# A couple of miscellaneous questions about special relativity

marschmellow
Hi all, I'm in a modern physics course and am surprised by how confused I am about special relativity. I know multilinear algebra quite well, and so I thought it would be pretty easy, but I'm having some conceptual difficulties. Any help would be greatly appreciated!

1. Why is a special formula needed for the addition of velocities? Aren't velocities 4-vectors, which transform via Lorentz Transformations? Are there situations in which you would simply add or subtract two velocities? I have in mind when you want to compute the relative velocity of two objects in a third object's rest frame.

2. What does it mean for an object's mass to increase with increased energy (but not an increased net momentum)? Does this mean that the same applied force will accelerate it less strongly? Or is it just an issue of conservation?

3. Why is it that the energy in the motion of a bunch of moving particles with no net momentum adds mass to the collection of particles? If you considered an individual particle, wouldn't it have a net momentum and thus no increase in mass? Where does this mass exist?

Staff Emeritus
Gold Member
1. Why is a special formula needed for the addition of velocities? Aren't velocities 4-vectors, which transform via Lorentz Transformations? Are there situations in which you would simply add or subtract two velocities? I have in mind when you want to compute the relative velocity of two objects in a third object's rest frame.
Yes, they're 4-vectors, and yes, they transform via Lorentz transformations; if you do this, and interpret the result as a velocity vector, you will recover exactly the velocity-addition formula you're talking about.

No, it doesn't really make sense to add 4-velocities. This bugged me too when I first learned it. It seemed like a vector you couldn't add was like food you couldn't eat or water you couldn't drink. But keep in mind that 4-velocities are by definition normalized in a certain way, so adding two v vectors will never give you a valid v vector. A nicer way to think about it is that a velocity vector is simply a specification of an observer. In that observer's own frame, the v vector is simply the unit vector in the time direction.

2. What does it mean for an object's mass to increase with increased energy (but not an increased net momentum)? Does this mean that the same applied force will accelerate it less strongly? Or is it just an issue of conservation?
You seem to be getting confused by the notion of relativistic mass. Nobody uses relativistic mass anymore, except maybe your professor. Please tell your professor to stop.

Increasing an object's speed, energy, or momentum increases its inertia, but nobody describes that anymore as an increase in mass.

3. Why is it that the energy in the motion of a bunch of moving particles with no net momentum adds mass to the collection of particles? If you considered an individual particle, wouldn't it have a net momentum and thus no increase in mass? Where does this mass exist?

The basic relationship (in units with c=1) is m2=E2-p2. This is simply the norm of the energy-momentum four-vector, and m is a constant. When you have a collection of particles with different energy-momentum vectors, their energy-momentum vectors are what add. In general, when you add two vectors, their norms do not add linearly.

Muphrid
1) The formula is for three-velocities.

2) I'm not sure what you're referring to. There is an invariant mass that is (within a factor of $c$) exactly equal to the magnitude of a particle's four-momentum. Naturally increasing the energy of a particle at rest increases the magnitude of its four-momentum.

3) I'm not sure what phenomenon you're referring to here, either.

Multilinear algebra should help you gain some insights into special relativistic dynamics. I dare say the usual explanations are not meant for someone with your background, and it might be more useful to focus on the math of four-vectors instead of the usual explanations, which are meant for people with less background to develop "intuition" about how stuff works. Good luck!

marschmellow
Thanks for your responses! Even if I don't comment on anything specifically, it was all helpful to hear.

Yes, they're 4-vectors, and yes, they transform via Lorentz transformations; if you do this, and interpret the result as a velocity vector, you will recover exactly the velocity-addition formula you're talking about.

No, it doesn't really make sense to add 4-velocities. This bugged me too when I first learned it. It seemed like a vector you couldn't add was like food you couldn't eat or water you couldn't drink. But keep in mind that 4-velocities are by definition normalized in a certain way, so adding two v vectors will never give you a valid v vector. A nicer way to think about it is that a velocity vector is simply a specification of an observer. In that observer's own frame, the v vector is simply the unit vector in the time direction.

Thanks, that was really helpful. I just want to ask one follow-up question. Is there a word for a vector that specifies the time derivative of one object using another object's clock? If I recall correctly, a 4-velocity is defined to use the object's own time, so how would a vector behave if it used another object's time? It's only a difference of scale by the chain, so it should still be a 4-vector, but I don't know what it means or what to do with it. Is it valid to subtract one of those vectors from another such vector to get the relative velocity of two objects in another's rest frame? I think you might have already answered that question, depending on whether or not those vectors count as "4-velocities."

Multilinear algebra should help you gain some insights into special relativistic dynamics. I dare say the usual explanations are not meant for someone with your background, and it might be more useful to focus on the math of four-vectors instead of the usual explanations, which are meant for people with less background to develop "intuition" about how stuff works. Good luck!

