Relative momentum and relative mass

[Amr Elsayed]

Hi all,
I feel like I have a misconception about that topic, so I hope I will get an answer for the question:

Momentum should be conserved from the same perspective, but does it have to from different perspectives or frames ? I mean we don't agree about sth's velocity, so we don't agree about momentum. What's wrong with that?
what I know is that mass changes from different frames is because each frame should observe the same momentum.

regards

 

[Orodruin]

What's wrong with that?

Nothing. Both momentum and energy are frame dependent quantities.

 

[Amr Elsayed]

Then why there is a need for mass to change due to special relativity laws ?

 

[Amr Elsayed]

I mean mathematically concerning momentum , Momentum and velocity are relativistic, then why mass is ?

 

[Janus]

Then why there is a need for mass to change due to special relativity laws ?

It's how the measurement of momentum changes as seen from different frames that distinguishes Newtonian physics and Relativity.

In the former, momentum can be measured as mass times velocity. This is frame dependent because an object's momentum as measured by any frame is dependent on its velocity with respect to that frame.

In Special Relativity, momentum is measured as mass times velocity times the Lorentz factor of $\frac{1}{\sqrt{1-\frac{v}{c^2}}}$

Since the Lorentz factor depends on the velocity, the increase of momentum with an increase of velocity is not linear under Special Relativity like it is under Newtonian physics.

 

[Amr Elsayed]

And why do we multiply it with Lorentz factor ?

 

[Orodruin]

I mean mathematically concerning momentum , Momentum and velocity are relativistic, then why mass is ?

It is not, at least not in the sense most physicists use the term. See our FAQ https://www.physicsforums.com/threads/what-is-relativistic-mass-and-why-it-is-not-used-much.796527/

 

[Orodruin]

And why do we multiply it with Lorentz factor ?

If we did not it would violate Lorentz invariance, i.e., the statement that the physics are described by the same equations in all inertial frames.

 

[Amr Elsayed]

violate Lorentz invariance, i.e., the statement that the physics are described by the same equations in all inertial frames

Would you please illustrate more ? why?

 

[Orodruin]

Would you please illustrate more ? why?

Which part of the statement do you have problems with?

In general, depending on your level, it may be more fruitful to think of things in terms of the classical expressions being only approximations to more general ones.

 

[Amr Elsayed]

I need to know what's different about momentum since we discovered time dilation. How this affects mass

 

[Nugatory]

I need to know what's different about momentum since we discovered time dilation. How this affects mass

it's not because we discovered time dilation - both time dilation and the relativistic momentum equation come from the Lorentz transformations, which are a consequence of the speed of light being finite and the same for all observers.

And it's not that anything was suddenly "different about momentum" after we discovered the Lorentz transforms. The momentum of a moving object has always been $p=\gamma{m}_0v$ and $p={m}_0v$ has always been just a very good approximation - so good, in fact, that it was several centuries before we figured out that it wasn't exact.

If the speed of light were infinite, then $p={m}_0v$ would be exact.

 

[Amr Elsayed]

Thank you, but I still want to know the mathematical and physical proof of mass change due to special relativity. I mean without mass changing " In my frame" momentum is conserved, and in the frame of any moving object it's conserved too. What is need for mass to change? or it's possible not necessary ?? You can tell me why P= gama*m*v
Sorry for bothering
regards,

 

[Orodruin]

Thank you, but I still want to know the mathematical and physical proof of mass change due to special relativity.

Did you read the FAQ I linked? Mass does not change in relativity.

 

[PeterDonis]

I still want to know the mathematical and physical proof of mass change due to special relativity

The "mass" that changes is relativistic mass, which is just another word for "energy". Do you understand why an object's energy changes when its velocity changes?

 

[Amr Elsayed]

Did you read the FAQ I linked? Mass does not change in relativity

I quickly did, but it was about how physics consider mass. I mean mass that is the resistance of the substance to accelerate.

The "mass" that changes is relativistic mass, which is just another word for "energy". Do you understand why an object's energy changes when its velocity changes?

I don't, sir. Just more kinetic energy is that i know, do you mean sth else ?

 

[Orodruin]

I quickly did, but it was about how physics consider mass. I mean mass that is the resistance of the substance to accelerate.

You should read it more carefully. It is about why relativistic mass is not a concept used by physicists. Therefore, we generally only talk about the rest mass. I also suggest you take post #12 to heart and think about it. There is a lot of useful material there.

 

[PeterDonis]

Just more kinetic energy

Kinetic energy is part of energy, so if kinetic energy increases with velocity, so does energy.

 

[PeterDonis]

I mean mass that is the resistance of the substance to accelerate.

