Welcome to the third installment in our video series, The Biggest Ideas in the Universe. Today’s idea is actually three ideas: Force, Energy and Action. There is a compelling argument to be made for splitting these up — and the resulting video is longer than it should be — but they kind of flowed together. So there you go.
The Biggest Ideas in the Universe | 3. Force, Energy, and Action
Major technological upgrades this time around, with improved green-screen setup, dark background on the writing, and a landscape format for the text. I am learning!
Hope everyone is staying safe and healthy during the lockdown.
Update: Here is the Q&A video. No cats make an appearance this time, sorry.
The Biggest Ideas in the Universe | Q&A 3 - Force, Energy, and Action
Thank you for this. I’m struggling with the intuition behind Galilean Relativity. If there was no rocket ship, and you were just a sentient particle accelerating in empty space, would you still know you were accelerating? More generally, why do we talk about this in terms of what a brain could detect as opposed to something simpler? For example, what if anything could we say if we were forced to explain why acceleration has a “preferred state” as opposed to position or velocity without referencing observers?
Could you perhaps comment on the following question: In what sense can the principle of least action be derived from quantum mechanics?
(Maybe it would be interesting to revisit the principle of least action in a later episode related to quantum mechanics and Feynman’s path integral formalism?)
I would also like to mention a physics textbook on the principle of least action: “The Variational Principles of Mechanics” by Cornelius Lanczos.
Another great and bite-sized lecture, thanks. My question is in regards to your mention of the Higgs field, mass and gravitational field. I promise I will read your book which I’m sure will explain this, but until then can you answer:
What is the relationship between the Higgs fields and the gravitational field? Is the gravitational field derived because there is a Higgs field? Or are they two independent phenomena?
Follow-up question because I dare: do fields already exist everywhere (beyond space-time) or do they expand as the universe expands? (My understanding is there’s no spooky action at a distance since the field is present, but is it independent of the new space being created?)
Hello would it be possible to consider a thought experiment where an object that have acquired potential energy by being lifted up from the ground and suddenly a”devil’makes the ground, gravity source disappear. what actually happens with that potential energy that has been acquired before the gravitational object has been removed ? Has it disappeared also and if so is potential energy just a way that is useful for us to describe deeper stages in nature and that’s why we created?
Hi, Sean.
About higher derivatives and negative derivatives of position, my post (a reference in wikipedia in the entry of Absement): http://www.thespectrumofriemannium.com/2012/11/10/log053-derivatives-of-position/
By the other hand, could you add/explain what apps and software are using for recording these videos and doing the blackboard? What about the background picture? I mean:
1st. What app did you use to do the blackboard inside the video?
2nd. What app did you use to record the video?
3rd. How did you manage to put those galaxies in the background?
I am interested since I have to give classes/lectures as well soon. How to handle them easily is a problem. And I need options. It seems your logistic is easy. I want to know how to do it…
Thanks!
Thank you Dr. Carroll for taking the time of explaining things so well.
The chronology of events was Newtonian mechanics first, and Least Action formulation later, was this sequence inevitable? Or do you think there was a way for Newton himself or someone else to think the other way around, assuming Calculus was available? It seems to me that the availability of the right mathematical tools in instrumental for Physics progress in some cases, even if you have the right physical intuitions.
Is it time yet to ask about relationships between symmetries of action and the principle of stationary action? Or about whether symmetries of quantities other than action would lead to other kinds of conservation?
Sorry if I’m becoming a broken record, but it just seems so tantalizing when all the Big Ideas so far seem to be approaching it from different angles without quite getting there.
You mentioned least time as similar principle to least action. Timelike geodesics might be another, all related to stationary phases reinforcing while shifting phases cancel, so is it because you have terms relating to a quantity and to its derivative? If so, what about other quantities and its derivative, or what about derivatives with respect to something other than time?
The Rice Krispie derivatives.
But position and velocity can’t be individually fundamental, when defining one undefines the other.
It could be possible that their joint distribution is fundamental, but you may have hinted at the right (biggest) idea at the end that it’s only the wave function of the universe that’s fundamental, rather than any particular way of decomposing it.
This is a question about changes in the rate of time & a potential ‘further’ test of General Relativity.
In 2010 NIST reported on their ground based precision clock Relativity tests comparing clocks at differing relative speeds (SR) & comparing clocks placed at differing positions within the gravity potential (GR).
These NIST Relativity tests prove beyond any doubt that a clock moving at greater velocity ticks slower & that a clock ticks faster higher up within a gravity potential – BUT has it been experimentally proven that time ticks slower in the greater gravity field? (think bigger galaxy)
Looking at the NIST 2010 experiments:
For SR NIST gyrated the comparison clock to obtain relative speeds.
