Time Contraction and length dilation in SR | Sociology in Pure Physics | N J Wildberger
Vložit
- čas přidán 22. 01. 2024
- We present a simplified Euclidean version of the mathematics behind Special Relativity, in which we are able to appreciate some of the seemingly remarkable consequences of the Lorentz transformations such as time dilation and length contraction. As the title of the video suggests, in the Euclidean case there is an interesting twist.
With a bit of geometry and linear algebra, we see that the heart of the matter is just a change of coordinates. Nothing "physical" is happening here, except that we are just developing the consequences of having different points of view on the same "underlying story". In fact we move towards letting go of there being a well-defined "underlying story": there are only points of view and the relations between them.
This is an elementary talk that hopefully will be accessible to a broad audience, even with only some basic high school mathematics.
This is part of the Sociology and Pure Maths playlist available at • Towards a Sociology of...
***********************
My research papers can be found at my Research Gate page, at www.researchgate.net/profile/...
My blog is at njwildberger.com/, where I will discuss lots of foundational issues, along with other things.
Online courses are being developed at openlearning.com. The first one, already underway, is Algebraic Calculus One at www.openlearning.com/courses/... Please join us for an exciting new approach to one of mathematics' most important subjects!
If you would like to support these new initiatives for mathematics education and research, please consider becoming a Patron of this Channel at / njwildberger Your support would be much appreciated.
Here are the Insights into Mathematics Playlists:
• MathHistory: A course ...
• WildTrig: Intro to Rat...
• Math Foundations
• Wild Linear Algebra
• Famous Math Problems
• Box Arithmetic
• Elementary Mathematics...
• Year9Maths
• Ancient Mathematics
• Wild West Banking
• Sociology and Pure Mat...
• Sociology and Pure Phy...
• Old Babylonian Mathema...
• Probability and Statis...
• Boole's Logic and Circ...
• Universal Hyperbolic G...
• Differential Geometry
• Algebraic Topology
• MathSeminars
• Playing Go
• Diffusion Symmetry: A ...
Here are the Wild Egg Maths Playlists (some available only to Members!)
• Algebraic Calculus One
• Classical to Quantum
• Algebraic Calculus Two
• Advice for mathematics...
• Solving Polynomial Equ...
• The Hexagrammum Mystic...
• Algebraic Calculus and...
• Dynamics on Graphs
• Playlist
• Triangle Geometry
• Explorations with q-se...
• Six: An elementary cou...
• Maxel Inverses and Ort...
************************
Excellent work! You should do one on Tullio Levi-Cevita's paper "On the analytic expression that must be given to
the gravitational tensor in Einstein’s theory."
14:16 "He wants to make the position measurements at the ends at the *same time*". Should we interpret not only this phrase, but the philosophy of, say, your physics as a defense of absolute time and space? To me, it is clearly the case; time contraction and length dilation being nothing more than parallax errors erroneously elevated to mysterious physical phenomena. As with the "real" numbers, once again the emperor is naked. I'm so glad someone with your intelligence and clarity of mind is speaking up. Keep up the good work!
This is a fundamental insight into how the effects of measurement in more than one concurrent domain are produced and are a natural consequence of it!
You exposes a complete and utter misunderstanding this phenomenon that has been with us for over a century now!
I have also dealt with this "sociology effect" in my own dealings with those who have a vested interest (both intentionally and unintentionally) in this confusion that you completely remove from Special Relativity.
You are very wise (and careful) in your use of the term "sociology". I understand why too. It can be very dangerous to speak of the full idea in the open. Those who "see" the charade know exactly what you mean.
It was about space that someone explained properly the difference between things happening in outer time and the way they look to us. In a way, it's much like you not really hanging from the bottom of my Earth and like me not hanging from the bottom of yours.
Many thanks from the antipodes. Looking forward (or is it sideways) to the next lecture on this.
Antipodal points identify for the rotation group SO(3).
North is dual to south -- magnetic fields.
Time contraction is dual to length dilation -- the twin paradox (duality).
Time dilation is dual to length contraction -- Einstein, special relativity.
