In the context of travel near c how is mass defined?

[Logician]

In the context of travel near "c" how is mass defined?

Hello all,

Brand new member here. I am trying to fully understand the different kinds of masses, i.e. rest mass, relativistic mass and inertial mass as well as any other kinds of relevant kinds of masses I am forgetting about, in the context of some theoretical travel at or near light speed. I have been to college before but I stopped to go help start up a business and so I have not yet been able to learn the the higher level math for physics. I am mostly self educated but I do have a very good grasp on the basics of physics and a decent amount of harder things, however, I am restarting school and going to earn my Education degree with at least a minor in physics.

I have a passionate love for science in general and physics in particular. I would appreciate any time and knowledge anyone has that they are willing to allow me to learn from. I am definitely not your average person who dropped out of college. I have the ability to truly understand these concepts if someone would be willing to explain things to me clearly. Please take mercy on this person who i temporarily ignorant f the higher math classes in college as well as missing the upper level physics classes in my first terms at college. I know that learning from the people on this forum will very much like me having Yoda whack m over the head until I get things right, but this I promise, I will not try to learn I will learn period and I will spend the time. Reading what is suggested.

As a very related question does anyone happen to be willing to share the information with me about when does a person start gaining mass and is the rise proportional once it starts or does the gain of mas happen in very short spurts as one crosses certain thresholds of sped? Also I would love to get a hold of the actual mathematical formula for the gaining of mas as on approaches the speed of light. I may not be able to use it now but I will in the future.

Thank you all for your kindness and welcome. I was somewhat apprehensive about posting here as I am not the most college-educated man but I hope that a lifetime of reading has helped me prepare in this most challenging of fields.


The Logician

 

[Bandersnatch]

Hi Logician, welcome to PF!

Well, tell us what you already know. We can address specific questions, but it's hard to describe broad subjects like those mentioned.

In very few words, classically there's inertial mass that is the measure of the resistance against acceleration(the m in F=ma), and gravitational mass that is the source of the gravitational force(the m's in the law of Gravity).
All experiments trying to compare them so far show that the two are equal(even though they appear conceptually not related, they seem to be the same property).

Within Relativity, you get the rest mass - the inherent mass of objects at rest. Then there's the relativistic mass - the increased mass moving objects seem to have.
To answer you question, the increase is continuous - any non-zero speed causes rel.mass to go up. It's just that at low speeds(as compared to the speed of light) the change is imperceptible.

If you don't know anything at all about relativity take a look at this site:
http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html (the relativistic mass section has got the relevant equation)
and come back when there's something you can't understand.

Be sure to read this bit:
http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html
as it's bound to come up sooner or later in the discussion(some members here are vehemently opposed to using the term relativistic mass at all as pedagogically misleading).

 

[DrStupid]

I am trying to fully understand the different kinds of masses, i.e. rest mass, relativistic mass and inertial mass as well as any other kinds of relevant kinds of masses I am forgetting about, in the context of some theoretical travel at or near light speed.

I'll try to give you a short overview:

1. classical inertial mass

This is m in Newton's definition of momentum p=m·a.
In classical mechanics this mass is invariant. In special relativity it becomes velocity depended (see relativistic mass).

2. gravitational mass

This is m in Newton's law of gravitation F=G·m1·m2·r/|r|³.
In classical mechanics this mass is assumed to be proportional to the classical inertial mass (supported by experimental observations) and therefore invariant too. In relativity this mass does not exists at all because Newton's law of gravitation is not compatible with relativity.

3. relativistic mass

This mass is identical to the classical inertial mass (see above) but by replacement of Galilei transformation with Lorentz transformation it becomes velocity dependent: m=mo/sqrt(1-v²/c²). In relativity is is also proportional to total energy: E=m·c².

4. rest mass or invariant mass

This is the mass (no matter what kind of mass) of a body at rest (p=0).

5. longitudinal and transversal mass

These are the Eigenvalues of M in F=M·a in relativity. Is the force acting parallel to the velocity than M = [I+v·v^T/(c²-v²)]·mo/sqrt(1-v²/c²) turns into the longitudinal mass mo/sqrt(1-v²/c²)³ and perpendicular into the transversal mass mo/sqrt(1-v²/c²).