For GR NIST raised the comparison clock 1 metre higher to obtain differing positions within the gravity potential.
The SR experiment incurs a difference in centrifugal force for the gyrated clock as well as difference in speed.
The GR experiment incurs a difference in both speed (centripetal) & centrifugal force for the raised clock as well as a difference in gravity.
I have not been able to find any evidence of a clock comparison experiment conducted ‘anywhere’ that holds speed & centrifugal force equal for both clocks & ONLY a difference in gravity occurs.
By isolating the variable of gravity from the variables of centripetal speed/centrifugal force & testing a difference in gravity ONLY* –
Q: Would a clock comparison experiment that measures ONLY a difference in gravity be a ‘further’ test of General Relativity?
*(by utilising NIST portable clocks at differing locations of same longitude & height above sea level to equalise centripetal speed/centrifugal force, but of differing geological density – or via placing clocks at existing grav.wave detectors – or via these similarly oriented proposals
https://arxiv.org/abs/1506.02853
https://arxiv.org/abs/1501.00996 )
Great talks! Thanks!
Would you mind revealing the secrets of your technological upgrades? As in what apps, programs and technology you are using? The more specific the better – like how you get yourself down in the corner, how you set up the green screen and background, and what you are using as the chalkboard-like text to the side, etc. For those of us who are occasionally called upon to create educational materials, this format is very effective and worthy of emulation.
In his book, E=mc^2, David Bodanis weaves Newton, Leibniz, du Chatelet, ‘sGravesande, religion, momentum, and energy together. According to Bodanis, Newton considered mv to be energy and noted that it was lost when two equal mass wagon, traveling at equal speeds in opposite directions, collided. Apparently, Newton depended on God to replace the lost “energy” from time to time–to wind the universe back up as it were. Leibniz thought the important quantity might be mv^2 and noted that in the collision, this “energy” was conserved in the noise and wreckage, so God’s intervention was not needed. ‘sGravesande provided the crucial data when he dropped balls onto a mud floor and saw that the depth of the depression in the mud depended on mv^2. du Chatelet put the idea and experiment together. I wonder if Galileo had made the observation when he studied pile driving technology in Venice? In any case, I thought you might be interested in Bodanis’s account.
I forgot to mention how much I am enjoying your series. Thank you.
Please could Sean Carroll explain the latest thinking on the cause-effect continuum and can he say if any event in Nature is truly random? It seems ludicrous to posit but in a deterministic universe can my choosing either a cup of tea or a cup of coffee be traceable to the Big Bang? Thanks.
Really enjoying your talks Sean. I’m pondering the question: Is time continuous or is it discrete?
As you pointed out there are a infinite number of real numbers between 0 and 1. Now let these numbers represent times between say 0 and 1 second. It stands to reason there are then an infinite number of times as you map the infinite set of real numbers to the infinite set of times. If time were discrete there would be a finite number of times between 0 and 1 second, right? So does this establish time as continuous?
A ball will start *rolling* down a frictionless plane? I think Aristotle is having a good laugh at us modern physics types right now. Sometimes it can be hard to keep your cows spherical: real world knowledge has a way of sneaking back in.
At the end, Sean points out that either Force or Least Action can be dispensed with in favor of the other, and we have a choice of ways of talking. But that doesn’t mean that force and action are unreal. Hmm, someone should invent a label for this philosophical point that we have multiple valid ways of talking about the universe – maybe “Poetic Naturalism”?
Sir, you are continuing without defining “Energy”.
1/2mv2 and mgh come from nowhere.
Our school texts defined it as Force x Distance which leads to 1/2mv2 and mgh in two steps.
Sean, what do you think about the analogy of molasses for Higgs interaction. If the Higgs interaction was like molasses won’t the “viscosity” of the Higgs field sap the kinetic energy of a particle moving through it at the constant velocity and thus contradicting the Newton’s First law about constant velocity motion. Instead I like the analogy of photon box (which confines the energy of the photons in a box with perfectly reflective inside walls) for the inertia of a massive object. IMO giving a wrong analogy is worse than giving no analogy. Thoughts?
Dear Gam Vije:
Yes, Sean has (non explictly) though, explained what is energy. But it is hidden in the lagrangian.
Let me enlighten you thanks so symmetries. Note that p and v are in general FUNCTIONS of time. However, energy is something that is “conserved”. For a free particle we want to guess something that when minimized (really it is more generally a critical point, but it is not important for this discussion), will give us the conservation of momentum for a free particle, i.e.,
p=constant (not a function of time!) and dp/dt=0.