Space is dual to time -- Einstein.
Simultaneity is dual to relativity -- Einstein.
Real is dual to imaginary -- complex numbers are dual.
Photons, light or probability waves are modelled with complex numbers or duality!
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality.
Sinh is dual to Cosh -- hyperbolic functions.
Syntax is dual to semantics -- languages.
If mathematics is a language then it is dual.
Concepts are dual to percepts -- the mind duality of Immanuel Kant.
Mathematicians create new concepts from their perceptions, observations, measurements or intuitions.
Symmetry is dual to conservation -- the duality of Noether's theorem.
"Always two there are" -- Yoda.
I am totally happy. Thankyou, Dr. Wildberger
Great video. You said: "Time dilation, length contraction, these are observational things. It's not as if anything is really happening". Yes, it's obvious for objects in non-accelerated (inertial) motion.
Einstein was undoubtedly a very creative person, with groundbreaking ideas/statements about reality. But he also made mistakes, the most wellknown is perhaps believing in a static universe, until he changed his mind 1931, long after the General theory of relativity was produced (1915). Another fairly wellknown mistake by Einstein, was to reject the existence of black holes. He did that in a published paper 1939. Slightly remarkable concerning this is that Karl Schwarzschild (clearly also an astrophysics genious) mathematically showed (1916) that black holes should exist, using Einsteins own basic equation system in General relativity! From Schwarzschild's equation (called the Schwarzschild metric) the gravitational time dilation equation, used since then, could be derived.
In the Special Theory of relativity (1905) there unfortunately seems to be a profound mistake (a thought error concerning the physical interpretation of certain equations), which actually can be proven without much mathematical acrobatics after understanding the conceptual aspect of it. I do that below. I think a big problem at that time for scientists was the lack of knowledge of the atom structure, and how physical time basically is relative rates of change in such structures. Physical time emanates from matter, not from a dimension in space.
1. When reasoning about clocks and time, one should use atomic clocks. Because physical time emanates from rate of change of the parts (electrons, quarks) atoms are made up of. Mechanical acceleration (not free fall with gravitational acceleration) cause time dilation, slowed rate of change in atoms. Einsteins equivalence principle is correct *, but not completely correctly interpreted physically by the mainstream science community imo.
* The rate of change in atoms in an object held fixed in a gravitational field (it is then mechanically accelerated upwards), is the same as the rate of change in atoms in an object similarly mechanically accelerated in a non-gravitational environment (and while accelerated, moving in an absolute sense).
2. It is a mathematical fact in the Lorentz factor, used in the Special Relativity (SR) time dilation equation, that the squared velocity variable there is equal to: 2 • acceleration • distance. Which is an equation originally derived by Torricelli (v squared = 2 • acceleration • distance, assuming initial velocity = 0), later incorporated among Newtons equations of motion. What that means is that the SR time dilation equation is not valid (can not be used) for calculating time dilation for time intervals where acceleration does not take place. Which means that the so called Clock hypothesis is false, which assumes that acceleration as such has no effect on time dilation. Instead, it is ONLY accelerated motion (but not gravitational acceleration with free fall) which creates physical time dilation.
In summary:
- Equations in Special relativity, containing the v^2 variable can not be used for inertial (non-accelerated) motions, they are only mathematically valid for accelerated motions. This seems to be a very common mistake by many physicists in calculation examples.
- The twin "paradox" can then be fully explained/understood, intuitively, logically, mathematically and physically. Physical processes in atoms slow down when a rocket is accelerated. Only one twin is accelerated and it does not matter how far he/she travels back and forth with a non-accelerated coasting motion, the physical time dilation will be the same. The so called distance proof, that velocity as such, in inertial or non-inertial motion, is the source of time dilation is false. It is an invalid use (interpretation) of the time dilation equation, which produces different time dilations for different travelled total distances for the travelling twin.
- If muons are time dilated (live longer in average before disintegrating) when decelerated in the atmosphere moving towards the ground of the earth, they can not be fundamental particles. They must in reality be constructed with parts which can change their rate of change when accelerated (decelerated), to fit with the pure mathematical
reasoning in point 2 above.