As there were no general convention for the use of the term "mass" in the beginning of relativity, these different concepts of mass lead to lot confusions. Today mass refers to rest mass only.

 

[Drakkith]

As a very related question does anyone happen to be willing to share the information with me about when does a person start gaining mass and is the rise proportional once it starts or does the gain of mas happen in very short spurts as one crosses certain thresholds of sped? Also I would love to get a hold of the actual mathematical formula for the gaining of mas as on approaches the speed of light. I may not be able to use it now but I will in the future.

A person never gains mass as they speed up because in that person's frame of reference they are always at rest. Also, if we look at the full form of einstein's equation:

e²=(mc²)²+(pc)²

We see that M is the mass, which is ALWAYS the rest mass of the object, while any relative velocity contributes to the momentum of the object, which is P in the equation.

Remember that velocity is always relative to something else. Both objects are equally right in saying that the other object is moving while they are stationary. In such a case, it is obvious that velocity can never add to the mass of a single object because you would never gain mass just because something else is moving very quickly relative to you. Some of the high energy cosmic rays are protons moving at 99.9999% c or greater. According to them, YOU are moving at that speed, yet you and the Earth do not collapse in on yourselves because you haven't gained any mass.

 

[Logician]

Drakkith,

I am confused. my understanding is that the reason we cannot travel at "c"in normal space is because of the gain of mass as one goes faster. Therefore more and more energy is needed to push the spaceship we are traveling in closer and closer to "c".

But if what you stated, "A person never gains mass as they speed up because in that person's frame of reference they are always at rest." is correct, then absent some outside force acting on the vessel then we could use an engine that, while not ultra powerful,has an extremely long life,(up to tens of thousands of years)and would give us the constant acceleration that could push us to "c".

However, I have read a lot of information from experts who all aver that without using some extremely high gravity and/or quantum gravity principles it is impossible for us to travel at or even very near "c"

Can we travel at or very near "c" or can we not and if we do not gain mass but we cannot travel that fast then please help me to understand what the reason is that even with an engine powerful enough to get us close to "c" that travel at that speed is impossible?

Bandersnatch,You advised me ,"Well, tell us what you already know. We can address specific questions, but it's hard to describe broad subjects like those mentioned."

So here is a specific question. If we assume that the idea that the faster an object goes the more mass it acquires, with travel at "c" equating to infinite mass, how does a particle that has almost zero mass,such as a neutrino, not become much more massive as it approaches the very high speeds associated with near "c" speed. Why does the neutrino not gain a sizable amount of mass while it is traveling at very high speeds?

As a corollary to the theory that things gain mass the faster they travel do they also lose mass when they slow down? If no why is that not the case?

DrStupid,

You advised me that,"Rest mass or Invariant mass. This is the mass (no matter what kind of mass) of a body at rest (p=0)." With that being said can an object ever actually be at rest? Here on Earth the Earth is moving both by spinning on it's axis and tracing an elliptical orbit around the sun. Beyond that of course our solar system orbits the center of our Milky Way galaxy and even further there is the expansion of the universe. So even if the Earth stopped all motion and our galaxy did the same, a person on Earth would still be in motion due to the expansion of the Universe.

Is this correct or am I missing something please help me to understand this.

A couple of other specific questions that relate to this subject. I will try and ask specific questions instead of very broad ones.

My first question is this, if the cosmic speed limit for almost everything is "c" and since we are not the center of the universe, how is it that when we detect the cosmic background radiation that no matter where in the sky we look the radiation is there and is almost completely uniform? The reason I ask is that the radiation appears to be coming from every point in space towards us. If there is a point "x" that is the absolute center of the universe, and the Big Bang originated there then it stands to reason that when we look out towards that point that we should see the radiation but when we look away from it then we either should not see it(if there hasn't been enough time for the radiation to pass us and travel huge distances past us) or we should see it hurtling away from us. Imagine yourself standing ten feet away from a firecracker suspended ten feet in the air. When the explosion takes place the energy is radiated in all directions. Your location will get hit with the energy radiation from the explosion some very small time in the future but if you turn around before the explosion and face away from it then in order for you to perceive the energy coming towards you that energy would have had to circle the Earth to come back towards you. However that energy that circled the Earth would take a measurable difference in time to get to you. Now with EM radiation having to go no faster than "c" the universe would have to be immensely older than we understand it to be in order for the radiation to travel completely around the universe to come at us from the side away from the direction the Big Bang happened.

