Does relativistic mass have a gravitational component?

[pmb_phy]

from Baez link: Philip Gibbs 1996 - If you go too fast do you become a black hole?

"M is contained within a sphere of radius 2GM/c2 (the Schwarzschild radius)"
Spherical symmetry is pretty much lost when the star is transformed by a very large boost.

"One way to avoid this is to not speak about relativistic mass and think only in terms of invariant rest mass"
How can one avoid something by merely not talking about it?

Don't ask me. I think some of those FAQs have a lot to be desired. That one is an example. It basically says "Don't think so much and don't ask probing questions unless you know as fact that you're right the first time." Dumb. Really dumb.

Whether you call relativistic mass the source of gravity or you call energy the source of gravity. Either way the same question arises. i.e. "The faster a particle goes the greater its energy. Energy is the source of gravity (one component of energy-momentum tensor, energyu in one frame is momentum and momentum flux in another etc). So why doesn't the partilce become a black hole?" - The answer is this - An object's rest mass confined within a given region of space is what determines ife a body is a black hole or not.

Pete

 

[pmb_phy]

from Baez link: Philip Gibbs 1996 - If you go too fast do you become a black hole?

"M is contained within a sphere of radius 2GM/c2 (the Schwarzschild radius)"
Spherical symmetry is pretty much lost when the star is transformed by a very large boost.

"One way to avoid this is to not speak about relativistic mass and think only in terms of invariant rest mass"
How can one avoid something by merely not talking about it?

Don't ask me. I think some of those FAQs have a lot to be desired. That one is an example. It basically says "Don't think so much and don't ask probing questions unless you know as fact that you're right the first time." Dumb. Really dumb.

Whether you call relativistic mass the source of gravity or you call energy the source of gravity. Either way the same question arises. i.e. "The faster a particle goes the greater its energy. Energy is the source of gravity (one component of energy-momentum tensor, energy in one frame is momentum and momentum flux in another etc). So why doesn't the partilce become a black hole?" - The answer is this - An object's rest energy confined within a given region of space is what determines ife a body is a black hole or not.

Pete

 

[j8hart]

from Baez link: Philip Gibbs 1996 - If you go too fast do you become a black hole?

"M is contained within a sphere of radius 2GM/c2 (the Schwarzschild radius)"
Spherical symmetry is pretty much lost when the star is transformed by a very large boost.

"One way to avoid this is to not speak about relativistic mass and think only in terms of invariant rest mass"
How can one avoid something by merely not talking about it?

My favourite quote from that page is the understated "It is actually quite difficult to define the correct conditions for a black hole to form."

But I think the point the article is making in the section you quote is that someone with my level of Physics might imagine a sphere smaller than the radius of the star 'at rest' but concentric with the star, and then say "even if you ignore the 'relativistic mass' and only consider the length contraction the mass within this sphere goes up". In this way of looking at it, the fact that the star itself is no longer a sphere does not matter, provided you can find a sphere with sufficient mass inside it.

However, the article then goes on to say "But this is based on a particular static solution to the Einstein field equations of general relativity, and ignores momentum and angular momentum as well as the dynamics of space-time itself."

In other words I guess, "it is not going to happen, but it's outside the scope of this article to explain why not in mathematical terms".

As for trying not to speak about "relativistic mass", one of the things I have learned from all this is that apparently people have been trying to avoid the term almost as long as it has existed.

The article links to another article "Does mass change with velocity" which has for example the following paragraph.

"In the final analysis the issue is a debate over whether or not relativistic mass should be used, and is a matter of semantics and teaching methods. The concept of relativistic mass is not wrong: it can have its uses in special relativity at an elementary level. This debate surfaced in Physics Today in 1989 when Lev Okun wrote an article urging that relativistic mass should no longer be taught (42 #6, June 1989, pg 31). Wolfgang Rindler responded with a letter to the editors to defend its continued use. (43 #5, May 1990, pg 13)."