Beautiful work.. Although, we see that the lengths for the B observer might be irrational number!! But since irrational numbers actually don't exist, how is that gonna be?! Is it possible to work with quadrance here too instead of lengths? Thanks professor ❤️🙂👍
wonderful teacher. Thank you
Video contents
17:59 Length Dilation In SR
Love this analogy!
I wonder how this translates to the 'bats in a 1D cave' example. Suppose we use this Euclidean model instead of the relativistic one.
Is there an equivalent concept that there's a 'maximum speed', and that's still the speed of sound? Would we need to live in a universe where things work quite differently to make it work? In other words, if we start with a speed of sound, will we always get a relativistic change of variables? (I think probably so.) If that's the case, then how would sound have to work in order for the universe to obey the Euclidean change of variables?
Or maybe it's just a case of measurement? In the relativistic framework, we had the bats holding mirrors and measuring distance by echoing sounds off of them. Is there some different kind of apparatus that could (perhaps 'magically') work in such a way that the resulting measurements would result in this Euclidean transformation instead?
I'm really having a hard time imagining this, so I really don't know, but it's fascinating to think about.
This is all hinged on whether light speed ALWAYS enters your eyes at one speed regardless of the speed of your eyes (postulate, assumption), so if you move your eyes, light speed compensates to YOUR new speed(???). ONLY if light speed is constant "RELATIVE TO THE (narcissistic) OBSERVER" does any of this make sense, and I would agree with you then, but I have never seen light speed constant relative to the observer be anything but an assumption used FIRST. Even the muon experiment assumes it to be true, so if it's not true and muons are traveling faster than light entering the atmosphere, this "test" becomes useless. I don't think velocity or relative conditions have any affect on nature at all, and time is nothing more than a sequence of events, SO IT CANNOT BE VARIABLE -just a sequence counter.
I will have to think about this for a while, but it seems to me that you have found a representation in which
1. speed is a spread
2. asking about a 'speed' of light makes no sense and amounts to a division by zero error
3. wondering why one can't go 'faster' than light is like wondering why nothing can be steeper than vertical
Everyone of these consequences looks at least promising.
For what sense there is in General Relativity, could one make E/m a quadrance that doesn't involve 'c'?
Could one put the result from that into quantum physics, get rid of the division by zero error of having 'c' and the 'infinities' that 'result' and in doing so find a formulation that works without needing renormalisation?
Hi Peter, Absolutely the connection with spreads and quadrangle ie Rational trig becomes important and I will be talking about that. And we will have to put the “speed of light is a universal constant “ idea to bed. GR is a whole new ball game though.
Have both observers decided on the shortest time interval, have they got the same limit/ scale/unit and how do they know that? Is it not a convention, agreed between observers, that the speed of light corresponds to a certain number (how exact is the measurement. Do we know precisely)? What if they use a different system all together or change the system (metric spaces? When they flip it like here? Spacetime/ Timespace?)...(Vague/silly questions to difficult for a "sociologist" concepts but trying to understand...).
17:35
Alright, if it's all just observational and nothing is actually changing in the internal workings, what about the aging ? What about the twins? If one of them takes a spaceship and comeback years afterwards. If he gonna be younger than his twin brother?
One of my life's aims Is to meet you in real life. If I ever go to Australia someday.
Professor, what is happening when someone is accelerating? Does that mean the spacetime around him changes?
That is a huge question. Which led Einstein to General Relativity. We will talk about that at some future point.
Why timespace as opposed to spacetime, isn't it time dilation & length contraction just by convention ?
Excellent demo. Nothing magical indeed. But how about Einsteins conclusions derived from the transformations, expressed in the principles of "relativity of simultaneity". According to these there is no designated universal "now" moment as we perceive it but rather a reality in which future and past is equally real and existent. This seems a profound and world shattering idea. For starters the A theory of time would be out of the picture with possible serious implications upon religion. I would love to hear your opinion about this.