Now I know that I am stating these things as if they are facts but I am not sure of this. If anyone knows the answer to these two things I would be greatly appreciative.
1: When we detect the cosmic background radiation on the side away from the Big Bang, are we detecting radiation coming towards us or are we detecting the remnants of that radiation as it has passed us?
2: If the radiation is actually coming towards us on the opposite side how is that possible?

Lastly, mu understanding of special and general relativity state that the only property of relativity that is, as Drakkith replied,"Remember that velocity is always relative to something else. Both objects are equally right in saying that the other object is moving while they are stationary." sort of dependent on the observer is time dilation. Are there more effects that are observer dependent like this or is Drakkith correct that all relativistic phenomena is observer dependent?

Thank you all for your time,effort and knowledge,
Logician

 

[Bandersnatch]

I'll just tackle the Big Bang question, if I may:

There isn't nor ever was a centre of the universe. The BB doesn't describe an explosion in space(like a grenade shrapnel exploding outwards from some particular point), but in time. 'Explosion' and 'bang' are generally terrible words to describe the theory, and are constantly misleading countless people curious enough to try and learn more about it.
BB happened everywhere at once, in the whole volume of the (possibly infinite) universe.
A bit more precisely, BB is an observation that when we extrapolate the expansion of space backwards in time, we have to arrive at a very dense and hot universe.
Somebody(Fred Hoyle, was it?) made a connection dense+hot+expansion -> kinda what an explosion is -> let's call it Big Bang. And it stuck.

Check out the "Balloon analogy" sticky thread in the cosmology section of the forum to learn more about expansion of the universe and the Big Bang, or read this:
http://www.mso.anu.edu.au/~charley/papers/LineweaverDavisSciAm.pdf
and this:
http://www.phinds.com/balloonanalogy/


Alright, and the neutrino question:
The neutrinos do gain mass(i.e., energy) as they travel faster. There's a whole range of neutrinos predicted to exist, with varying speeds(all very close to c) and correspondingly varying energies. These are, fundamentally, all the same particles, and the only difference between them is their speed.
If they were slowed down, they'd lose mass, yes.
Every time I used mass here I meant the equivalent of total energy, not rest mass(which doesn't change).

 

[Logician]

Thank you Bandersnatch,
That makes me feel more comfortable with an idea I have been thinking about a lot.I feel comfortable with the idea that the vast majority of educated people understand and agree that two of the fundamental properties of all matter is it has mass(I am not talking about anti-matter here) and it has potential energy.
However I think that when we talk about potential energy, we are almost always talking about kinetic energy and in some cases chemical energy. I also believe that each piece of matter has potential kinetic,chemical as well as nuclear energy. No matter what the matter is made of, when placed under the correct conditions, it will undergo a thermic reaction. Also with the right conditions if an atom of matter has the nucleus split it will obviously release energy.

I think most people understand what potential energy is. What I have been mulling is this though. Because of the equivalence of mass and energy as expressed in E=mc squared then shouldn't we talk about energy as having potential mass? Mass(which is in my opinion just another way to say matter) has the fundamental property of potentiality of energy so energy should have the fundamental property of potentiality of mass. With that accepted my brain went one step further and asked the following question," If energy has potential mass, then what process(es) would change energy to matter?" If we could figure this out then we could start to better understand what happened during the very early universe and also to better understand how the first matter coalesced out of all the energy released during the Big Bang.

If anyone know where I can find more information on the idea of potential mass and/or on how the first bits of matter came to be out of the gigantic sea of cosmic energy of the Big Bang.

