Can anyone deeply understand relativity?

[BadgerBadger92]

I’m still in school for physics, but I’m worried I will never deeply understand relativity. Can anyone understand these concepts, or is it something only a few people in the world can understand?

 

[Dale]

Can anyone understand these concepts, or is it something only a few people in the world can understand?

I think it is somewhere in between. Not anyone can understand it, but a whole lot of people do. It does take effort.

 

[BadgerBadger92]

I think it is somewhere in between. Not anyone can understand it, but a whole lot of people do. It does take effort.

I hope once I get to the level of relativity I can understand it. I know the facts, but I don’t have the understanding of the math yet.

 

[robphy]

I hope once I get to the level of relativity I can understand it. I know the facts, but I don’t have the understanding of the math yet.

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.

 

[sbrothy]

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".

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.

That should clear up how complex that topic is.

I can't say I understand it myself, but isn't Tensor Calculus a part of it. That's kinda where I gave up.

 

[robphy]

That should clear up how complex that topic is.

I am pointing out how along the way lots of attempts have been tried,
and how one can try to avoid the various potholes that some created.

There is a clean, more streamlined path using Minkowski spacetime diagrams,
which have unfortunately been avoided by many physics teachers
because, possibly following Einstein's initial reaction to "mathematicians" like Minkowski,
they feel it's too hard, too advanced.

 

[Matterwave]

I can't say I understand it myself, but isn't Tensor Calculus a part of it. That's kinda where I gave up.

For a modern treatment of GR, differential geometry would provide the broader perspective.

You can learn SR without much of that machinery, but you will appreciate the theory more if you do learn it.

 

[robphy]

That should clear up how complex that topic is.

I love this quote... (bolding mine)

To understand a subject, one must tear it apart and reconstruct it in a form intellectually satisfying to oneself, and that (in the view of the differences between individual minds) is likely to be different from the original form. This new synthesis is of course not an individual effort; it is the result of much reading and of countless informal discussions, but for it one must in the end take individual responsibility. Therefore, I apologise, if apology is necessary, for departing from certain traditional approaches which seemed to me unclear, and for insisting that the time has come in relativity to abandon an historical order and to present the subject as a completed whole, completed, that is, in its essentials.
In this age of specialisation, history is best left to the historians.

J.L. Synge in Relativity: The Special Theory (1956), p. vii

 

[sbrothy]

I know I know you guys. I was just yanking your chains. But in truth I gave up on this topic. I may just not be smart enough. Roger Penrose's Road to Reality might be a smarter startpoint but even there I didn't get far. (Probably I'm also lazy).

 

[robphy]

I can't say I understand it myself, but isn't Tensor Calculus a part of it. That's kinda where I gave up.

In college, I had a grasp of tensor calculus and could work problems in special relativity
...but there was a disconnect.

My "aha moment" in relativity came in grad school when
my instructor showed how operational definitions of radar-measurements led to
how an inertial observer splits up his view of spacetime.

From that point on, the geometry of spacetime, light cones, and causal structure
were more important than Lorentz transformations.

 

[sbrothy]

In college, I had a grasp of tensor calculus and could work problems in special relativity
...but there was a disconnect.

My "aha moment" in relativity came in grad school when
my instructor showed how operational definitions of radar-measurements led to
how an inertial observer splits up his view of spacetime.

From that point on, the geometry of spacetime, light cones, and causal structure
were more important than Lorentz transformations.

Aha, see now we're getting somewhere. I hope the OP can use some of it!

I'm still joking. I'm in way over my head!

EDIT: Lucky for me I recognize valid information when I see it. At least.

 

[wrobel]

I’m still in school for physics, but I’m worried I will never deeply understand relativity.

The whole point is what is meant by the word 'understand'.
For example, any qualified mathematician can understand the theory of relativity.

 

[sbrothy]

The whole point is what is meant by the word 'understand'.

Wow, now youre getting ontological. I don't think that's what the OP asked.

 

[Matterwave]

Wow, now youre getting ontological. I don't think that's what the OP asked.

Epistemological :)

 

[Ibix]

I’m still in school for physics, but I’m worried I will never deeply understand relativity. Can anyone understand these concepts, or is it something only a few people in the world can understand?

Special relativity is fairly simple, although @robphy is absolutely correct that you need a modern geometric approach to make it simple, IMO. Minkowski diagrams were the turning point for me. A former poster, @ghwellsjr, used to post them on every thread someone posted about SR, and they were what turned SR from something I could calculate into a coherent picture. I think if you can handle Pythagoras' Theorem and some simple algebra then you can handle the basics of SR when it's presented like this. You will need calculus to work problems including anything more complex than instantaneous acceleration, though.

I think a bit of the history is interesting in connecting this mathematical approach to what went before - the problems with Maxwell's equations leading to various approaches to fix them and finally Lorentz' work, then Einstein's insight that it was Galileo and Newton that needed fixing, not Maxwell. But, as Synge says in robphy's quote, the way we blundered into it is not necessarily the way to learn it.

General Relativity is a lot more complex mathematically and conceptually. In SR you can always fall back on a few fairly simple recipies and tools to solve a problem, but GR far more often requires you to fall back on the formal general approach, and usually then a knowledge of numerical techniques because the equations don't have analytical solutions. That is a real challenge and requires a solid grasp on the maths.

