
An explorable explanation of relativistic spacetime - mxfh
https://www.lucify.com/inside-einsteins-head/
======
chriswarbo
This is quite nice, although I think I'm too familiar with to topic to know
whether it's actually useful to someone trying to learn this stuff :)

Although I think that "explorable explanation" is a bit of a grand claim for
this sort of thing. The explanations (the connections, reasons, mechanisms,
etc.) are given in the text, which isn't any more "explorable" than a book;
whilst the "explorable" parts (the graphs/diagrams) aren't really
"explanations" since they have no depth: the relationships are hard-coded, so
the user can only adjust some parameters and get told what the "answer" is for
that parameter.

Many years ago I tried to make a more in-depth "explorable explanation" of
some concepts from thermodynamics, using the Smalltalk-based EToys system. I
did this by setting up many small simulations, e.g. of "atoms" bouncing around
in containers, of pistons and thermometers, of falling weights, etc. The
behaviour (scripts, methods, etc.) of each object could be inspected, and
mostly involved simple stepped-integral scripts like those in games (update
position based on velocity, update velocity based on acceleration, etc.).
"Measurements" would be taken of the aggregate behaviour, e.g. of a piston's
position, and plotted against some other parameter (time, pressure, velocity,
etc.), to produce graphs which (hopefully) demonstrate the relationship
described by the text. These were "explorable" since the user had complete
control to inspect and modify the dynamics of the simulations and see what
happens.

Unfortunately I ended up abandoning the project once it maxed-out the RAM of
my OLPC XO-1, since I was no longer able to load and save it :(

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z3t4
As a kid there was a military air base in the neighboring city, so fighter
jets passed our home regally. It was always fascinating that the sound and the
plane where on different locations. You could always hear them but it was hard
to spot the actual plane in the sky. Now imagine if they jets where going
faster then light and that light was a bit slower so we could see them. That
would mean the visible position would be different then the physical position
of the plane ...

~~~
SiempreViernes
Or you know, just things far away moving at normal speeds, like also happens
with sound. The sun for example is not physically in the place it appears to
be due to the 8 minutes of light travel time and the fact it moves about one
diameter in two minutes.

~~~
tgb
You've got two mistakes here. First, the sun appearing to move a diameter in
two minutes is due to the rotation of the Earth, not the relative motion of
the Earth to the sun (that's obviously a year). This can happen even if the
sun and the Earth are fixed in space (with the Earth still rotating about its
center). So this certainly does not make the sun look like it is in the
'wrong' spot.

Secondly, even the motion of the Earth in its orbit around the sun does not
cause an 8 minute delay in the apparent location of the sun. The location the
sun appears is determined by the direction that its light rays are travelling
when they hit Earth. Even with the Earth travelling through space, the light
rays that it hits at every point are always pointing back to the exact
physical center of the sun as if there Earth were not moving. Yes, those light
rays left 8 minutes ago from the sun, but the ones that Earth hits now are the
ones that were heading towards Earth's current position, not the ones heading
to Earth's position 8 minutes ago. Hence they are pointed from the sun to the
Earth's current position.