Thank You for all of your time, energy and knowledge,
Logician

P.S. I have asked a few questions I would like to know if the questions I am asking are stupid. I would like to know if my basic ideas and knowledge about the physical laws that govern our universe is pretty close or not even close to what the facts are. If I am way off in most or al of my thinking then I know it is time to go back to the early physics books.

Thanks again.

 

[DrStupid]

can an object ever actually be at rest?

Yes of course. In it's rest frame it is at rest per definition. If this frame of reference is locally free falling the object would even be at rest in an inertial system.

My first question is this, if the cosmic speed limit for almost everything is "c" and since we are not the center of the universe, how is it that when we detect the cosmic background radiation that no matter where in the sky we look the radiation is there and is almost completely uniform?

It is not completely uniform. For example there is a dipole anisotropy that tells us that we are moving relative to the rest frame of the radiation field.

If there is a point "x" that is the absolute center of the universe

There is no such point.

and the Big Bang originated there

The Big Bang originated everywhere.

Imagine yourself standing ten feet away from a firecracker suspended ten feet in the air.

If the universe would be a firecracker we would be part of it.

1: When we detect the cosmic background radiation on the side away from the Big Bang

There is no side away from the Big Bang.

2: If the radiation is actually coming towards us on the opposite side how is that possible?

It is possible because the Big Bang was everywhere in the universe. That's why the background radiation comes from everywhere too - from any direction and from any distance.

It appears that you have some general misconceptions in regard to the Big Bang and the expansion of the universe.

Are there more effects that are observer dependent like this or is Drakkith correct that all relativistic phenomena is observer dependent?

There are relativistic effects that can be observed by a single observer (e.g. the increase of inertia with velocity) but without observations from another observer it would be hard to realize that these effects are relativistic.

 

[Bandersnatch]

Well, you do need to do some proper reading, that's for sure. There's a lot of misunderstood terminology and concepts here.

Just randomly taking potshots at your last post:
Mass and energy are the essentially the same thing. Mass and matter are not the same thing. Antimatter does have mass. Potenial energy contributes to mass(=energy) of matter. Potential energy is a special subset of the concept of energy, entirely different from kinetic energy. Chemical energy is fundamentally just potential energy(binding energy). PE is hardly a fundamental property of matter. Fundamental properties of matter include rest mass, charge, spin, colour etc.

It pays to learn what the terms you're using mean precisely, so that you know exactly what you're talking about, can be understood, and not end up reinventing the wheel.


For how matter and energy change into one another, google 'pair production' and 'annihilation'. Googling 'big bang nucleosynthesis' will net you some more relevant information. Wikipedia articles are a very good start.
A good book to read about the stuff is S.Weinberg's "First three minutes".


Remember that science is not about personal opinions and vague speculation, but observation, precision, predictions and testing.

 

[Mordred]

The first three minutes although excellent at its time is somewhat misleading to cosmology today. Simply due to the understanding at the time of its writing. Particle physics has come a long way since then. The chronology of the universe described in modern textbooks no longer go into the Planck epoch, lepton epoch, electro-weak epoch etc. I recommend a relatively current article covering the history of thermodynamics of the universe. This at least correlates to Scott Dodleson's Modern cosmology 2nd edition.

http://www.wiese.itp.unibe.ch/lectures/universe.pdf :" Particle Physics of the Early universe" by Uwe-Jens Wiese Thermodynamics, Big bang Nucleosynthesis.

This is a now free Cosmology textbook, the author explains that he has released it as much has changed since his writing of it.

http://arxiv.org/pdf/hep-th/0503203.pdf "Particle Physics and Inflationary Cosmology" by Andrei Linde

this article is similar to a textbook for its collection of formulas and explanations however doesn't go into as much detail as the typical textbook

http://arxiv.org/pdf/hep-ph/0004188v1.pdf :"ASTROPHYSICS AND COSMOLOGY"- A compilation of cosmology by Juan Garcıa-Bellido

numerous more article, covering common misconceptions and teaching articles can be found on my signature. There is even a historical article section, such as the "great debate" and “The Waters I am Entering No One yet Has Crossed”: Alexander Friedman and the Origins of Modern Cosmology"