I was excited about being exposed to the "usual" explanations for things, to descend from the ladder of elegant tensor analysis, and to get my hands dirty with spacetime diagrams and other stuff for people who aren't good at math, but I'm finding it surprisingly hard. Do you know any good online explanations of special relativity using tensors?

Staff Emeritus
Gold Member
Do you know any good online explanations of special relativity using tensors?

It's not free online, but: https://www.amazon.com/dp/9810219296/?tag=pfamazon01-20 I wouldn't recommend it as an intro to SR, but for someone mathematically minded like you who's already had an intro to SR, it might be of interest. The point of view is very, very different from what you see in most intro texts -- which is good, if you can wrap your mind around it.

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One point is that while 4-velocities, being tangent unit vectors, don't add in meaningful way (that is: the sum of two 4 velocities is a vector, but it isn't 4-velocity since it is no longer a unit vector; similar to basis vectors - sum of two basis vectors is not a basis vector) - 4 momentum does add like a vector. Change the normalization so the norm is physical (invariant mass) and you recover a vector with all the normal properties (sum of two 4-momenta is 4-momentum of the total system). So just multiply unit vector by mass, and add with meaning.

harrylin
Hi all, I'm in a modern physics course and am surprised by how confused I am about special relativity. I know multilinear algebra quite well, and so I thought it would be pretty easy, but I'm having some conceptual difficulties. Any help would be greatly appreciated!
There are different ways to teach special relativity, and regretfully some ways of teaching can be confusing (compare Bell "how to teach special relativity", you can Google for related discussions).
1. Why is a special formula needed for the addition of velocities? Aren't velocities 4-vectors, which transform via Lorentz Transformations? Are there situations in which you would simply add or subtract two velocities? I have in mind when you want to compute the relative velocity of two objects in a third object's rest frame.
The special formula is for a reference system transformation and the standard Lorentz transformations refer to 3-velocity. In classical mechanics many books don't distinguish between a velocity addition/subtraction and a frame transformation, because the result are the same. However, physically those are very different things, and in SR also the results are different because different systems measure differently.
Simply put, adding apples + apples is not the same as adding apples + oranges. 2. What does it mean for an object's mass to increase with increased energy (but not an increased net momentum)? Does this mean that the same applied force will accelerate it less strongly? Or is it just an issue of conservation?
Firstly, it depends on the definition of "mass" that you use. Regretfully the classical definition (corresponding to counting elementary particles) could not be maintained in a useful way, and nowadays two competing new definitions are in circulation. Relativistic mass mr and momentum p equally increase with increased energy. With that definition, expressed in 3-velocity:

p = γ m0 v = mr v
3. Why is it that the energy in the motion of a bunch of moving particles with no net momentum adds mass to the collection of particles? If you considered an individual particle, wouldn't it have a net momentum and thus no increase in mass? Where does this mass exist?
 Mass is proportional to energy, in this case the combined energy of the particles. In his first(?) paper on that, Einstein phrased it rather neatly: the mass of a body is a measure of its energy content.
- http://www.fourmilab.ch/etexts/einstein/E_mc2/www/

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sarsonlarson5
Hello!

I apologize if I am hijacking this post. I am new to Physics Forums, and I do not know how to start a new thread to ask my question. I was hoping to ask a few questions related to the above questions - special relativity, general relativity, etc.

I am currently a twelve-year old, and I am interested in learning about Relativity. I have heard that there are two main fields relativity is divided into - special relativity, and general relativity. Although some people have said that general relativity is considerably harder than special relativity, I am not sure if I even have the knowledge required to understand either fields in the first place.

I have taken a Calculus I/II course and a rather limited AP level Physics course. I have an excellent Physics teacher to help me. Would this level of knowledge in mathematics and Physics be enough to get me started on Special relativity? My goal for Physics is to learn all the math and Physics necessary needed for learning Quantum Physics/accelerator Physics by the time I am fourteen.

Once again, I apologize for rudely interrupting this thread. (It would be very helpful if you gave me some tips on how to start a new thread.) Please excuse me if I am making unplausible or ignorant claims, as I am not familiar with Modern Physics.

Thank You.

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Mentor
I do not know how to start a new thread to ask my question.

When you are viewing the list of threads in a forum, click the "New Topic" button at the top or bottom of the list of theads. Please feel free to do so for this question. I suggest you also look at the Physicsforums FAQ which you can find here:

and our rules, which you can find by clicking the "Rules" link at the top of any page.

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sarsonlarson5
Thank you for the information - however, I do not have a "new thread" button were it should be. Looking at the images in the FAQ, it appears that there is something strange going on with my computer.

Thank You for the tips, though.

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