If you can't visualize this, then imagine a alien sitting in its space ship
hovering fixed in space (relative to the sun) just next to Earth's orbit as
Earth passes by. Both you and the alien shoot a bullet such that its velocity
is going exactly back along a light ray from the sun. Clearly the aliens'
bullet will reach the sun. But then so too must your bullet: both you and the
alien matched the same ray of light so your bullets are actually travelling in
exactly the same direction from the same starting point. So in that sense the
ray of light that you see is pointed right at where the sun is right then.

~~~
SiempreViernes
Ok fine: The sun for example is not physically in the place it appears to be
due to the 8 minutes of light travel time and the fact it moves about one
diameter _across the sky_ in two minutes.

I just feel that if you talk about where the sun appears to be you can infer
from context that I talk about actual sky positions and motions, not the
inferred global state of the solar system, but here we are.

Anyway, the central fact in my argument is that the light that reaches us from
the sun has travelled for 8 minutes, and so shows us now^1 where the sun would
have been on the sky^2 8 minutes ago if we then had observed it by
superluminal means.

Now, that there are no superluminal means to observe the sun with (as far as
we know) is not crucial to the point because there are other situations where
one mode of observation carries older information than other. That is the
example of observing^3 the distant air plane with sound waves, and then with
the supersonic means of light waves. The sound waves emitted at a given point
in time reach you after the light waves emitted at that same instant, and so
it looks^4 like the plane is at a different position than the sound is coming
from. Since most of us rely mostly on sight, the sound is judged to be
erroneous, but clearly we could just as well pick the sound position to be the
baseline, which we do for instance if the plane is hidden by a cloud.

Also, in the spirit of your objection, you are almost completely wrong, in an
exact technical way, when you say that

> the light rays that it hits at every point are always pointing back to the
> exact physical center of the sun

Because the sun is so close, it is in fact NOT a point, but it resolves to a
disk, and since most of the disk is not on the one (infinitesimally thin)
light ray that points back trough "the exact physical centre of the sun", most
of the light does not point exactly there.

1: By which I mean a given instant in time, and due to how close the sun is
and our reasonable velocity relativity of simultaneity is not important 2:
Here "on the sky" means what position in an earth co-moving sky coordinate
system 3: Again, I am talking about the motion of objects across the sky, in
an earth co-moving sky coordinate system. The various and important ways that
the example objects differ are neglected if they do not matter for the
observed motion in the sky. 4: That is, you see optical waves with your eyes
and infer from that a specific sky position different from the one
reconstructed by your brain based on the sound waves detected by your ears.

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adamnemecek
If you are interested in this, check out the formalization of space time in
geometric algebra. It kinda makes sense! (i.e. more sense than in other
formalisms)
[https://en.wikipedia.org/wiki/Spacetime_algebra](https://en.wikipedia.org/wiki/Spacetime_algebra)

------
xixixao
I have to add that although there are quite a few great resources explaining
special relativity, resources explaining general relativity, beyond focusing
on describing the math behind it, are few and far between. Anyone's got a
contradicting example?

~~~
jperras
General relativity is, fundamentally, the study of the metric tensor: it is
the object that connects the geometric representation of space and time with
mass and energy. Different metric tensors can produce wildly different
results, from flat space to black holes to bubbles in space time that provide
inertial reference frames that can travel "faster" than than the speed of
light.

You could spend your entire life studying small variations in metric tensors.
Some people do.

More simply:

1\. Light travels from point A to point B via the shortest path.

2\. In the absence of mass (aka where the Minkowski metric applies), the
shortest path is a "straight line".

3\. In the presence of mass (where a non-trivial metric exists), space itself
curves. Light, which follows the shortest path between two points, must follow
this curved space.

As a lower-dimensional analogy, think of what a "straight" line means on the
surface of a sphere[1]. The shortest path between two points is, by
definition, a "straight" line, but if the space itself that the path is
embedded in is curved, then when viewed from a higher dimension (i.e. on the
surface of the sphere it's straight for you, but if you were in space) it
would appear curved.

The analogy fails when you attempt to think about it in higher dimensions,
since humans have a very difficult time perceiving anything greater than the
standard 3+1 spatial & time dimensions. But the math still holds.

Light will always travel in a straight line, it just so happens that gravity
redefines what "straight" actually means.

1\.
[http://mathworld.wolfram.com/GreatCircle.html](http://mathworld.wolfram.com/GreatCircle.html)

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CuriousSkeptic
I’ve been trying to learn all this stuff lately. One thing that bothers me
though, that perhaps someone here can help with.

In Minkowski diagrams you set things up to have rays as diagonal lines. But
this way of illustrating things suggests that light moves in time. But they
should really not be interpreted as that since they essentially define the
limit of the time dimension.

So I was thinking that I would like to have another illustration of things
where light moves in 90-degree angles instead to fix this.

