Can anyone deeply understand relativity?

[BadgerBadger92]

"Understand" at what depth? There is a big difference between developing an intuitive acceptance and being able to do all the calculations. GR is an extreme example of that. Even Einstein had trouble doing specific calculations for examples to support GR. SR is much easier. Intuitive acceptance of SR is very easy if you understand (and consider) the relativity of simultaneity. That is especially true for the well-known "paradoxes" of SR. Even in SR, the calculations can get tricky if the problems introduce unnecessary complications.

I want to deeply understand it on a mathematical level. Unfortunately I’m not quite there yet in school, but looking forward to it makes me feel unconfident. I know the facts, but I don’t “understand them.”

 

[ohwilleke]

Advice:
-Work in natural units so that c = (3e8 m/s) doesn't show up in calculations.

Strong disagree on this point. A novice should be meticulous about fully displaying all units and continue to do so until the person intuitively knows the units of everything in the equations being used.

As a math and science tutor when I was in college (and afterwards), one of the most common mistakes I saw was confusion about what units a quantity has which leads to stupid mistakes because the student is confused about which dimensions something has and makes assumptions that are dimensionally obviously wrong.

Reserve natural units until finishing a course in graduate school.

 

[Ibix]

Strong disagree on this point. A novice should be meticulous about fully displaying all units and continue to do so until the person intuitively knows the units of everything in the equations being used.

You just work in seconds and light seconds (or years/light years or whatever). Then $c$ shows up but the numerical work is easy.

 

[robphy]

Advice:
-Work in natural units so that c = (3e8 m/s) doesn't show up in calculations.

Strong disagree on this point. A novice should be meticulous about fully displaying all units and continue to do so until the person intuitively knows the units of everything in the equations being used.

As a math and science tutor when I was in college (and afterwards), one of the most common mistakes I saw was confusion about what units a quantity has which leads to stupid mistakes because the student is confused about which dimensions something has and makes assumptions that are dimensionally obviously wrong.

Reserve natural units until finishing a course in graduate school.

Take the next sentence in my quote.

Advice:
-Work in natural units so that c = (3e8 m/s) doesn't show up in calculations.
-Use seconds and light-seconds, not seconds and meters.

A "light-second" is a perfectly good unit of length,
and on the road to truly natural units where (say) length is also measured in "seconds".

My point is: use "4 light-seconds" instead of "1.2e9 m".

 

[ohwilleke]

Take the next sentence in my quote.


A "light-second" is a perfectly good unit of length,
and on the road to truly natural units where (say) length is also measured in "seconds".

My point is: use "4 light-seconds" instead of "1.2e9 m".

A frequently meaning of "using natural units" is to omit c=h=1 from the notation entirely.

 

[Herman Trivilino]

N. David Mermin likes to use the nanosecond for time and what he calls the phoot for distance, which is really just a light-nanosecond.

c equals one phoot per nanosecond.

The distance light travels in a nanosecond is 0.3 m. A foot is 0.3048 m. So the foot and the phoot differ by less than 2%.

 

[GIASAL]

The essential mathematics underlying special relativity
is a
- mild generalization of Euclidean Geometry (called Minkowski spacetime geometry)
- mild generalization of circular trigonometry
(called hyperbolic trigonometry [not to be confused with hyperbolic geometry])
- mild generalization of vector algebra in Euclidean space (using the Minkowski dot-product).

Learn relativity from a modern relativist,
who thinks first in terms of constructions on position-vs-time graphs (a.k.a. spacetime diagrams),
rather than "effects" and Lorentz transformation formulas.
(You don't learn high-school geometry by studying rotation matrices.
You learn by drawing geometric figures then understanding relationships among them,
like scaling, intersections, parallelism, and tangency [related to perpendicularity].)

Advice:
Quickly learn to translate between the physics,
the "words in a problem", the geometry in a spacetime diagram,
and the associated vectorial expressions.

Advice:
- Try to appreciate "operational definitions" of things,
e.g. "radar methods" involving light-signals and light-cones.
- Appreciate Minkowski's characterization of "normal" or "perpendicular":
"the tangent-line to a circle is perpendicular to the radius".
- Appreciate that the geometry underlying the familiar PHY101 position-vs-time graph
is already a flat[=not-curved] non-Euclidean geometry.

Advice:
-Work in natural units so that c = (3e8 m/s) doesn't show up in calculations.
-Use seconds and light-seconds, not seconds and meters.
-Use arithmetically convenient values like v=(3/5)c and v=(4/5)c
[in the beginning, avoid v=(1/2)c, v=(0.99)c, v=(0.999)c..
because these lead to unnecessary and distracting arithmetic,
obscuring geometric and physical understanding].

Use the geometry (not just words or formulas)
to scaffold your understanding and intuition of the physics.

Many introductory problems in special relativity
are essentially hyperbolic-trigonometric analogues
of problems involving solving for some unknown feature in a right-triangle,
which arise by drawing a spacetime diagram of the situation.

Avoid:
- approaches involving Loedel diagrams and Epstein diagrams
(and any other attempt to use Euclidean geometry because they claim
that Minkowski spacetime geometry is too hard)
- asking about the frame of a light-signal (there is no such frame)
- the phrase that "we are all traveling through spacetime at the speed of light" (it sounds profound, but it's making a mountain out of a molehill... and is not useful to be elevating the notion of a "unit vector")
- thinking only in terms of "space" ... using moving boxcars.

Sorry, but Einstein was an engineer—how did he know all that math? Was he self-taught? In fact, he did work with a mathematician. So, did he theorize the imaginary part?
Apologies to Einstein if I'm mistaken.

 

[Matterwave]

Sorry, but Einstein was an engineer

What? What is your source for this claim?

 

[Dale]

Sorry, but Einstein was an engineer—how did he know all that math? Was he self-taught?

That is the sort of thing that Wikipedia is really good for: https://en.wikipedia.org/wiki/Albert_Einstein

He studied at the

Swiss federal polytechnic school(teaching diploma, 1900) University of Zurich (PhD, 1905)

So not self taught, although definitely self motivated.

So, did he theorize the imaginary part?

The imaginary part of what?

 

[Sagittarius A-Star]

So, did he theorize the imaginary part?

In Einsteins lectures about relativity in 1921 at Princeton University, he took over in SR the $ict$ from Minkowski.
See for example after equation $(22b)$:
https://en.wikisource.org/wiki/The_Meaning_of_Relativity/Lecture_2

 

[robphy]

Sorry, but Einstein was an engineer—how did he know all that math? Was he self-taught? In fact, he did work with a mathematician. So, did he theorize the imaginary part?
Apologies to Einstein if I'm mistaken.

Often, in a discovery or in the development of an idea,
one should distinguish
"the way it was discovered" from
"the way it could be (better) understood by others"
(especially if the idea is part of a bigger picture).

In the description of relativity I gave above,
the "math" and the "geometric way of thinking" came from others
who reinterpreted Einstein's original presentation of relativity,
which did not have this mathematics or this geometric way of thinking.

Indeed, we typically learn physics,
not from the original papers or articles of those ideas,
but from textbooks that have been revised over the years.

 

[PeterDonis]

Einstein was an engineer

I don't know where you're getting that from. He got a Ph.D in physics. His work at the Swiss patent office was what we would now call a day job, to enable him to support himself while he worked towards his degree and getting a job in academia, which was where he spent the rest of his career.