When I was at university I picked up the idea from somewhere or someone I don't remember, that it was better to consider it as "relativistic inertia", which is part of what triggered my original question.

 

[pmb_phy]

"In the final analysis the issue is a debate over whether or not relativistic mass should be used, and is a matter of semantics and teaching methods. The concept of relativistic mass is not wrong: it can have its uses in special relativity at an elementary level. This debate surfaced in Physics Today in 1989 when Lev Okun wrote an article urging that relativistic mass should no longer be taught (42 #6, June 1989, pg 31). Wolfgang Rindler responded with a letter to the editors to defend its continued use. (43 #5, May 1990, pg 13)."

That article by Okun is filled with flaws. In fact each and ever argument he gave is wrong. Granted, they seem correct if you don't take the time to really look into it, but I had plenty of time and picked each one apart.

Pete

 

[quartodeciman]

...imagine a sphere smaller than the radius of the star 'at rest' but concentric with the star, and then say "even if you ignore the 'relativistic mass' and only consider the length contraction the mass within this sphere goes up". In this way of looking at it, the fact that the star itself is no longer a sphere does not matter, provided you can find a sphere with sufficient mass inside it.

I was thinking that removing spherical symmetry alters the metric and requires different solutions.

As for trying not to speak about "relativistic mass", one of the things I have learned from all this is that apparently people have been trying to avoid the term almost as long as it has existed.

That wasn't my impression. I started investigating the ideas of SR back when Einstein still lived and worked. I don't recall any complaints about relativistic mass then. People who accepted SR gladly used the Lewis-Tolman style of derivation for SR dynamical relations. I can't speak for those who objected to SR theory in the first place. And I can't relate what was complained in journal notes, because I didn't know any. For me, the energy-first (invariant-mass-only) school was just suddenly there with a vengeance. Maybe it was the mid-1960s that I first noticed it.

When I was at university I picked up the idea from somewhere or someone I don't remember, that it was better to consider it as "relativistic inertia", which is part of what triggered my original question.

My thought on that is of a strong objection to reification (treating something abstract as if it were real and substantial). Elementary masses are real existents. On the other hand, energy is an abstract concept from the beginning, so it doesn't bear that onus. The answer to "What is energy made of" is easily answered "It's not made of anything because it is a quantitative abstraction, not a substance". So, particle physicists readily convert masses to energies and deal exclusively and freely in that abstract domain. Masses are thereby filled in as final conclusions to HEP problems. That's just my supposition of the situation. The GR (what makes gravity?) situation seems to be another major battlefield for this opposition.

 

[DW]

That article by Okun is filled with flaws. In fact each and ever argument he gave is wrong. Granted, they seem correct if you don't take the time to really look into it, but I had plenty of time and picked each one apart.

Pete

You are the one who has long sense been shown to be wrong in your attempts to ressurect the speed varient mass concept that is even manifestly inconsistent with Einstein's first postulate.

 

[j8hart]

You are the one who has long sense been shown to be wrong in your attempts to ressurect the speed varient mass concept that is even manifestly inconsistent with Einstein's first postulate.

Guys, I can see a fight brewing here. I know that just because I started the thread that does not mean I own it, but I think this is going to put people off any further contributions to my questions, if any are possible, so can I ask you both as a special favour to continue this part of the discussion on a different thread? Let me know where and I'll read some of it, though I doubt I'll understand much!

 

[Chronos]

This is what Hawking said about creating black holes in particle accelerators.
.
"If one had a particle with an energy above what is called the Planck energy, 10 million million million Gev (1 followed by 19 zeros), its mass would be so concentrated that it would cut itself off from the rest of the universe and form a little black hole."

"Of course, the Planck energy is a very long way from the energies of around a hundred GeV, which are the most we can produce in the laboratory at the present time. We shall not bridge that gap with particle accelerators in the foreseeable future!"

p225 The Illustrated A Brief History of Time [1996 edition].