In other words, you mean inertia (or "inertial mass"). If that's what you're interested in, you shouldn't be looking at just energy and momentum; you should be looking at force, to see how much acceleration a given applied force imparts to an object which is already moving at a given velocity. In other words, you should be looking at the relativistic equivalent of $F = m a$. See, for example, here:

http://en.wikipedia.org/wiki/Four-force

What you will find is that "relativistic mass" is not a useful concept for understanding inertia in general in relativity; it's better to focus on invariant mass (i.e., rest mass, i.e., what relativists mean when they say "mass") and how it appears in the various equations involved.

 

[Orodruin]

Just to add that this ...

What you will find is that "relativistic mass" is not a useful concept for understanding inertia in general in relativity; it's better to focus on invariant mass (i.e., rest mass, i.e., what relativists mean when they say "mass") and how it appears in the various equations involved.

... is also covered in the FAQ.

 

[Amr Elsayed]

why relativistic mass is not a concept used by physicists.

Thank you but it's not my problem, I want to learn about relativistic mass, I heard sth like we know it changes because of relativity and conservation of momentum

When an object's in a free fall for example due to gravity it will accelerate but will not reach c, They say because mass increases. I think this relates

What you will find is that "relativistic mass" is not a useful concept for understanding inertia in general in relativity; it's better to focus on invariant mass

Well, I want to understand what relativistic mass is if i will not waste your time

 

[Orodruin]

Well, I want to understand what relativistic mass is if i will not waste your time

It is not our time you would be wasting, it is your own. If you want to learn why resistance to acceleration changes with velocity and is different in different directions, the answer lies in the relativistic expression for momentum - as outlined in the FAQ. It is generally not what is called relativistic mass, which as noted before is just another way of saying "total energy" in relativity.

 

[PeterDonis]

When an object's in a free fall for example due to gravity it will accelerate but will not reach c

This has nothing to do with "relativistic mass". First of all, an object in free fall is not accelerating in any physical sense--its proper acceleration is zero. Its "acceleration" is coordinate acceleration, which is, as the name implies, coordinate-dependent; you can eliminate it by picking appropriate coordinates (those of a local inertial frame). So this is not at all analogous to an object undergoing proper acceleration in flat spacetime, but never reaching c relative to some fixed inertial observer.

 

[Amr Elsayed]

as outlined in the FAQ.

Unfortunately FAQ doesn't contain the proof for the increasing relativistic mass. I need a proof like: C is constant so there is length contraction and time dilation with equations. I think trying to understand sth is not time wasting :)

 

[Amr Elsayed]

coordinate acceleration

I just meant velocity increasing for a fixed frame of reference like the ground. If I fall in a planet that gives enough height and time to reach C ? increasing velocity will continue because of acting force. Why wouldn't it reach C then ?

 

[Orodruin]

Unfortunately FAQ doesn't contain the proof for the increasing relativistic mass. I need a proof like: C is constant so there is length contraction and time dilation with equations. I think trying to understand sth is not time wasting :)

It contains perfectly fine argumentation that based on $\vec p = m\gamma\vec v$, the force required to accelerate an object will depend on both direction and speed. It also states why relativistic mass is obsolete as a concept and not directly related to inertia. The reason that momentum appears the way it does is that it is the spatial part of a 4-vector, which transforms in a very particular way under Lorentz transformations.

I just meant velocity increasing for a fixed frame of reference like the ground. If I fall in a planet that gives enough height and time to reach C ? increasing velocity will continue because of acting force. Why wouldn't it reach C then ?

You should leave gravity out of the discussion for now. Gravity is not a force in relativity and requires general relativity for a proper treatment. Why an object does not reach c even after having a constant force acting on it for a very long time is explained in the FAQ.

 

[Nugatory]

Thank you, but I still want to know the mathematical and physical proof of mass change due to special relativity...You can tell me why P= gama*m*v

You will find a bunch of derivations if you google for "relativistic momentum derivation". Try reading a few, and if you still can't see why $p=\gamma{m}_0v$ is the correct expression for momentum, try asking some more specific questions and we can help you over the sticking point.

Note that these will be derivations of the relationship $p=\gamma{m}_0v$. More than a century ago, physicists took a wrong turn when they interpreted that equation as saying that there's a "relativistic mass" defined by $m=\gamma{m_0}$ and you just have to use the relativistic mass instead of the rest mass to make the old $p=mv$ work. It took many decades to get back on track after that wrong turn, and during those decades many textbooks were written and many popularizations were published... So to this day, you'll hear people saying "mass increases with speed". But that is not what modern physics says.

 

[PeterDonis]

I just meant velocity increasing for a fixed frame of reference like the ground.

But this "fixed frame of reference" is not an inertial frame, and does not work like an inertial frame in SR. (And acceleration relative to it is coordinate acceleration, as I said.)

If I fall in a planet that gives enough height and time to reach C ?

"Reach C" has no meaning as it stands, because, as above, this "fixed frame of reference" is not an inertial frame.

