What's the secret behind v < c

Mar 2019
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One of the premises of traditional 3D space SR is v < c. What does this hint? What secret / philosophical principle is behind it?
 
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3D verses 4D

v < c is actually a core feature of 4D space!

It is the 4D connection between time and space that dictates this limit.
 
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That's an interesting question ..

A trivial reply would be "everything is light" or that light is of supreme import ...

But it's misleading to call c the speed of light ... it's the speed of Electromagnetic Radiation light is just one infinitesimal part of the ER band ...

But why should this speed be the link between Energy and Mass ?? E= m c squared???
 

topsquark

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A trivial reply would be "everything is light" or that light is of supreme import
Not everything is light. Electrons are certainly not photons! Or did you mean something else?

But it's misleading to call c the speed of light ... it's the speed of Electromagnetic Radiation light is just one infinitesimal part of the ER band .
We can use the Maxwell equations to write EM phenomena as a wave with speed \(\displaystyle \dfrac{1}{\sqrt{ \mu \epsilon}}\), where \(\displaystyle \mu\) is the permeability and \(\displaystyle \epsilon\) is the permittivity of the material. In empty space \(\displaystyle \dfrac{1}{\sqrt{\mu _0 \epsilon _0}} = c\).

But why should this speed be the link between Energy and Mass ?? E= m c squared???
In SR the momentum 4-vector is \(\displaystyle p = \left ( \dfrac{E}{c}, -\vec{p} \right )\). Any 4-vector times itself is a constant. In this case it works out that \(\displaystyle p^{ \mu } p_{ \mu } = m^2c^2\). We get the relationship \(\displaystyle E^2 = (pc)^2 + (mc^2)^2\) from this.

-Dan
 
Mar 2019
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@dragon:
Can you say something more interesting and fascinating...
 

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What are permeability and permittivity?
What sets the properties of "empty" space such that its permeability and permittivity are these specific values?

@oz
We have had this conversation before...
c= the speed of light is a generally accepted short way of saying the longer (but more complete)
c= the speed of all Electro-Magnetic radiation.
 

topsquark

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What are permeability and permittivity?
What sets the properties of "empty" space such that its permeability and permittivity are these specific values?
The permittivity is a property of materials in an electric field. Probably the best introduction to this is a parallel plate capacitor. The formula for the capacitance is \(\displaystyle C = \epsilon \dfrac{A}{d}\). (This is actually something of an idealization, but it works.) \(\displaystyle \epsilon\) is the (electric) permittivity. If the capacitor is "empty" then \(\displaystyle \epsilon\) is a measure of the permittivity of air, which is going to be very close to that of empty space. (And, of course, you could put the capacitor in a vacuum if you like.)

The (magnetic) permeability is somewhat more complicated. So far as I know there is no such thing as a magnetic capacitor. But we define \(\displaystyle \vec{H} = \mu \vec{B}\) where H is the magnetic field and B the magnetic induction. The best idea that I have here is an inductor (or perhaps an electromagnet with an inner core.) H represents the field inside the inductor and B the field outside the conductor. For the free space \(\displaystyle \mu _0\) can be got by using an inductor (or electromagnet) with no core.

ln the derivation of the EM wave equation we get that the speed of light in a material is \(\displaystyle v = \dfrac{1}{\sqrt{ \epsilon \mu }}\). Obviously in empty space we use \(\displaystyle \epsilon _0\) and \(\displaystyle \mu _0\).

-Dan
 
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topsquark

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@dragon:
Can you say something more interesting and fascinating...
Hey, you haven't heard my lecture on Moller scattering near the Z channel in Electroweak theory yet (I did my Master's thesis on it.)

-Dan
 
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Mar 2019
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v vs c

Permittivity, permeability, light speed...That dragon seems to be talking something interesting: light speed (motion of photon) is determined by electromagnetic interaction.
In traditional 3D space SR, we care about only the motion of objects in 3D space but care no about electromagnetic interaction.
Does this mean that the issues of motion of photon should not mixed with other objects in traditional 3D SR?
Electroweak theory? Fascinating...but not the topic here...
 

topsquark

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Apr 2008
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On the dance floor, baby!
Permittivity, permeability, light speed...That dragon seems to be talking something interesting: light speed (motion of photon) is determined by electromagnetic interaction.
In traditional 3D space SR, we care about only the motion of objects in 3D space but care no about electromagnetic interaction.
Does this mean that the issues of motion of photon should not mixed with other objects in traditional 3D SR?
Electroweak theory? Fascinating...but not the topic here...
Hey, you were the one that wanted fascinating!

Yes, the constancy of the speed of light first cropped up in the Maxwell equations... some 20 to 30 years before Einstein. At the time it was thought that this was an effect of only the EM equations. Einstein adopted the Lorentz transformations for all of Physics, not just EM in his 1905 paper on SR.

-Dan