I do not interpret this as meaning you can accelerate a particle less massive than the Planck energy into a black hole. Rather, that if you could accelerate and collide less massive particles at high enough energies, a Planck energy size particle could be created [i.e., a tiny black hole result]. This is the same process CERN used to discover the W+, W- and Zo particles. They did not convert less massive particles to these highly massive particles through acceleration, it was through the energy-matter conversion resulting from collisions between highly accelerated less massive particles

 

[pmb_phy]

Guys, I can see a fight brewing here. I know that just because I started the thread that does not mean I own it, but I think this is going to put people off any further contributions to my questions, if any are possible, so can I ask you both as a special favour to continue this part of the discussion on a different thread? Let me know where and I'll read some of it, though I doubt I'll understand much!

dw is always anxious to post provoking comments like that.

Try using the "ignore" list and you'll see a prettier board.

Pete

 

[Nereid]

pmb_phy and DW:

I would like to second j8hart's sentiment - many of us here at PF have noticed that the two of you have different opinions, and sometimes the expression of those differences gets personal.

It may well be very interesting to have a discussion of what the content of those differences of opinion are, please start a separate thread to hold that discussion.

My impression is that you are both very knowledgeable about GR, and so can contribute a great deal to increasing the understanding of the many others here at PF who don't have as deep an understanding as you, as well as their feeling of comfort with the subject.

Let's keep this thread on topic, please.

 

[pmb_phy]

pmb_phy and DW:

I would like to second j8hart's sentiment - many of us here at PF have noticed that the two of you have different opinions, and sometimes the expression of those differences gets personal.

Hence my placing him on used the "ignore list". He's been posting so much after all my posts that I wanted to see if he was posting without insulting. I tok a gander to see if he's changed his tone. I saw that he hadn't. I corrected him, and placed him back on the ignore list.

It may well be very interesting to have a discussion of what the content of those differences of opinion are, please start a separate thread to hold that discussion.

The purpose of this thread is to discuss why solme relativists (e.g. Rindler, D'Inverno, Mould, Sartori, Peacock, etc.) use relativistic mass. I have not strayed from that.

Let's keep this thread on topic, please.

What did you think the topic was?

Pete

 

[pmb_phy]

Hence my placing him on used the "ignore list". He's been posting so much after all my posts that I wanted to see if he was posting without insulting. I tok a gander to see if he's changed his tone. I saw that he hadn't. I corrected him, and placed him back on the ignore list.

The purpose of this thread is to discuss why solme relativists (e.g. Rindler, D'Inverno, Mould, Sartori, Peacock, etc.) use relativistic mass. I have not strayed from that.

What did you think the topic was?

Pete

Oops! I see what I did. I mistook this thread for the thread I started on mass. Sorry dude!

Pete

 

[DW]

Hence my placing him on used the "ignore list". He's been posting so much after all my posts that I wanted to see if he was posting without insulting. I tok a gander to see if he's changed his tone. I saw that he hadn't.

You never had me on any ignor list. Often you claim to kill file people, but never do. You read everything I write.

I corrected him, and placed him back on the ignore list.

Correction, I corrected you.

 

[DW]

pmb_phy and DW:

My impression is that you are both very knowledgeable about GR, and so can contribute a great deal to increasing the understanding of the many others here at PF who don't have as deep an understanding as you, as well as their feeling of comfort with the subject.

I'm quite sure that everyone else posting here is far deeper in understanding of GR than pmb is. For example he is still fighting against the invariance of mass. How deep an understanding can one have and still not even understand what mass is.

 

[Zanket]

Whether you call relativistic mass the source of gravity or you call energy the source of gravity. Either way the same question arises. i.e. "The faster a particle goes the greater its energy. Energy is the source of gravity (one component of energy-momentum tensor, energy in one frame is momentum and momentum flux in another etc). So why doesn't the partilce become a black hole?" - The answer is this - An object's rest energy confined within a given region of space is what determines ife a body is a black hole or not.

Can you please elaborate on that answer? If energy of motion is a source of gravity, how can that gravity not contribute to a body becoming or being a black hole?