The articles I have posted is based on a few years of searching, and from my resultant database. I'm always interested in other recommended articles to add to the site.

there is also two articles on GR, though the one from Einstein is not the best to learn from, the Mathius Blau article is highly advanced

 

[Logician]

Thank you all for your links and suggested readings. I will be reading them soon. After doing some thinking I think I can more clearly illustrate what I am trying to understand by saying that I am trying to get a handle on the MECHANISM by which relativistic effects happen and how to picture in my mind things like exactly what relativistic mass is and the reasons why Newtonian physics appear to break down at relativistic speeds. An example I can give is that in my mind it is perfectly logical that if you can provide a constant unending acceleration then eventually you should be able to get to light speed. I understand it is not true but I do not understand the WHY of the fact that it is not true. I haven't yet read anything that tells me how to think of relativity exactly. I am trying to say what I man but it is hard. If the rest mas does not increase but relativistic mas does what is the actual real world difference between the two. Yes that is it. I somewhat understand the concepts of relativity in an abstract way but not in a real world way. Time dilation as well, I understand it in an abstract way but not in a real world way. I understand that time observed time pass slower at higher speeds but what is the actual reason or mechanism by which time slows down? These are the things I am trying to wrap my mind completely around in a real world way.

Thank You,
Logician

 

[Logician]

Eureka!

With the help form those who replied as well as som more reading I think I understand now the idea of relativistic mass and why relativity says you cannot get to "c"

Please tell me if I am understanding this correctly.
If we take the classic equation E=mcsquared then we can restate the equation as M=E/csquared. With that equation it means for every amount E of energy that is put into a system the M is increased. Therefore in relativity you can pump more and more energy into the system but all that does is increase the Rmass of the system. I hope I got that basically right

P.S. Will someone please explain to me how to put mathematical notation like squared or cubed for example?

Thank You,
Logician

 

[Dale]

my understanding is that the reason we cannot travel at "c"in normal space is because of the gain of mass as one goes faster. Therefore more and more energy is needed to push the spaceship we are traveling in closer and closer to "c".

That is one common explanation which is used (and abused) in the pop-sci literature. The thing is that this explanation is circular. It is based on the concept of "relativistic mass" which is simply another word for energy in natural units. So, the pop-sci explanation in essence goes "you gain energy therefore more energy is needed". The use of the word mass is just a red herring.

It takes an infinite amount of energy to accelerate a massive object to c because the KE becomes unbounded as v approaches c. This, in turn, is due to the fact that c is invariant, although the explanation for that is beyond what I can put in a post other than to reference Einstein's 1905 paper.

 

[ghwellsjr]

P.S. Will someone please explain to me how to put mathematical notation like squared or cubed for example?

I suspect that you are using the "Post Quick Reply" button at the bottom of this page. Instead, use the "Go Advanced" button. Then you will see a bunch of icons at the top of the Message window. One of these is the X² button. So if you want it to say X², you first type X2 and then highlight just the 2 and click the X² button.

 

[Mordred]

the method already described is one way, another is using latex the steps and examples of latex commands can be found here

https://www.physicsforums.com/showpost.php?p=3977517&postcount=3

 

[ghwellsjr]

the method already described is one way, another is using latex the steps and examples of latex commands can be found here

https://www.physicsforums.com/showpost.php?p=3977517&postcount=3

The built-in PF way works on all devices, at least all of mine, whereas the latex way does not. If all you're doing is something simple, the PF way is better, if you want it to be understandable by more people.

 

[Drakkith]

Drakkith,

I am confused. my understanding is that the reason we cannot travel at "c"in normal space is because of the gain of mass as one goes faster. Therefore more and more energy is needed to push the spaceship we are traveling in closer and closer to "c".

As far as I know that isn't correct.

But if what you stated, "A person never gains mass as they speed up because in that person's frame of reference they are always at rest." is correct, then absent some outside force acting on the vessel then we could use an engine that, while not ultra powerful,has an extremely long life,(up to tens of thousands of years)and would give us the constant acceleration that could push us to "c".

No, it is not possible to reach c.