Are there some specific, established ways to do this?

~~~
philipov
If you were to imagine photons having an internal clock, that clock never
ticks. That's not the same as light not moving through time. Light does not
experience _proper_ time.

Any statement that combines the word "move" with the claim that it shows how
light lacks time has already contradicted its goal.

I think it is better to imagine light as line-like objects that extend through
time, not point-like objects that move through time, and do away with speaking
of light as something that moves, if you wish to express that it lacks a clock
(proper time).

~~~
CuriousSkeptic
But still, that line, in a minkowski diagram, represents a boundary, so it
still seems more intuitive to me to eliminate the area beyond it completely
from the diagram.

~~~
QAPereo
A _causal_ boundary, but not a real boundary. That is to say, events separated
by that line can never meet, never interact, never observe each other, but
they can still exist.

For a “real world” example see:
[https://en.m.wikipedia.org/wiki/Kruskal–Szekeres_coordinates](https://en.m.wikipedia.org/wiki/Kruskal–Szekeres_coordinates)

~~~
fjsolwmv
They exist in the sense that the uncountable real numbers exist. The reals you
can name are like your light cone. By symmetry it seems that the other reals/
spacetime points should exist, but there's no way to prove they actually do if
you're not already "nearby". :-)

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quadcore
Can someone explain that to me. When the observer is in the train, he sees the
right tower future? Or maybe another way of saying it, the left and right
tower time are not the same? Or yet another way of saying it would be, two
points of an observed moving object are not at the same position AND not at
the same time? Two points of a moving object got different delta positions and
delta times?

~~~
jules
From a modern perspective the relation of the time axis to the x axis is
roughly the same as the relation between the x axis and the y axis. Saying
that two events happened at the same x coordinate (delta x is 0) is a
statement that depends on the observer. Another observer who is rotated with
respect to the first will see the events at different x coordinate (delta x is
not 0), because some of the x axis has been rotated into the y axis.
Similarly, saying that two events happened at the same t coordinate (delta t
is 0) depends on the observer. Another observer who is moving with respect to
the first will see the events at different t coordinate (delta t is not 0),
because some of the t axis has been "rotated" into the x axis.

This notion of "rotation" only differs from ordinary rotation by a minus sign.
If you rotate the xy plane around the origin then points move in a circle x^2
+ y^2 = const. If you rotate in the xt plane then points move in a hyperbola
x^2 - t^2 = const.

Physically you "rotate" in the xt plane by accelerating in the x direction.
This causes points to move along that hyperbola, which means that some delta x
manifests as delta t and vice versa. We don't see that in ordinary life
because we only move with respect to one another at incredibly low speeds,
which corresponds to very small "rotations".

Imagine a world where all creatures are facing along the x axis with only very
small deviations. These creatures may think that the x coordinate has physical
meaning independent of the y and z axes. They might think that there is
something special about their x axis. Similarly, we are all moving with
roughly the same velocity through the universe. We initially thought that
there was something special about our velocity, namely, we thought that we
were at rest with respect to the aether.

~~~
quadcore
_edited 10 thousand times_

Very good explanation, thanks for your time.

Still something that's unclear to me. Let say I'm observing a star moving away
from me at speed of light. Will I see the star? If yes, it means that, those
photon-things, have a property which _I_ haven't. Some "move at speed of
light" type of property. So now, we have two points on a line, the "value" of
a photon (speed of light) and my "value" (speed x), do we? What about that? We
cant call that speed because we are saying speed is relative. How do you call
that property then? And what is the other bound? Cant be x right?

Or another solution is that, light depends on the observer. Ok got it. 100%
got it. Light both has a speed (speed being always dependent from the
observer) and both has a property I havent. The key is that light has a
special property I havent, but it isnt about speed. Am I correct?

~~~
jules
You can't move with the speed of light. If you move at 99% of the speed of
light away from the star you do see the star but it will look red because the
light waves get stretched out.

Photons indeed have the special property that they move at the speed of light.
This speed is special. If A is moving relative to B and they both measure the
speed of the same light ray they both measure the same speed. This is not the
case if they measure a particle C going at less than the speed of light. In
some sense the speed of light acts like an infinite speed in the sense that "c
plus v = c" just like "infinity+x = infinity". But the plus in this equation
is relativistic velocity addition: v plus w = (v + w)/(1 + vw/c^2) where c is
the speed of light. You can see that if you set w=c you get v plus c = (v +
c)/(1 + vc/c^2) = (v + c)/(1 + v/c) = c(v + c)/(v + c) = c. The speed of light
has this property but the speed is definitely finite approximately 300,000,000
m/s.