Let two lumps of clay collide at relativistic velocity. After the collision they are at rest relative to you. The collision released heat, which has mass. The mass of the heat wasn’t created in the collision; it was the energy of motion. The mass of the system didn’t change, and after the collision that mass--now all rest energy--could be enough to make the system a black hole. If before the collision we could have contained the system within the Schwarzschild radius for the mass of the system, then we’d have a region of space in which a black hole spontaneously forms even though no mass was added to it. Strange. What am I missing? (Maybe heat is not rest energy?)

 

[DW]

Can you please elaborate on that answer? If energy of motion is a source of gravity, how can that gravity not contribute to a body becoming or being a black hole?

Let two lumps of clay collide at relativistic velocity. After the collision they are at rest relative to you. The collision released heat, which has mass. The mass of the heat wasn’t created in the collision; it was the energy of motion. The mass of the system didn’t change, and after the collision that mass--now all rest energy--could be enough to make the system a black hole. If before the collision we could have contained the system within the Schwarzschild radius for the mass of the system, then we’d have a region of space in which a black hole spontaneously forms even though no mass was added to it. Strange. What am I missing? (Maybe heat is not rest energy?)

No one said that system mass was rest energy. It is center of momentum frame energy. Heat or kinetic energy of the constituents with respect to the center of momentum contributes to that. In the case that the system is reduced to a single particle the center of momentum frame energy reduces to simply rest energy.

 

[pmb_phy]

Can you please elaborate on that answer? If energy of motion is a source of gravity, how can that gravity not contribute to a body becoming or being a black hole?

When I was using the terms "mass" and "energy" in this thread I'm speaking of T⁰⁰/c² and T⁰⁰, respectively, where
T is the stress-energy-momentum tensor. When I say that mass is the source of gravity I use it to mean mass is gravitational charge. Mass in one frame is momentum in another frame etc. Momenutm contributes to the gravitational field.

Let two lumps of clay collide at relativistic velocity. After the collision they are at rest relative to you. The collision released heat, which has mass. The mass of the heat wasn’t created in the collision; it was the energy of motion.

The mass of the heat was in the form of relativistic mass before the collision. Therefore the mass of heat came from the mass of motion.

The mass of the system didn’t change, and after the collision that mass--now all rest energy--could be enough to make the system a black hole.

That is very true.

If before the collision we could have contained the system within the Schwarzschild radius for the mass of the system, then we’d have a region of space in which a black hole spontaneously forms even though no mass was added to it. Strange. What am I missing? (Maybe heat is not rest energy?)

You're speaking of two different situations. In one case you're examining a black hole from two different frames and in the other case, i.e. the one you've just explained, you're examining a collapsing black hole from two different frames.

Pete

 

[Zanket]

You're speaking of two different situations. In one case you're examining a black hole from two different frames and in the other case, i.e. the one you've just explained, you're examining a collapsing black hole from two different frames.

I don’t see how it’s two different frames in the “case I’ve just explained.” It seems to me that it’s all happening in my frame. Things can move in my frame right?

Say I’m floating in deep space, watching a light-year-in-diameter sphere of space. Within that sphere are the two lumps of clay (let’s say boxcar-sized) approaching each other. They collide in the middle of the sphere, releasing enough heat energy to create a black hole with an event horizon that encloses the sphere. Didn’t I witness a black hole spontaneously form in a region of space to which no mass was added? All in one frame?

 

[Zanket]

The mass of the heat was in the form of relativistic mass before the collision. Therefore the mass of heat came from the mass of motion.

Isn’t mass of motion the same as energy of motion? Making relativistic mass the same as energy of motion?

 

[DW]

Isn’t mass of motion the same as energy of motion?

No, mass is center of momentum frame energy. Translational kinetic energy is energy of translational motion. Relativistic system energy is the sum of the two.

Making relativistic mass the same as energy of motion?

Relativistic mass is a missnomer, and no it is not kinetic energy.