However, I have read a lot of information from experts who all aver that without using some extremely high gravity and/or quantum gravity principles it is impossible for us to travel at or even very near "c"

You can reach in arbitrary speed less than c, even one very, very close to c, but you will never reach c.

Can we travel at or very near "c" or can we not and if we do not gain mass but we cannot travel that fast then please help me to understand what the reason is that even with an engine powerful enough to get us close to "c" that travel at that speed is impossible?

I don't know if there is an underlying reason other than that is just the way it works.

Now I know that I am stating these things as if they are facts but I am not sure of this. If anyone knows the answer to these two things I would be greatly appreciative.
1: When we detect the cosmic background radiation on the side away from the Big Bang, are we detecting radiation coming towards us or are we detecting the remnants of that radiation as it has passed us?
2: If the radiation is actually coming towards us on the opposite side how is that possible?

Note that radiation that passes us cannot be seen for the simple reason that it has passed us. Radiation doesn't emit more radiation that then comes towards us that we see, it simply passes us by and we don't see it.

 

[xox]

I'll try to give you a short overview:

1. classical inertial mass

This is m in Newton's definition of momentum p=m·a.

No, it isn't, $p=mv$ , more correctly said $\vec{p}=m \vec{v}$

In classical mechanics this mass is invariant. In special relativity it becomes velocity depended (see relativistic mass).

"Relativistic mass" is deprecated in contemporary physics. The correct approach is to use the notion of relativistic energy-momentum, $\gamma(mc^2, m\vec{v})$.

2. gravitational mass

This is m in Newton's law of gravitation F=G·m1·m2·r/|r|³.

If you write it the way you wrote it above, the correct expression is $\vec{F}=\frac{Gm_1m_2 \vec{r}}{r^3}$. Writing F=G·m1·m2·r/|r|³ is no different from writing F=G·m1·m2/|r|^2

3. relativistic mass

This mass is identical to the classical inertial mass (see above) but by replacement of Galilei transformation with Lorentz transformation it becomes velocity dependent: m=mo/sqrt(1-v²/c²). In relativity is is also proportional to total energy: E=m·c².

Nope, total energy is $E=\gamma mc^2$. m·c² is the rest energy.

5. longitudinal and transversal mass

These notions have been long expunged from physics.

the force acting parallel to the velocity than M = [I+v·v^T/(c²-v²)]·mo/sqrt(1-v²/c²) turns into the longitudinal mass mo/sqrt(1-v²/c²)³ and perpendicular into the transversal mass mo/sqrt(1-v²/c²).

This is total nonsense, for multiple reasons:

- the "longitudinal" and "transverse" mass have been expunged from modern physics

- even worse, M = [I+v·v^T/(c²-v²)]·mo/sqrt(1-v²/c²) is total gibberish

- the modern way is to express $\vec{F}=m\frac{d \gamma \vec{v}}{dt}=m(\gamma \frac{d \vec{v}}{dt}+\vec{v} \frac{d \gamma}{dt})=m(\gamma \vec{a}+\vec{v} \frac{d \gamma}{dt})$, so the force has a component along the acceleration and one along the velocity.

 

[DrStupid]

No, it isn't, $p=mv$ , more correctly said $\vec{p}=m \vec{v}$

Yes, you are right. That was a typo.

The correct approach is to use the notion of relativistic energy-momentum, $\gamma(mc^2, m\vec{v})$.

That's correct for another concept of mass.

Writing F=G·m1·m2·r/|r|³ is no different from writing F=G·m1·m2/|r|^2

Only in the one-dimensional case. In two or three dimensions it is different.

Nope, total energy is $E=\gamma mc^2$. m·c² is the rest energy.

You again refer to a different concept of mass. Please read my posts more carefully.

the "longitudinal" and "transverse" mass have been expunged from modern physics

My post was not about modern physics but about different concepts of mass as used in the past.

- even worse, M = [I+v·v^T/(c²-v²)]·mo/sqrt(1-v²/c²) is total gibberish
- the modern way is to express $\vec{F}=m\frac{d \gamma \vec{v}}{dt}=m(\gamma \frac{d \vec{v}}{dt}+\vec{v} \frac{d \gamma}{dt})=m(\gamma \vec{a}+\vec{v} \frac{d \gamma}{dt})$

The resulting equations are identical.

