with it. A hammer is one of the most useful tools ever invented, but it works terribly for driving screws.

The OP made the (common) mistake of asking for a metric without specifying what it will be used for. That's roughly the same as walking up to a carpenter and asking, "Hey, can I borrow a tool?"

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41 minutes ago, mikegarrison said:

That's roughly the same as walking up to a carpenter and asking, "Hey, can I borrow a tool?"

HA! Just yesterday I had a guy come up to me and asked to buy a nonspecific "controller" produced by multiple companies in significantly different variations, with different prices. After asking him "Which one?" for about five times I just gave up and told him to go away and come back when he decides what he wants.

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Well, sometimes it is not that easy to exactly specify what you need because nobody knows about all the tools that exist. You want to hang up a ceiling lamp in a 500 year old house ? Do you know that there are sort of anchor or straddling dowels that fit through a small borehole and brace up when tightened so that you can fasten like 5kg at the hollow between a clay/wood ceiling and thin cover of gypsum ? Explain this to the poor guy in the hardware store ... :-))

That's why we have division of work (and a language or pen & paper). In this case it was quite obvious what the OP wanted because there aren't that many tools, but speaking of hammers, it can be easier abused than a screwdriver :-)

 

Edited by Green Baron
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Lorentz factor is great if you are trying to quantify how much quantum-mechanical weirdness is going on as a result of your velocity. It works well in all reference frames.

Rapidity is great if you are trying to quantify how the observer in the reference frame of interest is going to experience the rate of their transit. It works best for the reference frame that has been accelerated to a high velocity.

Edited by sevenperforce
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34 minutes ago, sevenperforce said:

Lorentz factor is great if you are trying to quantify how much quantum-mechanical weirdness is going on as a result of your velocity. It works well in all reference frames.

Rapidity is great if you are trying to quantify how the observer in the reference frame of interest is going to experience the rate of their transit. It works best for the reference frame that has been accelerated to a high velocity.

As long as by "quantum-mechanical" you mean "relativistic", then yeah I agree :P

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