Relativity 104f: Special Relativity - Relativistic Dynamics and 4-Vectors (E=mc^2)

0:00 Intro (4-vectors and Invariance)
3:38 4-velocity derivation
9:07 4-velocity example
14:44 4-momentum derivation
21:25 4-momentum for light
24:07 4-momentum example
25:38 Conservation of 4-momentum
28:20 4-acceleration
31:42 4-force
32:44 Summary

[jonasmanuel]

The dislikes are from people who are currently bouncing around in a frictionless vacuum in perfectly elastic conditions and now have one more may of calculating a momentum vector.

[eigenchris]

A number of people have asked why I'm not using the imaginary unit i to get i(et).i(et) = -1, or i(ex).i(ex) = -1. The short answer is that this "i" trick is a "hack" that works in some coordinate systems, but not others. In particular, this trick fails if our spacetime basis is non-orthogonal. Allowing dot products of basis vectors to be negative is a better solution because it works it absolutely all coordinate systems.


To sum up, the "ict" or "ix" trick is more of a "hack" that works nicely in some specific cases, but not other cases. But using a metric (defining dot products between all basis vectors) is the more "correct" way to handle things, and it will work in any coordinate system, no matter how crazy it is.

[HighWycombe]

I've been working my way through these Relativity videos slowly and making notes as I go. I'm now up to 98 pages of hand written notes in a ring binder. Really looking forward to General Relativity, but first I'm going to enjoy learning about Special Relativity with Acceleration. (Relativity 105.)

[karkunow]

Thank you SO MUCH! I've watched everything from SR playlist here and this stuff is AMAZING!
The best intro to Special Relativity, personally for me. It wins my attention even comparing to Susskind's, Feynmann's or Penrose's books!
Your work is just a diamond! :)

[imaginingPhysics]

1:39 from a mathematical point of view the 4velocity and 4acceleration are just usual differentials along a world line (that is a path) in a 4-dimensional space with minkowski metric. Differentiating with respect to proper time means using the length of the path (natural choice even without a physical interpretation) as a path parameter. All the invariances follow from general mathematical principles (without reference to physics).
14:00 Btw. Using path length (proper time is by definiton the path length) as parameter also implies the "differential vector's (4velocity)" length is automatically the constant unit length c.

This is to point out that we would pretty naturally end up with these definitions by copying their abstract counterparts in mathematical vector analysis and differentiable curves.
Of course physical interpretations are important. I just think it's useful to understand which results come from mathematical rules.

See next videos for more examples of mathematical analysis.

[warrenchu]

3:29 marks the first appearance of contravariant vectors in superscript notation.
Contravariant vectors are objects acted upon by covariant vectors (such as vector bases) 34:55.

[patriciacosson]

Always perfect as usual a very great job thanks a lot for your hard work

[DavideLibenzi]

Again, another very good video !
10:00 It seemed clearer to me to bring home the invariance of U by substituting the e.t and e.x transformation from e.t' and e.x' into the U formula, which leads directly into U being (c * e.t') when expressed in primed frame.

[rk]

I don't understand how e=mc2, a physical reality, can be discovered from what appears to be the "happenstance" of measuring time in units related to c and then just also multiplying 2 sides of the momentum equation by C again? In any case all your video series show your hard work, passion and teaching skills. I hope you have received tangible rewards as as a result. You deserve them. I see these professors on youtube teaching tensors and they don't care if anyone understands now that they do.

[metrictensor]

Hi Chris, I am so impressed with your videos. They are clear and obviously took a great deal of effort to produce. I wanted to point out that I think you missed an opportunity at 28.16. In your example the collision was inelastic so the energy from the components went in to binding them together (i.e. the electric potential energy that could be liberated in fission) or heating up the resultant and thus the increase in rest mass. If the collision had been elastic there would have been a different answer for the sum of resultant rest masses.

[namesurname]

Very helpful video but I have a question. In the minute 30:13 I can not understand the third step. I can not understand the logic why the derivative of (1-v**2/c**2)**-1/2 is equal to this term. Thank very much for your time and your amazing videos!

[damski]

I just noticed: the implication of 4-velocity beeing equal to c is that everything, including matter, moves in spacetime with a speed of light. Just, if you move faster in space, you move slower in time; and if you move slower in space, you move faster in time.

[abcdef-rfxt]

4:41 If the derivative of the basis vectors with respect to tau is zero, then why isn't dS/dτ always equal to zero?

[ashwinvishwakarma]

Why do we take the plus sign transformation at 5:24? Why couldn't we take the opposite sign? Also why is beta=u/c, where u is the same as the spatial velocity of the spacetime event S? Shouldn't it be just any velocity of the other frame relative to the first one?

[teezettsb]

Great explanations, but I have one question:
At 3:20 in the Lorentz-Transformation only the x components changes, y and z stay unchanged.
I could not find an explanation for that and intuitively I don't understand that.
Could you explain that? Or is it just the special case for that example?

[gregglind]

@eigenchris. Thank you again for the very deep and thoughtful explanation of the mathematics underlying relativity.

at 10:41, I realize that I don't really understand what it "means" to be traveling at 0.5 in that diagram. Are spacetime diagrams an "omniscient frame"?

I felt confused and unable to properly interpret the U vector coefficents of:

2/root(3) : car seconds per einstein second?
1/root(3) : car (meteres) / per einsten meter?

[cinemaclips]

At 30:19, when you differentiate -u^2, shouldn't dγ/dt be +γ^3 instead of -γ^3 because the derivative of -u^2 is -2(a.u)?

[laonza]

Question please: At 10:54 I see a “gamma-tilde” term vice the regular gamma term in previous videos. I don’t recognize this term. Is this the same as the regular gamma or is it maybe the gamma from the perspective of the car, or…? BTW – Relativity 104f is very informative video jam packed with tons of information, so thanks.

[MajaMarczak-ub]

I have a question. If now we consider an acceleration of an object, why we didn't calculate the basis vectors derivatives by chain rule? I assume it's because we are considering 4-vector of acceleration in "Einstein's" inertial frame?

[eugenioguarino]

Hi Chris (I suppose this is your first name), I know special relativity rather well, but this vector approach is quite impressive. Furthermore your explanations are really clarifying, as usual. When I think of the amount of work needed to prepare these lessons, I believe everyone of us must be grateful to you: great job! Nevertheless I have some doubts that I hope you will clarify. The first one refers to video 103e where you show the incompatibility of Galilean transform rules with electrmagnatism: if we accept the idea that positive charges in the frame of the lone proton undergo a contraction of the distance between them, then the same thing should happen to negative charges in frame of the room; so an attractive electrical force towards the wire would arise, balancing the Lorentz force, isn't it? Second question, I read somewhere that the components of the position 4-vector could be s=(x, y, z, ict), so that when one calculates the square of s, what one gets is: s^2 = x^2+y^2+z^2-(ct)^2. Is this choice compatible with the vector approach in some way? Third, in this video, when you calculate the derivative d/dt(gamma), it seems to me that a minus sign disappears: I mean that d/dt(1 - v dot v) should be -2(v dot a), neglecting the factor 1 over c squared, isn't it? This would only affect the sign + or - the gamma cubed factor in the expression of the acceleration, not a big deal, but any way it seems important to me. Fourth and last, this term is meaningful only when a dot v is non zero, i.e. when a and v are not right angles one to the other; well this happens only for straight forward motion, because for curving motion a and v are perpendicular; it sounds weird that the additional term, which makes A and a not aligned, arises only for straight motion, isn't it? Thanks a lot for you answers, Eugenio. Looking forward 105 video series!