 

[pmb_phy]

I don’t see how it’s two different frames in the “case I’ve just explained.” It seems to me that it’s all happening in my frame. Things can move in my frame right?

I made a mistake. I meant that the situation of having one lump of clay (which isn't a black hole) at rest in a frame is different than having two lumps of clay approaching each other.

Say I’m floating in deep space, watching a light-year-in-diameter sphere of space. Within that sphere are the two lumps of clay (let’s say boxcar-sized) approaching each other. They collide in the middle of the sphere, releasing enough heat energy to create a black hole with an event horizon that encloses the sphere. Didn’t I witness a black hole spontaneously form in a region of space to which no mass was added? All in one frame?

It takes more than energy to create a black hole. There is enough mass in my body to create a black hole. All that needs to be done to create a black hole is to have all my matter within a much smaller region of space.

In what you've just described, that energy must be compressed into a region of space so small as to have all the matter all within the Schwarzschild radius. However the mass remained constant in the situation you just described. Because mass is a conserved quantity the total mass in the end was identical to the total mass in the beginning. You forgot that there is mass just due to the motion of the clay lumps, i.e. the fast a lump of clay, the greater its mass is.

Pete

 

[Zanket]

In what you've just described, that energy must be compressed into a region of space so small as to have all the matter all within the Schwarzschild radius. However the mass remained constant in the situation you just described. Because mass is a conserved quantity the total mass in the end was identical to the total mass in the beginning. You forgot that there is mass just due to the motion of the clay lumps, i.e. the fast a lump of clay, the greater its mass is.

By "spontaneous" I mean "unconnected to a cause," not "immediately."

Yes, the mass remained constant (I didn’t forget; in the example I’m using that fact to make a point). In my frame, this constant mass was compressed to within a Schwarzschild radius when the lumps of clay were one light year apart. But not until they collided at least six months later did the black hole form (with an event horizon diameter of one light year).

Unlike the case of an imploding star which surface falls through its Schwarzschild radius, in which case a black hole exists immediately, in the lumps example the black hole did not form until at least six months after all the mass fell through the Schwarzschild radius. The six-month delay seems strange.

All that needs to be done to create a black hole is to have all my matter within a much smaller region of space.

It took more than that in the case of the lumps. The lumps' relativistic mass had to be converted to rest mass (heat energy). That’s what’s strange.

 

[j8hart]

I do not interpret this as meaning you can accelerate a particle less massive than the Planck energy into a black hole. Rather, that if you could accelerate and collide less massive particles at high enough energies, a Planck energy size particle could be created [i.e., a tiny black hole result]. This is the same process CERN used to discover the W+, W- and Zo particles. They did not convert less massive particles to these highly massive particles through acceleration, it was through the energy-matter conversion resulting from collisions between highly accelerated less massive particles

Thanks for that Chronos. I think that was the quote. And I am sure you are right, so now I finally understand it

So he is talking about creating theoretical particle whose rest mass is more than 10²² times greater than an electron?

Sounds like a “don’t try this at home kids” kind of experiment!

But wouldn’t the particle also have to be contained within a radius of about 10^-35 of a meter?

 

[pmb_phy]

In short, does relativistic mass have a gravitational component?

Yes.

And if so, does it increase in proportion to the inertia?

No. The gravitational force, G on a moving particle has componentsG_k is given by

G_k = m\Gamma^{\alpha}_{k\beta}v_{\alpha}v^{\beta}

where the relativitic mass m is given by

m = \frac{dt}{d\tau}

which is a function of both speed and gravitational potential. For a derivation of this result, and the definition of the terms used, please see

http://www.geocities.com/physics_world/gr/grav_force.htm

So while you've included the speed factor of the mass you've neglected the functional dependence of m on gravitational potential. You've also incorrectly assumed that the velocity dependence of the gravitational force depends only on m.

F = GM2/r2

From this relationship I assume that you're referring to the gravitational force on one particle due to the mass of the other body.

