Beginner Resources for Special & General Relativity

[DeltaForce]

TL;DR Summary

I'm completely new to Special and General Relativity. I have some background in newtonian/classical physics. But in class, special and general relativity is mind-boggling to me. I have a hard time learning/wrapping my head around it. So I'm looking for good online resources to strengthen my understanding of it.

Like I said, I'm new to Special and General Relativity. It really sucks when I space out for 2 minutes in class and for the next 30 minutes I'm completely lost until the teacher brings up a new topic. I'm looking for supplementary materials (online preferably as I'm away from home) that can strengthen my understanding of the topic. I've looked at khan academy and youtube videos but it still doesn't quite click for me

I don't want to sit in class and pretend to understand the material; I actually want to understand it. So if you have any advice or tip you can give me for learning this sort of physics, I will also take with gratitude. Also, currently, I'm a bit hazy in the Lorentz transformation and the Minkowski diagram department, so if anyone have some resource that focus on explaining that would be great as well.

 

[Pencilvester]

To get a basic grasp and intuition for SR, a great resource is simply playing around with the Lorentz transformation and Minkowski diagrams. Just start by evaluating time plus a single spatial dimension (that’s the most that hand-drawn Minkowski diagrams will allow you to easily depict anyway). Also start by making yourself comfortable with geometric units where $c = 1$. This will be helpful in the long run.

In many beginner texts you will often see the Lorentz transformation like this: $$x’ = \frac{x-vt}{\sqrt{1-v^2}}\\
t’ = \frac{t-vx}{\sqrt{1-v^2}}$$ All this does is take an input—the coordinates of an event (something that happens at a particular time in a particular place) in one inertial frame of reference—and outputs the coordinates of that same event for a different inertial frame of reference.* As an example, you could start with a situation where one inertial observer witnesses a bunch of simultaneous flashes of light spaced all along his $x$-axis and evaluating what a different observer traveling with velocity $v$ relative to him would see.

*There are some caveats that go along with that, such as the two frames must share an origin (the event labeled as $(0,0)$), and all observers must compensate for travel time of light when they assign times to events. The interesting results of SR most clearly manifest themselves when travel time of light is compensated for.

 

[DeltaForce]

For the Minkowski diagram, my teacher used "ct" as the unit y-axis and "x" as the unit of x-axis
where x is distance.
I presume that c is light speed and t is time for the y-axis. I think they are multiplied together? I thought that speed times time is distance/displacement
I'm a bit confused as to why both the x-axis and y-axis are "distances" or is it something else.

Before I can grasp the set-up of the graph, my teacher went on to talk about worldlines, time intervals, and space-time. So I'm confused about the Minkowski diagram, like very confused...

Can I ask you for your help to explain these stuff?

 

[pervect]

It's convenient to have a light ray on a space-time diagram be a line at a 45 degree angle.

Scaling the time t by the universal constant "c" is just a a way of doing this. There are a number of equivalent ways of thinking about this, I don't want to confuse you too much by introducing one that's incompatible with your class, if you're struggling a little bit with it.

 

[Pencilvester]

I'm a bit confused as to why both the x-axis and y-axis are "distances" or is it something else.

There are members here more knowledgeable than I who can really explain the significant profundity of geometric units, but the superficial gist of it is that it’s typographically convenient for the math formulas to work in units where the speed of light, $c$, equals 1, and as @pervect mentioned, convenient for the Minkowski diagrams as well.

Note that in post #2 the units in the Lorentz transformation don’t work out if you’re using e.g. the metric system of units. You can strategically insert $c$’s (which in the metric system has units of meters per second) to make the units work out, but then you have a bunch of $c$’s all over the place. Geometric units avoids this.

 

[SiennaTheGr8]

For the Minkowski diagram, my teacher used "ct" as the unit y-axis and "x" as the unit of x-axis
where x is distance.
I presume that c is light speed and t is time for the y-axis. I think they are multiplied together? I thought that speed times time is distance/displacement
I'm a bit confused as to why both the x-axis and y-axis are "distances" or is it something else.

Don't confuse the physical quantities with the units we use to express them.

If you walk at a speed of 4 miles per hour, then you can say that the grocery store 1 mile down the road is "a 15-minute walk away." You're still talking about a distance, but you're expressing that distance in units of time. Your speed serves as a unit-conversion factor here: divide the distance (in miles) by the speed to express the distance in time-units.

Of course, your speed isn't a universal constant. Even if you managed to maintain a perfectly steady pace, different observers would disagree on your velocity. We can't all use @DeltaForce's walking speed as an agreed-upon standard for converting between distance- and time-units.

But $c$ is both constant (unchanging) and invariant (everyone agrees on it). It's nature's ready-made distance/time unit-conversion factor, and using it to express times and distances in the same unit turns out to be extraordinarily convenient. To do this, either divide all distances by $c$ or multiply all times by $c$. (Or set $c = 1$.)

So $c \Delta t$ isn't "a distance"; it's a time-interval expressed in units of distance.

 

[kent davidge]

to express times and distances in the same unit turns out to be extraordinarily convenient

and I think, is mandatory if you are summing up terms

 

[Mister T]

Before I can grasp the set-up of the graph, my teacher went on to talk about worldlines, time intervals, and space-time. So I'm confused about the Minkowski diagram, like very confused...

Can I ask you for your help to explain these stuff?

What textbook are you using for the course? You can also ask your instructor for a list of sources such as textbooks and websites.