 

[xox]

Yes, you are right. That was a typo.

Based on the rest on your post, I very much doubt that it "was a typo", it is an outright mistake.

That's correct for another concept of mass.

It is the MODERN, MAINSTREAM concept.

Only in the one-dimensional case. In two or three dimensions it is different.

No, it isn't since you wrote your equation in terms scalars only. In other words, in your posted equation, $r$ is the SAME thing as $|r|$.

You again refer to a different concept of mass. Please read my posts more carefully.

I did, they are riddled with mistakes and misconceptions.

My post was not about modern physics but about different concepts of mass as used in the past.

"Transverse" and "longitudinal" mass have been expunged from physics for a long, long time. No one uses these "concepts of mass" , so why do you bring them up? Besides your formulas are wrong.

The resulting equations are identical.

Prove it, your M = [I+v·v^T/(c²-v²)]·mo/sqrt(1-v²/c²) is total gibberish, care to explain what is v·v^T, for example? I gave you the mainstream derivation (of force), use it as a starting point to prove that your formulas are equivalent to the mainstream view.

 

[DrStupid]

It is the MODERN, MAINSTREAM concept.

Once again: my post was not about modern mainstream.

you wrote your equation in terms scalars only

Vector arrows are optional (and not available in plain text).

"Transverse" and "longitudinal" mass have been expunged from physics for a long, long time. No one uses these "concepts of mass" , so why do you bring them up?

For the sake of completeness.

Besides your formulas are wrong.

What exactly is wrong (apart from the typo) and why?

Prove it, your M = [I+v·v^T/(c²-v²)]·mo/sqrt(1-v²/c²) is total gibberish.
I gave you the mainstream derivation (of force), use it as a starting point.

With

$$\gamma = \frac{1}{{\sqrt {1 - \frac{{v^2 }}{{c^2 }}} }}$$

and therefore

$$\frac{{d\gamma }}{{dt}} = \frac{1}{{c^2 }} \cdot \frac{{v^T \cdot a}}{{\sqrt {1 - \frac{{v^2 }}{{c^2 }}} ^3 }}$$

your

$$F = m_0 \cdot \left( {\gamma \cdot a + v \cdot \frac{{d\gamma }}{{dt}}} \right)$$

turns into

$$F = m_0 \cdot \left( {\frac{a}{{\sqrt {1 - \frac{{v^2 }}{{c^2 }}} }} + \frac{v}{{c^2 }} \cdot \frac{{v^T \cdot a}}{{\sqrt {1 - \frac{{v^2 }}{{c^2 }}} ^3 }}} \right) = \frac{{m_0 }}{{\sqrt {1 - \frac{{v^2 }}{{c^2 }}} }} \cdot \left( {I + \frac{{v \cdot v^T }}{{c^2 - v^2 }}} \right) \cdot a = M \cdot a$$

 

[xox]

Once again: my post was not about modern mainstream.

It was not mainstream at all, modern or otherwise. This is why I corrected it.

Vector arrows are optional.

That was not the point, the point was that your formula made no sense as posted.

With

$$\gamma = \frac{1}{{\sqrt {1 - \frac{{v^2 }}{{c^2 }}} }}$$

and therefore

$$\frac{{d\gamma }}{{dt}} = \frac{1}{{c^2 }} \cdot \frac{{v^T \cdot a}}{{\sqrt {1 - \frac{{v^2 }}{{c^2 }}} ^3 }}$$

your

$$F = m_0 \cdot \left( {\gamma \cdot a + v \cdot \frac{{d\gamma }}{{dt}}} \right)$$

turns into

$$F = m_0 \cdot \left( {\frac{a}{{\sqrt {1 - \frac{{v^2 }}{{c^2 }}} }} + \frac{v}{{c^2 }} \cdot \frac{{v^T \cdot a}}{{\sqrt {1 - \frac{{v^2 }}{{c^2 }}} ^3 }}} \right) = \frac{{m_0 }}{{\sqrt {1 - \frac{{v^2 }}{{c^2 }}} }} \cdot \left( {I + \frac{{v \cdot v^T }}{{c^2 - v^2 }}} \right) \cdot a = M \cdot a$$

This is better than the gibberish you posted originally but it still contains a bunch of errors since you are not being able to deal with the vectors $\vec{F}, \vec{v}, \vec{a}$ that intervene in the formula.