Now imagine an observer "o" at exactly the mid point between the two spheres. Also imagine that o observes the spheres as moving with a velocity close to the speed of light along a course exactly perpendicular to a line drawn between the two spheres and through o.

Does "o" remain stationary while the two paricles move?

The diagram below shows the situation, o in the middle, the two spheres are represented by *, and the arrows show the direction on motion.

*-------------->

o

*-------------->


If the velocity of the spheres is chosen so that the relativistic mass is 10 times the rest mass, then we should still be able to use Newton’s law of Universal Gravitation to calculate the force exerted by one sphere on the other as observed by o.

Newton's Law is inadequate here since Newton's Law is not a function of the speed of the particle etc.

To observer o, M' = 10Kg. r is still 1m (obviously no contraction, since it is perpendicular to the line of motion).

Thus theoretically the gravitational force (F') as observed by o is 100xG Newtons.

In other words F' = 100xF.

I don't follow. The gravitational force on what? And where did the factor of 100 come in? I thought you said gamma = 10.

If an observer on the spheres sees no acceleration towards the other sphere because of the electrostatic force, then nether can o.

If therefore the Gravitational force increases 100 fold, then so must the electrostatic force.

The electric field does increase by a factor of gamma. I'm not clear on the specifics of your question though

More later

Pete

 

[Zanket]

Pmb_phy (or anyone),

Any comment on my post above? Can you confirm whether or not it is widely accepted (among experts in general relativity) that the theory supports the spontaneous formation of a black hole an arbitrarily long time after all the mass fell through the Schwarzschild radius? (Where the time is measured by a distant observer at rest with respect to the region in which the hole eventually forms.)

 

[DW]

Yes.

No. The gravitational force, G on a moving particle has componentsG_k is given by

G_k = m\Gamma^{\alpha}_{k\beta}v_{\alpha}v^{\beta}

where the relativitic mass m is given by

m = \frac{dt}{d\tau}
...

Now you're just getting rediculous.

The electric field does increase by a factor of gamma. I'm not clear on the specifics though

Don't tell me you are about to propose un-invariant charge now?!
Mass is invariant the same way that charge is. Factors of \gamma in the dynamics equations come from time dilation in proper time derivatives, not from the mass, and just the same not from the charge.

 

[pmb_phy]

Pmb_phy (or anyone),

Any comment on my post above? Can you confirm whether or not it is widely accepted (among experts in general relativity) that the theory supports the spontaneous formation of a black hole an arbitrarily long time after all the mass fell through the Schwarzschild radius? (Where the time is measured by a distant observer at rest with respect to the region in which the hole eventually forms.)

Yes. That's correct.

Pete

 

[pervect]

Now you're just getting rediculous.

Don't tell me you are about to propose un-invariant charge now?!

First I want to point out that I am _not_ a fan of relativistic mass. I prefer to use the term energy, it leads to a lot less arguments and confusion.

Secondly, I want to point out that the transverse electric field of a moving charge *DOES* increase by a factor of gamma.

The easiest way to show this is to boost the Faraday tensor

<br /> F_{ab}= \left[ \begin {array}{cccc} 0&amp;-{\it Ex}<br /> &amp;-{\it Ey}&amp;-{\it Ez}\\\noalign{\medskip}{\it Ex}&amp;0&amp;{\it Bz}&amp;-{\it By}<br /> \\\noalign{\medskip}{\it Ey}&amp;-{\it Bz}&amp;0&amp;{\it Bx}\\\noalign{\medskip}{<br /> \it Ez}&amp;{\it By}&amp;-{\it Bx}&amp;0\end {array} \right] <br />

