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Longitudinal Mass, Please Explain...


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#1 devin553344

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Posted 04 July 2020 - 09:01 AM

This was vaguely discussed in another post, but unsatisfactorily in my oppinion. (http://www.sciencefo...ffect-is-right/)

 

I read the article on mass in special relativity, and it discusses transverse and longitudinal mass. https://en.wikipedia...ngitudinal_mass

 

My question is this, why is it a power of 3 for parallel to the direction of the velocity and a power of 1 for the perpendicular velocities, ie:

 

fx = m . γ^3 . ax = mL . ax

fy = m . γ . ay = mT . ay

fz = m . γ . az = mT . az

 

I've always been taught that relativistic mass is:

 

m = m0/γ

 

Thanks for any explanation in advance.

 

The reason I'm asking is that I found that gravitation expands itself with a power of 3 and might relate to longitudinal mass, but it's unclear how it relates until I understand why they're using a power of 3 above. Here's the theory that includes power 3 for gravity:http://www.sciencefo...-of-the-forces/


Edited by devin553344, 05 July 2020 - 02:12 PM.


#2 AnssiH

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Posted 06 July 2020 - 05:28 AM

Well, the exact form of the equation depends on the exact definition one assigns to "mass".
 
Over the history, people have defined mass as whatever they feel is most convenient expression. Nothing wrong with that of course, so long that they are also consistent with how they use it (and it's easy to make mistakes here if one takes input from multiple sources, without realizing they might define their meaning of "mass" differently).
 
It mentions, in that wikipedia page you cite, that the power to three version, mass is "the ratio of force to acceleration" not "the ratio of momentum to velocity", thus the need to differentiate between longitudal and transverse forces (and thus "masses" by that definition of mass).
 
I can see that this related page is indeed relevant to what we've been discussing in another thread:
https://en.wikipedia...ransverse_mass¨
 
Notable points
 
* J.J. Thomson 1881 noting apparent mass effect by self-induction (perhaps indeed explainable by simply feedback between magnetic field and the charged particles generating said field).
* Thomson 1893 deriving the absolute speed limit of C to massive objects (..."when v = c its energy becomes infinite")
* Lorentz theory applying length contraction in the direction of motion yields that power of three Lorentz factor. 

[math]m_L = \frac{ m_{em} }{ \left ( \sqrt{1-\frac{v^2}{c^2}} \right )^3}[/math]

If you follow the citations you should find exact original derivations (I didn't trace it but I'm sure the original logic can be found from the derivation)
 
There's interesting comments about the usual definition of "relativist mass" [math] m = \frac{m_0}{\gamma} [/math] in that page you linked. Basically another example of making something so succint that you obfuscate the logic and the meaning behind it. That concept relates to "total energy" of a body, and discards (or "conceals") the concept of longitudal vs transverse momentum. I think Einstein's criticism (in that same page) to the idea of "relativistic mass" is very astute.
 
So important difference between the older forms and the modern "relativistic mass" definition is "...[relativistic mass] applies the name mass to a very different concept, the time component of a 4-vector". The increase in energy in the framework of special relativity should be viewed via the time relationship between inertial frames. Basically it is a consquence of consistent coordinate system transforms. (I would not personally teach it as "arising from the geometric properties of spacetime itself" as that's just one (metaphysical) way to view the relevant relationships. Also famously criticized by Einstein.)


EDIT: I have no idea why the latex math is not working... EDIT: Had to use full editor, simple mode and try three times :shrug:


Edited by AnssiH, 06 July 2020 - 05:41 AM.


#3 devin553344

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Posted 06 July 2020 - 06:37 AM

Well, the exact form of the equation depends on the exact definition one assigns to "mass".
 
Over the history, people have defined mass as whatever they feel is most convenient expression. Nothing wrong with that of course, so long that they are also consistent with how they use it (and it's easy to make mistakes here if one takes input from multiple sources, without realizing they might define their meaning of "mass" differently).
 
It mentions, in that wikipedia page you cite, that the power to three version, mass is "the ratio of force to acceleration" not "the ratio of momentum to velocity", thus the need to differentiate between longitudal and transverse forces (and thus "masses" by that definition of mass).
 
I can see that this related page is indeed relevant to what we've been discussing in another thread:
https://en.wikipedia...ransverse_mass¨
 
Notable points
 
* J.J. Thomson 1881 noting apparent mass effect by self-induction (perhaps indeed explainable by simply feedback between magnetic field and the charged particles generating said field).
* Thomson 1893 deriving the absolute speed limit of C to massive objects (..."when v = c its energy becomes infinite")
* Lorentz theory applying length contraction in the direction of motion yields that power of three Lorentz factor. 

[math]m_L = \frac{ m_{em} }{ \left ( \sqrt{1-\frac{v^2}{c^2}} \right )^3}[/math]

If you follow the citations you should find exact original derivations (I didn't trace it but I'm sure the original logic can be found from the derivation)
 
There's interesting comments about the usual definition of "relativist mass" [math] m = \frac{m_0}{\gamma} [/math] in that page you linked. Basically another example of making something so succint that you obfuscate the logic and the meaning behind it. That concept relates to "total energy" of a body, and discards (or "conceals") the concept of longitudal vs transverse momentum. I think Einstein's criticism (in that same page) to the idea of "relativistic mass" is very astute.
 
So important difference between the older forms and the modern "relativistic mass" definition is "...[relativistic mass] applies the name mass to a very different concept, the time component of a 4-vector". The increase in energy in the framework of special relativity should be viewed via the time relationship between inertial frames. Basically it is a consquence of consistent coordinate system transforms. (I would not personally teach it as "arising from the geometric properties of spacetime itself" as that's just one (metaphysical) way to view the relevant relationships. Also famously criticized by Einstein.)


EDIT: I have no idea why the latex math is not working... EDIT: Had to use full editor, simple mode and try three times :shrug:

Thanks for your reply, I'll study it some more. And I couldn't get the latex math to work either, thanks for the info :)