# Confusing points in relativistic dynamics

1. Jun 13, 2014

### Jeffrey Yang

Hi everyone:

I just started to learn special relativity but totally being confused about "velocity", "momentum" and "force".

1. The relativistic momentum is defined by "rest mass * ordinary velocity * gamma". There are 2 kind of explanations. The first one is combining "rest mass * gamma" to be "relativistic mass", it is easily to be accepted because high speed motion own high energy and can be treated as large mass. The second one is combining "ordinary velocity * gamma" to be "proper velocity", and I cannot see any physical meaning of this definition. However, some books said "in recent years, researcher have already abandoned the definition of 'relativistic mass' ". Why about this? And what's the physical meaning of "proper velocity"?

2. Considering in the lab's reference system, we (rest observer in lab's RS ) measure the velocity of a particle as "u", not "gamma * u". Why we still use relativistic momentum to construct momentum conservation law? The only explanation is the mass of particle is changed due to relativistic effect. In this story, we can see the "relativistic mass" is reasonable. How to explain this using "proper velocity"?

3. Using the relativistic momentum "p" here, we can define the ordinary force F = dp/dt, where t is measured in lab's system (This one can also be explained by relativistic mass but not in proper velocity, how to do that????). Also, we can define another kind of force K = dp/dτ, where τ is the proper time. Force K can be transformed by Lorentz Transformation directly but F cannot be. Does this imply that, the whole story of special relativity and 4-vector is trying to find the "what can be transferred by Lorentz Transformation"? And what the physically meaning of this 4-force ? Just to define a "force" can be transferred ??

2. Jun 13, 2014

### Orodruin

Staff Emeritus
1. There is nothing here that needs an explanation. The momentum for a massive particle is simply mvγ, where m is the rest mass, v is the velocity, and γ is the gamma factor. The only limit where it has to be (approximately) equal to mv is when v << c. One reason to abandon "relativistic mass" is that in Newtonian mechanics you have two types of mass, gravitational mass and inertia - neither of which is strictly related to the relativistic mass (there is no gravity in SR and the acceleration of an object works different from classical mechanics). Calling vγ "proper velocity" is also a misnomer since "proper" typically refers to something as seen from an observer's rest frame (proper time, proper length, etc). That being said, vγ is a part of the 4-vector V = γ(c,vx,vy,vz) usually referred to as the 4-velocity, which is tangent to an observer's world line and related to the 4-momentum P = (E/c, px, py, pz) through multiplication with the rest mass, hence the relation P = mV.

2. Because relativistic momentum is what is conserved. The mv you have in classical mechanics is simply an approximation valid for small velocities. In relativity, the conservation of energy and momenta are simply special cases of the conservation of the total 4-momentum, which contains both energy and momentum. In fact, it has to be this way for conservation of momentum to hold for observers in different inertial frames.

3. The definition of F is referring to a particular frame (namely the frame you are using t in) while K does not (the proper time is well-defined regardless of the frame). It is always the case that you are looking for statements which are frame-independent.

3. Jun 13, 2014

### Jeffrey Yang

Ok, thanks! It's more clear now.

What I need to accept is the true/intrinsic/most correct definition of momentum is just "mvγ", right? It is just the physical rule. Ok, I accept that. However, could you please give me more illustrations why it is? As you said, we are always looking for something frame-independent. The meaning of gamma factor here is just convert the "measured momentum in a specified frame" into a frame-independent result. And this is also the correlation between F and K. Can I understand it in this way?

4. Jun 13, 2014

### pervect

Staff Emeritus
Proper velocity is a vector with 4 components, a 4-vector. The 4 components are: the rate of change of coordinate time with respect to proper time, and the rate of change of the three spatial components, each with respect to proper time.

Proper time is (as always) what a clock measures, in this case it's what the clock on the moving point mass measures.

I don't really find your "derivation" of the relativistic momentum that convincing personally, you seem unreasonably attached to it. It's your choice, I suppose, but the "derivation" is not really very solid, it's more of a plausibility argument at best. I don't quite see the reason for becoming attached to relativistic mass, though.

Goldstein ("Classical Mechanics) has a at least two derivations of the correct formula for relativistic momentum, one in particular considers a simple elastic collision from various frames and insisting that momentum must be conserved by the collision in all frames. It's a bit detailed to reproduce here, but it's more convincing (IMO) than just assuming that one can replace mass with relativistic mass and somehow get the right answer.

There are other easy ways that easily suggest correct conclusion. X = d(t,x,y,z) forms what is known as a 4-vector, which has the property that it transforms via the Lorentz transform. As a consequence it has an invariant "length", the Lorentz interval, given by -c^2*t^2 + x^2 + y^2 + z^2

dX/dt, t being coordinate time is not a four vector, it does not have the above two properties needed for it to be a four vector. dX/dtau, where tau is equal to proper time, is a 4-vector, the proper velocity. Multiplying the proper velocity by the invariant mass gives another 4-vector, the energy momentum 4-vector (E,p)

There are some generalized theories about "geometric objects", which consist of world scalars, (such as proper time) that are independent of the observer, 4-vectors which transform according to the Lorentz transform and have an invariant length, higher rank tensors (which may not be needed much for a basic treatment), plus rules for how geometric objects can be combined to form other geometric objects. In particular taking the derivative with respect to a world scalar is an allowed way to generate a new geometric object from an old one.
Actually this is one of the cases where people commonly get the wrong answer by using relativistic mass.