if you boost around the z axis you should get

<br /> F_{a&#039;b&#039;} = \left[ \begin {array}{cccc} 0&amp;{\frac {-{\it Ex}+{\it By}\,v}{\sqrt {1<br /> -{v}^{2}}}}&amp;-{\frac {{\it Ey}+{\it Bx}\,v}{\sqrt {1-{v}^{2}}}}&amp;-{\it <br /> Ez}\\\noalign{\medskip}-{\frac {-{\it Ex}+{\it By}\,v}{\sqrt {1-{v}^{2<br /> }}}}&amp;0&amp;{\it Bz}&amp;{\frac {{\it Ex}\,v-{\it By}}{\sqrt {1-{v}^{2}}}}<br /> \\\noalign{\medskip}{\frac {{\it Ey}+{\it Bx}\,v}{\sqrt {1-{v}^{2}}}}&amp;<br /> -{\it Bz}&amp;0&amp;{\frac {{\it Ey}\,v+{\it Bx}}{\sqrt {1-{v}^{2}}}}<br /> \\\noalign{\medskip}{\it Ez}&amp;-{\frac {{\it Ex}\,v-{\it By}}{\sqrt {1-{<br /> v}^{2}}}}&amp;-{\frac {{\it Ey}\,v+{\it Bx}}{\sqrt {1-{v}^{2}}}}&amp;0<br /> \end {array} \right] <br /> <br />

My favorite link for the relativistic E-field of a moving charge is at

http://www.phys.ufl.edu/~rfield/PHY2061/images/relativity_14.pdf

you can of course do gauss's intergal on the resulting E-field in accordance with gauss' law to get a conserved charge.

 

[DW]

First I want to point out that I am _not_ a fan of relativistic mass. I prefer to use the term energy, it leads to a lot less arguments and confusion.

Secondly, I want to point out that the transverse electric field of a moving charge *DOES* increase by a factor of gamma.

The easiest way to show this is to boost the Faraday tensor

<br /> F_{ab}= \left[ \begin {array}{cccc} 0&amp;-{\it Ex}<br /> &amp;-{\it Ey}&amp;-{\it Ez}\\\noalign{\medskip}{\it Ex}&amp;0&amp;{\it Bz}&amp;-{\it By}<br /> \\\noalign{\medskip}{\it Ey}&amp;-{\it Bz}&amp;0&amp;{\it Bx}\\\noalign{\medskip}{<br /> \it Ez}&amp;{\it By}&amp;-{\it Bx}&amp;0\end {array} \right] <br />


if you boost around the z axis you should get

<br /> F_{a&#039;b&#039;} = \left[ \begin {array}{cccc} 0&amp;{\frac {-{\it Ex}+{\it By}\,v}{\sqrt {1<br /> -{v}^{2}}}}&amp;-{\frac {{\it Ey}+{\it Bx}\,v}{\sqrt {1-{v}^{2}}}}&amp;-{\it <br /> Ez}\\\noalign{\medskip}-{\frac {-{\it Ex}+{\it By}\,v}{\sqrt {1-{v}^{2<br /> }}}}&amp;0&amp;{\it Bz}&amp;{\frac {{\it Ex}\,v-{\it By}}{\sqrt {1-{v}^{2}}}}<br /> \\\noalign{\medskip}{\frac {{\it Ey}+{\it Bx}\,v}{\sqrt {1-{v}^{2}}}}&amp;<br /> -{\it Bz}&amp;0&amp;{\frac {{\it Ey}\,v+{\it Bx}}{\sqrt {1-{v}^{2}}}}<br /> \\\noalign{\medskip}{\it Ez}&amp;-{\frac {{\it Ex}\,v-{\it By}}{\sqrt {1-{<br /> v}^{2}}}}&amp;-{\frac {{\it Ey}\,v+{\it Bx}}{\sqrt {1-{v}^{2}}}}&amp;0<br /> \end {array} \right] <br /> <br />

My favorite link for the relativistic E-field of a moving charge is at

http://www.phys.ufl.edu/~rfield/PHY2061/images/relativity_14.pdf

you can of course do gauss's intergal on the resulting E-field in accordance with gauss' law to get a conserved charge.

Just to be clear, I was not saying that it didn't. I was just saying that the factor of \gamma does not come from the charge which like mass is invariant. It comes instead ultimately from time dilation.

 

[Zanket]

Yes. That's correct.

Seems strange! Thanks much for the confirmation.