 

[wrobel]

I don't think that's what the OP asked

I think OP suffers from a lack of intuitive clarity regarding relativity

 

[sbrothy]

I think OP suffers from a lack of intuitive clarity regarding relativity

I wanted to respond with a laughing emoji but that would be unfair. After all, highschool is where you make your choices regarding your future. (As I understand looking from Denmark.)

EDIT: And in my experience intuition is of little help in physics.

 

[wrobel]

A general note (just for historical fairness): the theory of relativity originated with Poincaré, not Einstein. And it was developed not only through Einstein's efforts—far from exclusively his.

 

[sbrothy]

Yeah, as history goes, without those mathematical "geniuses" close to him. As you say Pointcare et al. who knows where it would have ended up?

EDIT: I need to update my "philosophy/history"-thread with the mathematicians behind Einstein (and Claude Shannon), but man I'm busy! :)

 

[sbrothy]

I wanted to find the others but you know who they are. This is a good history reference :

(History of Special Relativity - Wiki)

 

[sbrothy]

I can't believe I got the last word in a thread in the relativity forum?! I'm sure you're yanking my chain now!

This wont stand!

EDIT: Ah, phew! I got nervous there!

 

[robphy]

A general note (just for historical fairness): the theory of relativity originated with Poincaré, not Einstein. And it was developed not only through Einstein's efforts—far from exclusively his.

Just for historical fairness, one should also mention Galileo
( https://en.wikipedia.org/wiki/Principle_of_relativity#In_Newtonian_mechanics )
who used the idea of (but not the word) relativity in his story of https://en.wikipedia.org/wiki/Galileo's_ship .


One should also mention Felix Klein who formulated the Cayley-Klein geometries (1870s)
https://en.wikipedia.org/wiki/Cayley–Klein_metric#Klein
which essentially has Minkowski spacetime geometry as a special case.

Undergrad Post in thread 'Why Is Minkowski Spacetime Non-Euclidean?'

Jul 1, 2022

I want to know, what people think when they say "space-time is non-euclidean". Do they have a mental image in which space-time is represented?

This is a tough question to answer because the Minkowski metric does not seem imaginable or representable to human minds.

A further, possibly interesting, comment:

---

First, the setup:

Felix Klein (Minkowski's grand-PhD-advisor​

• Klein https://www.genealogy.math.ndsu.nodak.edu/id.php?id=7401
• Lindemann https://www.genealogy.math.ndsu.nodak.edu/id.php?id=7404
• Minkowski...

• robphy

• 

From Torretti's Philosophy of Geometry from Riemann to Poincare, p 129 [via Google books]

In his posthumous Lectures on Non-Euclidean Geometry (1926) Klein briefly examines the other four degenerate cases. He does not pay much attention to the resulting geometries because angle-measure in them is not periodic - a fact that, in Klein's opinion, makes them inapplicable fo the real world, since "experience shows us that a finite sequence of rotation [about an axis of a bundle of planes] finally takes us back to our starting point". [Torretti references Klein's lectures [in German], p. 189]

I believe this is referencing the fact that the Galilean and Lorentz Transformations are not periodic.
It's possible that Klein (in the 1890s) in his study of hyperbolic and elliptical geometry could have uncovered, by analogy, the mathematics of special relativity before Einstein (1905) and Minkowski (1907).

I think it's fair to say that all of these were maybe one or two steps away from a correct formulation of special relativity.


update:
As JL Synge says, history is best left to the historians.
Let's move along....

 

[wrobel]

history is best left to the historians.

I doubt that this statement is applicable to the history of science.

 

[Dale]

My "aha moment" in relativity came in grad school when
my instructor showed how operational definitions of radar-measurements led to
how an inertial observer splits up his view of spacetime

My “aha moment” came when I drew my first spacetime diagram complete with coordinate lines for both frames via the Lorentz transform. This was 7 years after first being introduced to SR through a hodgepodge of thought experiments and formulas. Suddenly I could see how time dilation was reciprocal and the other little confusing things that trip all students.

I agree wholeheartedly with Synge “time has come in relativity to abandon an historical order and to present the subject as a completed whole”. The historical approach (focused on thought experiments) is counterproductive for learning SR. The focus should be on spacetime diagrams, the Lorentz transform, the spacetime interval, and four-vectors.

 

[weirdoguy]

I doubt that this statement is applicable to the history of science.

Well, I read one book on history of QM (Duncan, part 1) and I don't know how non physicist could understand any of it.

 

[Mike_bb]

My “aha moment” came when I drew my first spacetime diagram complete with coordinate lines for both frames via the Lorentz transform. This was 7 years after first being introduced to SR through a hodgepodge of thought experiments and formulas. Suddenly I could see how time dilation was reciprocal and the other little confusing things that trip all students.

I agree wholeheartedly with Synge “time has come in relativity to abandon an historical order and to present the subject as a completed whole”. The historical approach (focused on thought experiments) is counterproductive for learning SR. The focus should be on spacetime diagrams, the Lorentz transform, the spacetime interval, and four-vectors.

Nice answer! It's an example that persistent matters.
"It's not that I'm so smart, it's just that I stay with problems longer" Albert Einstein.

 

[wrobel]

Well, I read one book on history of QM (Duncan, part 1) and I don't know how non physicist could understand any of it.

That's exactly what I mean.
And I was reading a book by a philosopher of science, and he claimed that Einstein canceled Newton.

 

[Sagittarius A-Star]

I’m still in school for physics, but I’m worried I will never deeply understand relativity. Can anyone understand these concepts, or is it something only a few people in the world can understand?

I think those people can understand relativity, who had good understanding of and good grades in math and physics high school.