It is correct to say that F = dp/dt, as you did, however it would be wrong , for instance to say that F = m_relativistic * a.

Writting it out in full, and using non-relativistic mass m (which is constant and independent of time) we can see that

F = d/dt (gamma(t) * m * v(t)) = (d gamma / dt) * m * v(t) + gamma * m * dv/dt

The definition of a 4-vector is that it transforms via the Lorentz transform, yes. It is very convenient to have equal and opposite forces, this is a property that 4-forces have, and that regular forces do not. It's much more convenient to use 4-forces, then convert them back to 3-forces as necessary, because of the standardized way in which they transform.

5. Jun 13, 2014

### Jeffrey Yang

The reason why I feel proper velocity is uncomfortable is it define by 2 parameters not in a same frame. The spatial displacement in lab's frame but the time in particle's frame. I cannot see a clear physical meaning in it.

I know there are also know clear and solid reason for "relativistic mass". And this is why I confuse about the relativistic momentum. Both the "relativistic mass" and "proper velocity" explanation have problem.

But as you said, we just need to consider what can be conversed in all the frame, and therefore construct the right form of momentum. Maybe this is the best way to understand the relativistic momentum.

6. Jun 13, 2014

### pervect

Staff Emeritus
If you consider coordinate time, t, and proper time tau, the first is specific to a given coordinate system, the later is a world scalar, agreed on by everyone. It may take some getting used to , but it's very convenient to plot the trajectory of a body through space time by the parameter tau, rather than some frame dependent time. Tau has a physical interpretation as the time on a clock moving with the body, it's not any big abstract mystery. I don't think there's really any argument that it's "not physical", it may be a matter of what one is used to.

Thus you describe the motion of a body by

t(tau), x(tau), y(tau), z(tau)

Coordinates have their usual meanings, as labels to tell where and when a body are.

4-vector notation makes conversions to different coordinate systems are all easy and standardized, and generalizations such as the 4-velocity and 4-acceleration are straightforwards (just take the derivative of the 4-positions with respect to tau), and in a relativistic context, much easier to work with than 3-velocities and 3-accelerations. The 3-acceleration is in particular much more complex than the 4-acceleration, and the physical interpretation of the magnitude of the 4-acceleration vector is just the felt force in the bodies own frame of reference via its own clock.

As an example working out the worldline of a body moving at constant proper acceleration is a two-line derivation (well, it helps to know that the dot product of the 4-acceleration and 4-velcoity is zero, in advance, i.e. that they are orthogonal). It's a real mess trying to do it with 3-vectors.

7. Jun 13, 2014

### pervect

Staff Emeritus
Oh, one other useful snippet. You don't need a clock synchronization system to measure proper velocity, this makes defining the concept much more direct. Thus if you set out a course that's a kilometer long (in some rest frame), your proper velocity in kilometers / second is just the proper distance of the course (as measured in its restframe) divided by the time it took you to traverse the course (measured by your clock). If it were easier to construct moving clocks and not worry about issues such as vibration, conceptually proper velocity is a much easier way to define velocity, due to the fact that you don't have to worry about any "clock synchronization" issues.

8. Jun 13, 2014

### jkl71

I think these are important points.

I’m not sure if this will help or not, but I think it helps to think in terms of spacetime geometry. The 4-vectors are really the fundamental entities in special relativity since spacetime is 4 dimensional. The 3-vectors are just an artifact of how one slices up spacetime into space and time. From this perspective it’s just 4-velocity equals the derivative of position with respect to proper time, (4)momentum is just (rest) mass times 4-velocity and (4)force is just the derivative of (4) momentum wrt proper time.

To actually measure something you need to eventually pick coordinates, but I believe talking about 3-vectors too early obscures some things. As an analogy, imagine doing ordinary vector analysis in 3-dimensional Euclidean space. However, as a starting point you slice 3-D space up into 2-D xy-planes. Then you take your 3-vectors break them up into 2-vectors, their projection into your xy-planes, and a z component. Sure, you could do this, but it might give the impression that there’s something special about the 2-vectors and might make you miss some insights into 3-D geometry. Obviously time is not a spatial dimension, so it's not exactly like a z-axis (because of the minus sign in the metric mentioned in the previous post), but there are a lot of similarities.

9. Jun 14, 2014

### Jeffrey Yang

Oh, yes, yes. This is the key points. In my mind, I just think the proper velocity use 2 parameters which is "not in a same standard". But yes, as you said, I should understand the tau as a world scalar and every thing under a specific measurement under certain frame can be described by this same world scalar. It's clear now, thanks very much.