~~~
quadcore
_If you move at 99% of the speed of light away from the star_

What is the light ray speed I measure in that case? 1% speed of light?

~~~
Viliam1234
The speed of light ray (in vacuum) is always 100%. Other things keep changing
in a way that keeps the speed of light constant from everyone's perspective.

~~~
quadcore
Thank you that helps.

Question: if I move away from A near speed of light, will I be seen moving
near speed of light in all other frames?

~~~
jules
What about another person walking around in your spaceship?

~~~
quadcore
Thank you.

So the fact that you always see light at speed of light whatever is the frame,
isnt because light is going at speed of light but because light is light.
What's different between a photon and my spaceship is not so much that my
spaceship cant reach a full 100% speed of light, but that youll always see a
photon at speed of light because of some "magic" property which cause isnt
speed. Is that correct?

~~~
jules
It is speed, the speed of light. Particles that have zero mass go at the speed
of light. This is a property of the particle. That there is a maximum speed is
a property of spacetime however. Even if there were no massless particles we
could still discover special relativity, for example by noticing that when you
speed up particles they can't get past a particular speed. In the large hadron
collider they can speed up particles to 99.9999991% of the speed of light.

~~~
quadcore
Following this discussion, I've started to study this:
[https://oyc.yale.edu/physics/phys-200](https://oyc.yale.edu/physics/phys-200)

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dvt
This is amazing! I remember having to study and draw some of these diagrams in
an undergraduate Philosophy of Space & Time class.

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catnaroek
Explanations of special relativity, explorable or otherwise, are dime a dozen.
Try explaining general relativity in this fashion. Try explaining cosmology in
this fashion.

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mrfusion
I always wondered how a database would know which transactions to commit first
in a relativistic world?

~~~
fjsolwmv
You don't need relativity, this problem is already answered by distributed
systems that lack TrueTime clock guarantees. Today have to make arbitrary and
inconsistent choices. (Or "eventually consistent", but then you lose some
level of resolution in your ordering of moments)

Related, this classic economic article:
[https://www.princeton.edu/~pkrugman/interstellar.pdf](https://www.princeton.edu/~pkrugman/interstellar.pdf)

------
mjfl
beautiful work. To anyone who is interested in really understanding relativity
deeply without intimidating math, I recommend Tim Maudlin's book: "Philosophy
of Physics: Space and Time".

~~~
Koshkin
> _without intimidating math_

While having nothing against the book or its author, I must point out that
ultimately it is "math in which we trust" (even experimental science leaves
room for interpretation); philosophy, on the other hand, gives us no choice
but to trust the philosopher!

~~~
dvt
Not exactly sure what you mean. The foundation of philosophy is logic. Also
the idea that "math in which we trust" is kind of bizzare, as mathematics is
an axiomatic system. So really, it's the axioms in which we trust. But that
also opens a whole can of worms, because some axioms are kind of weird and
controversial (like the Axioms of Choice, Replacement, Regularity).

Not to mention that any for any system (> Peano Arithmetic) you can't have
your cake and eat it, too; sound, complete, consistent: pick two.

~~~
hackinthebochs
I think the point was simply that there is no understanding of physics without
the math, as the math is the only accurate description of the physics
involved. Admittedly, "trusting math" was a poor phrasing.

~~~
Koshkin
We can trust math because it is trivially _verifiable_. Its relation to
physics is somewhat complicated (e.g. verifying physics is not as trivial),
and yes - you _can_ understand a lot about physics without math - by combining
direct observation with trusting the "philosopher".