Hi Adam I will for sure be discussing these important questions.
If two similar objects were dropped from the same height for the same vertical distance, and one of them was inside a moving train at the time, would they complete the falls in unison, meaning neither would hit bottom first? According to the laws of gravity, the answer is yes. Now what if time were seen to be passing at half the rate inside the train as outside of it, would both objects have the same velocity, meaning distance divided by time? They both fell the same distance but half as much time passed in the train during the fall as outside the train.
Wait a minute... shouldn't that be quadrance dilation and something squared time something contraction?
thank you Professor. does this pertain to one of the 3 Chromatic inner products, also is there a such a thing as a Chromatic outer product?
Yes, there is a three-fold symmetry of chromogeometry, which is 2D geometry, and that is reflected in the three fold symmetry between blue red and green complex numbers. The story here is the blue (Euclidean) version: the red and green versions are the more usual relativistic variants in which we have time dilation and length contraction instead. But mathematically they are all pretty close, but of course we have a lot more familiarity with this Euclidean version, as it corresponds to a notion of "rotation" with respect to a circle rather than a hyperbola.
@@njwildberger Time contraction is dual to length dilation -- the twin paradox (duality).
Time dilation is dual to length contraction -- Einstein, special relativity.
Space is dual to time -- Einstein.
Simultaneity is dual to relativity -- Einstein.
Real is dual to imaginary -- complex numbers are dual.
Photons, light or probability waves are modelled with complex numbers or duality!
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality.
Sinh is dual to Cosh -- hyperbolic functions.
Syntax is dual to semantics -- languages.
If mathematics is a language then it is dual.
Concepts are dual to percepts -- the mind duality of Immanuel Kant.
Mathematicians create new concepts from their perceptions, observations, measurements or intuitions.
Symmetry is dual to conservation -- the duality of Noether's theorem.
"Always two there are" -- Yoda.
@@hyperduality2838 And there is also the cool way that the dualities of time and space get merged into a oneness in this theory, same with energy and momentum, and with electric and magnetic fields. Einstein the Yoda of physics!?
@@WildEggmathematicscourses The teacher (Yoda, Socrates) is dual to the pupil (Padawan, Plato).
The master (lordship, client) is dual to the slave (bondsman, server) -- the Hegelian dialectic.
"The master is dual to the apprentice" -- The rule of two, Darth Bane, Sith Lord.
The master is dual to the emissary -- Iain McGilchrist, the divided brain.
Mathematical problems lead to mathematical thinking or reflection (reaction) lead to mathematical solutions.
Problem, reaction, solution -- the time dependent Hegelian dialectic.
Thesis is dual to anti-thesis creates the converging or syntropic thesis, synthesis -- the time independent Hegelian dialectic or Socratic method.
The Socratic method of asking primary questions to synthesize answers is equivalent or dual to the Hegelian dialectic -- questions are dual to answers.
Rational is dual to empirical (irrational) -- numbers.
Rational, analytic (mathematics) is dual to empirical, synthetic (physics) -- Immanuel Kant.
Deductive inference, reasoning (mathematics) is dual to inductive inference, reasoning (physics) -- Immanuel Kant.
Noumenal (rational, "A priori") is dual to phenomenal (empirical, "A posteriori") -- Immanuel Kant.
Gravitation is equivalent or dual (isomorphic) to acceleration -- Einstein's happiest thought, the principle of equivalence (duality).
Injective is dual to surjective synthesizes bijective or isomorphism -- Group theory (Hegel).
Subgroups are dual to subfields -- the Galois correspondence.
Duality is a symmetry and it is being conserved according to Noether's theorem.
Einstein (physics) is dual to Yoda (metaphysics).
"Physics is what we know and metaphysics is what we do not know" -- Bertrand Russell.
Synthetic a priori knowledge -- Immanuel Kant, knowledge is dual according to Kant.
There are patterns of duality hardwired into physics, mathematics and philosophy.
Yoda is the Einstein of metaphysics!
Absolute truth (universals) is dual to relative truth (particulars) -- Hume's fork.