Moreover, as written, the formula is WRONG. Way WRONG.

You must have copied the above from somewhere because of the presence of "v \cdot v^T" which makes no sense whatsoever.

 

[DrStupid]

You once again claim errors without sufficient justification. End of discussion.

 

[xox]

You once again claim errors without sufficient justification. End of discussion.

Let me give you a hint, you claim $F=M \cdot a$.
But the above is easily provable wrong since , as explained, $\vec{F}$ has components BOTH along $\vec{a}$ AND $\vec{v}$.

Try doing the complete derivation (don't copy and paste from elsewhere) and you may get the correct answer. As it stands, you have it wrong.

 

[Bill_K]

You must have copied the above from somewhere because of the presence of "v \cdot v^T" which makes no sense whatsoever.

Isn't this a dyadic, the outer product of v with itself?

Let me give you a hint, you claim $F=M \cdot a$.
But the above is easily provable wrong since , as explained, $\vec{F}$ has components BOTH along $\vec{a}$ AND $\vec{v}$.

I think the point is that M is a dyadic, and therefore F and a will not in general be parallel. The I part of M yields the part of F parallel to a, while the "v \cdot v^T" part yields the part parallel to v.

 

[xox]

Isn't this a dyadic, the outer product of v with itself?

I think he's copied it from someplace, look at his "derivation", especially the gibberish:

$$\frac{d\gamma}{dt}=\frac{1}{c^2}\frac{v^T a}{(\sqrt{1-(v^2/c^2})^3}$$

I asked him to explain the $v^T$, no answer.

The I part of M yields the part of F parallel to a,

Yes

while the "v \cdot v^T" part yields the part parallel to v.

I don't think it yields the part parallel to v, the outer product is a tensor product of two vectors, the LHS of the expression is a vector ($\vec{F}$). Besides, there is no reason to use this type of formalism, simple vector algebra suffices. Let him work through the math by himself, he has a history of posting errors and refusing to own up to them. This is the only way he'll learn.

 

[DrStupid]

Isn't this a dyadic, the outer product of v with itself?

Exactly.

 

[xox]

Exactly.

Why don't you do the complete derivation, rather than cutting and pasting? Here, start with:

$\vec{F}=m(\gamma \vec{a}+\vec{v} \frac{d \gamma}{dt})$

This is simple vector algebra, all is left for you is to calculate $\frac{d \gamma}{dt}$ and plug it into the above.

 

[dauto]

With the help form those who replied as well as som more reading I think I understand now the idea of relativistic mass and why relativity says you cannot get to "c"

Please tell me if I am understanding this correctly.
If we take the classic equation E=mcsquared then we can restate the equation as M=E/csquared. With that equation it means for every amount E of energy that is put into a system the M is increased. Therefore in relativity you can pump more and more energy into the system but all that does is increase the Rmass of the system. I hope I got that basically right

P.S. Will someone please explain to me how to put mathematical notation like squared or cubed for example?

Thank You,
Logician

This is probably the most common explanation given but if you scratch the surface you will see it is not a very insightful explanation since relativistic mass is just another name for energy it is no surprise that increasing the energy will increase the relativistic mass. That's just a tautology, not a significant statement. A better explanation is that v>c is not possible not because it is unachievable, but because it simply doesn't exist. An imperfect but insight analogy is the statement that the sine of an angle is restricted: sinθ <= 1. Sinθ > 1 is impossible not because it is unachievable, but because it does not exist. There is not way to make an angle θ that satisfy sinθ > 1 in Euclidean geometry. There is no way to create a speed v that satisfy v > c in space-time geometry - Not if causality is imposed as a principle anyways...

 

[berkeman]

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