Well, the quantity is indeed known as Vis Viva since Descartes times…But it predates as well the Noether theorem and the modern physics:
We want a quantity (generally depending on time) that when DERIVED with respect to time (indeed also with respect to q or position, but as the particle is free, no depending on x in principle) will give us the equations above.
First procedure:
1) Rewrite the two equations in a piece of paper: p=constant (we do know that p=mv but it is not important for the rest until the end) and dp/dt=0
2) Multipliply dp/dt by p, to get pdp/dt=0.
3) Use the chain rule (or Leibniz rule for the derivative of a product) to guess that
p(dp/dt)=d(p²/2)/dt=0
4) Divide by the constant m
5) d(p²/2m)/dt=0
6) Define the KINETIC energy OR VIS VIVA (Descartes name) as E_k=VV=p²/2m.
7) Recover the standard definition plugging the linear momentum definition p=mv into 6), so
E_k=mv²/2. Note that in general E_k will depend on time…BUT, if the particle is really free, there is invariance under translations in time. Then, v will be “constant”, and then p or E_k.
Second procedure (subliminal in Sean’s lecture).
You have an action, S. S generally dependes on x(t) and its derivatives untill order k, but physicists generally work out with FIRST order actions for simplicity. So, we presume the action for a free particle es something like this:
S(x, dx/dt; t)==\integral “Lagrangian” (x(t), dx/dt;t) dt
The question is now, what should we choose as lagrangian for the free particle? Well, there is nasty trick in Mechanics, known as Jacobi last multiplier. The Lagrangian function must be something such as
m=partial²(L)/partial(v²), and then L will be L=mv²/2!
Equivalently, supose the lagrangian is some arbitrary function L(x, dx/dt). For the free particle it can NOT depend on q, since momentum IS conserved, a translation in x should leave the action invariant up to a boundary term. Mathematically, differentiating:
dL=partial(L)/partial x dx+partial(L)/partial (dx/dt)=partial (L)/partial(dx/dt)). Then the lagrangian can only dependes on velocity v=dx/dt. We usually want quadratic functions…And then…
L(dx/dt)=A(dx/dt)²+C, where C is any constant non depending explicitly on dx/dt but it could depend on x(t)!!!
Now, derive twice the lagrangian and again you will get L=mv²/2 if A=m/2…But it is of course conventional…
Third procedure. The Noether procedure.
How to derive the Lagrangian for a given set of symmetries?
You need indeed the characteristic equations for the operators of the invariance (more generally quasiinvariance, invariance modulo a boundary term) of the lagragian and the action integral.
For the free particle, you have two symmetries: invariance under translation in time and invariance under translation in space.
Space translations are generated by the operator partial/partial (x)=D_x
Time translations are generated by the operator partial/partical (t)=D_t
You WANT to build up some action or LAGRANGIAN, depending on x and t such as, when VARIED with respect to time and space, will conserve
a) D_x ( L(x, dx/dt))=0
b) D_t (L (x, dx/dt))=0
dL=(partial L /partial x) dx + (partial L/partial (dx/dt)) d ( dx/dt)+ partial L/ partial t dt.
The first term in the right handed side is zero due to a) Condition. Condition b) means that the last term (the third) is zero. So we are left with the condition
dL=(partial L/partial (dx/dt)) d(dx/dt)=d L( dx/dt)=0 [it can not depend on x explicitly due to space translations!]
But this implies that d/dt(partial L /partial (dx/dt))=0
Integrating once: partial L/partial (dx/dt))=f(x)+C(dx/dt), note that x and dx/dt are “independent” as variables (and that d/dt(dx/dt)=0 by our equations of motion and time translation symmetry)
Integrating again: L=f(x)(dx/dt)+C/2 (dx/dt)^2 up to a constant. Again, que must set up f(x)=0 as L can NOT depend on x by SPACE translational symmetries, so we are left L=A/2 (dx/dt)²=Cv²/2, again our Vis Viva or kinetic energy if we define C as the mass, or as the last Jacobi multiplier
C=partial^2(L)/partial (v²)=m
The lagrangian function can ONLY depend on the velocity quadratically in standard theories. Indeed, requiring that the function is always positive and “simple”, and that if reproduces the Newton second law (in fact that is important too!), it simplifies this further. L(v)=Av²+constant. BUT, many interesting properties of gravity are due to the fact General Relativity is A SECOND order theory (it can be rewritten as first order with nasty tricks). Theories beyond standard theories or standard gravity(General Relativity) like string theory, M-theory, brane theory do contain KINETIC energies beyond the quadratic usual term. BUT, due to some technical aspects (yet topic of research!) use to contain unstabilities, so quadratic energies do it better (yet), excepting a few known cases! I hope these lines help you to understand why Sean did it …
The only thing that confused me was when you say ‘god willing’
No response needed. I love these videos.