 

[Amr Elsayed]

"relativistic momentum derivation

0:52 for example
It's talking about 2 different frames
I think like" You are considering 2 different times for 2 different frames, P must be equal for only the same frame !" That's my story with derivations

 

[Amr Elsayed]

I have searched about it again in Haliday. I found that relativistic momentum is measuring velocity with distance measured by an observer (at rest) and time is measured by the moving object itself, how come ? I mean velocity should be determined as a single observer observe it.
It's also saying that total momentum is not conserved if we use classical law of momentum. I think I need an example for this, or I will give it my self
Suppose a car moving with 0.8 C with a mass of 100 kg crashes into a wall with a mass of 1000 kg
If I'm at rest according to the wall,then momentum before crash is 100kg*0.8 C, and after crash the car will go backward with a momentum of 100kg*0.8 C
If I am at the car, Momentum of moving wall" according to me" is 1000kg* 0.8 C and the same after since I will see the wall going forward
I will need to change mass if I calculate velocity as d/t where d is distance measured by me if I'm at rest according to wall and t is time needed to cut the distance according to car !

 

[Nugatory]

It's also saying that total momentum is not conserved if we use classical law of momentum. I think I need an example for this

An object of mass m is approaching you from the northwest, moving to the southeast at a speed of .5c relative to you. A second object, also of mass m, is approaching you from the southeast, moving to the northwest at a speed of .5c relative to you. They collide, and the first object rebounds to the northeast at a speed of .5c while the second one rebounds to the southwest at that speed. Both $mv$ and $\gamma{m}v$ are conserved in this collision (they're both zero before and after).

Now, consider this exact same momentum-conserving collision as described by an observer traveling due east at a speed of $\sqrt{2}c/4$ relative to you. He will describe the first object as approaching him directly from the north and rebounding back to the north, while the second object approaches him from a direction somewhere between east and southeast, and rebounds in a direction between southwest and west. Use the relativistic velocity addition law to calculate the velocities of the two objects as described by the second observer - you will find that quantity $mv$ is not conserved but the quantity $\gamma{m}v$ is conserved.

 

[Amr Elsayed]

you will find that quantity mvmv is not conserved but the quantity γmv\gamma{m}v is conserved.

Thank you for response, but there is sth concerning axis and angular directions in the example above. I will try to apply it on a simpler example, and I hope you will tell me where I have mistaken
Assume that those 2 objects with the same mass are both moving with a speed of 0.5C according to the origin, First object is coming from west and second one is coming from east.According to point of origin: both objects will crash then rebound in opposite direction with a speed of 0.5C and mv is conserved
According to a third object who is moving due east with speed of 0.5C according to" with respect to" origin : first object is stationary, and second object is coming from west with speed of 299999999.8 m/s (using relativity addition law) , after collision the second object is stationary and first object is moving due west with a speed of 299999999.8 m/s.

For both frames momentum will be observed the same before and after collision , but momentum itself will differ, and there is nothing about that 'cause momentum is frame dependent property
When we apply m*v*gamma we will also conserve momentum since change will be zero, but again momentum of each frame is different
I want to know what was wrong about that .
I want to know why we use distance measured by an observer watching but the time measured by the object itself to calculate velocity

Thank you, and I'm grateful for your care

 

[Amr Elsayed]

I can conclude the increase in mass if I consider time dilation, for example a spacecraft moving from our perspective with 0.9 C will shoot with a right angle from the direction of motion. Since the action of shooting will slow down but when the shot ball hits another ball , from my perspective it will go faster " from my perspective" since it's not affected by time dilation and I conclude that I should observe more mass from the moving ball for instance. Is that right ? but with shooting in the same direction of motion we shall need no increase in mass to keep the conservation of momentum.
regards

 

[Nugatory]

I can conclude the increase in mass

You can conclude an increase in mass, but we've spent the best part of two pages now trying to convince you that this is a bad idea, that you should not be thinking of momentum as $m_Rv$ where $m_R$ is something that increases with speed ($m_R=\gamma{m}$), but rather as $\gamma{m}v$ where $m$ does not change with speed.

 

[Nugatory]

or both frames momentum will be observed the same before and after collision , but momentum itself will differ, and there is nothing about that 'cause momentum is frame dependent property
When we apply m*v*gamma we will also conserve momentum since change will be zero, but again momentum of each frame is different
I want to know what was wrong about that .

Nothing is wrong with that, but it also doesn't tell you anything - a conservation law has to work always, not just for carefully selected special cases, or it's not a conservation law. It's easy to construct special cases in which both $mv$ and $\gamma{m}v$ happen to be conserved for some observers in some situations, and the case of two equal masses moving on the same axis as the observers is one of those special cases. (If you try it with different masses you will find that $mv$ isn't conserved; the reason I didn't use that example is that the calculations are much more difficult).