Could we make time a force? What I mean each clock could pre determine how much dilation each clock
Professor, if it was possible to make time have magnitude how could we do the rate of change?
Is SR "just" an observational effect? Well, yes and no. There is no sense in which one would observe changes in co-moving objects because it is all relative. The Lorentz transformations are, indeed, just coordinate system transforms. However, traveling near c will have you arrive at your destination faster than you would predict before you left. An observer would say that's because your time dilated, but you would say your time was normal but the rest of the universe got shorter. The whole point is that frames can disagree on the orientation of the space and time axes, but ultimately they all agree on the physical result.
This is a little bit confusing to me. You're talking about listening to clicks and looking at rulers? That means that both the speed of sound and the speed of light are involved? Shouldn't you at least specify what you are assuming these speeds to be?
@erniefrom... No speed of sound or light is involved. Like in SR, each observer has sufficiently many helping mates, namely one at each position where something interesting happens. These helping mates are at rest w.r.t. to the aforementioned observer, and they are equipped with clocks synchronized with each other. Their job consists in noticing the events which happen exactly at the place where they are standing, in reading off their clock at the very moment of such an event and in recording all these pieces of information in a protocol. After the end of some experiment (i.e. some series of events) these protocols are collected in the observer's office and evaluated appropriately. The speed at which this information (these protocols) are transmitted is completely irrelevant.
If they're 90 degree rotated from each other this method doesn't work, whatever time and space of A, the B would see always as 0 and viceversa.
I think that would require B to be travelling at the speed of light with respect to A.
@@brendanward2991 you're assuming there's any travelling at all, but time independently makes no sense to be 0. Visual artist here, if I'm missing anything with the fact the observers are rotating their axis radially while their metric comparisons are straight-linear please let me know
@@IsaacRC If A and B were at rest with respect to each other, their axes would be aligned with one another. The axes are not rotating. They're fixed, but at an angle due to the relative velocity. That's my understanding.
@@brendanward2991 definitively i lost it, thanks anyway :) without explaining it's confusing the geometric logic of overlaying A and B rotating them radially and then just comparing metrics with a perpendicular straight line instead of also radially, then you have this 90 degree cancelation of time
17:00 - "Time dilation, length contraction ... these are observational things. It's not as if anything is really happening ..." - But what about things like the Twin Paradox, in which one twin really does age at a different rate than the other and ends up younger than his twin?
Time contraction is dual to length dilation -- the twin paradox (duality).
Time dilation is dual to length contraction -- Einstein, special relativity.
Space is dual to time -- Einstein.
Simultaneity is dual to relativity -- Einstein.
Real is dual to imaginary -- complex numbers are dual.
Photons, light or probability waves are modelled with complex numbers or duality!
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality.
Sinh is dual to Cosh -- hyperbolic functions.
Syntax is dual to semantics -- languages.
If mathematics is a language then it is dual.
Concepts are dual to percepts -- the mind duality of Immanuel Kant.
Mathematicians create new concepts from their perceptions, observations, measurements or intuitions.
Symmetry is dual to conservation -- the duality of Noether's theorem.
"Always two there are" -- Yoda.
that is not special relativity. that is general relativity. When you are accelerating, that means you are not moving in the unit circle here. The geometry around you changes so you can move faster.
What you are talking about is general one, not special one. The spaceship itself does not change the realspace when it moves, it only changes when it accelerates.
Right off the bat I can tell this is not going to work. Your B frame is rotated in a standard euclidean fashion, and that is not what a Lorentz transform does. It's a _hyperbolic_ rotation, so the x and t axes get closer together or farther apart. It's particularly clear that this is correct in Cl(1, 3), geometric algebra, or spacetime algebra.
This is why you are getting length contraction and time dilation. A magnitude (proper time) is given by t^2 - x^2, and for any massive object's velocity, it is 1 (assuming c = 1). The B frame axes need to be scaled to reflect that, in addition to moving the x axis into the correct angle. If you did that, you would indeed see that _both_ frames appear time dilated and length contracted from one another.