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〈 Berger |🎄| Dillon 〉 Profile picture
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Jul 16, 2019 • 3 tweets • 1 min read
i'm not sure if this should be obvious, but it surprised me to find that both are equal to π
Plot of the error between the integral and the sum as a function of the limits of integration/sum.
Here's my guess: Since the frequency of sin(x) gets arbitrarily large, its peaks get smooshed arbitrarily thin.

👉Area under sin(x)/x → its "heights" when x∈[X₁-ε,X₁+ε] for sufficiently large X₁ and ε>0

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〈 Berger |🎄| Dillon 〉 Profile picture
Aug 5, 2020
The curves traced out by the intersection of two lines rotating at different speeds
i'm not sure why i did this but i'm glad i did
A neat fact i found was that if the ratio of the speeds ω is not rational then the the curves traced out are non-periodic (and hence make for bad gifs)
Read 5 tweets
〈 Berger |🎄| Dillon 〉 Profile picture
Jan 26, 2020
Prof. Claudia de Rahm's theory of 'Massive Gravity' to explain Dark Energy is getting a lot of press as of late.

So let's dig into it. (thread: 1/n)

theguardian.com/science/2020/j…
Before getting to gravity, recall that forces are transmitted by a 'mediator' which are always bosons

EM: the photon, strong force: gluon, weak force: W± & Z.

When the symmetry corresponding to the force carrier is unbroken the force carriers are massless (m=0).

(2/n) Image
What about Gravity?

The symmetry corresponding to gravity is the most respected symmetry in all physics & goes by many names

It's the one underlying Einstein's General Theory of Relativity:

General Covariance/Gauge-Lorentz Symmetry/ Diffeomorphism Invariance

(3/n) Image
Read 7 tweets
〈 Berger |🎄| Dillon 〉 Profile picture
Jan 15, 2020
Noether's Theorem is perhaps the most beautiful mathematical theorem in physics

•It states that: Every continuous symmetry (T) of the Lagrangian has a corresponding conservation law Image
Caveats (1/2)

The astute reader will notice that technically Noether's theorem applies to invariance of the action, which is the spacetime integral over the Lagrangian density

Hence, you will also need to make sure that the measure is invariant under the transformation as well
Caveat (2)

Since surface terms do not affect the Euler Lagrange equations, the equations of motion are always trivially invariant upon adding a 4-divergence to the Lagrangian density
Read 7 tweets
〈 Berger |🎄| Dillon 〉 Profile picture
Nov 9, 2019
(1) The Weak Force is extremely short ranged: its potential is *almost* Coulomb's Law (2), but exponentially suppressed due to the fact its force carriers W⁺⁻, Z⁰ are massive (i.e. m≠ 0)

(3) The second term in the last equation (kr) is what makes the strong force so "strong"
How do we get the potentials of the 3-fundamental forces from first principles? Quantum Field Theory

(1) Specifically, we take the Fourier Transform of the Feynman diagram that describes each force

We can *almost* get all 3 this way. Everything except the Strong Force (2/N)
For the Gluons, we *do* obtain the "Coulomb" term (1/r) but we don't get the linear term (kr).

To find out what’s going on, we’ll need a couple facts.
Read 8 tweets
〈 Berger |🎄| Dillon 〉 Profile picture
Aug 2, 2019
A changing Magnetic flux induces an Electric field. This can create a current in a wire to power a💡!

No battery needed!
This is the idea behind power generators:

• Spin a magnet with a mechanical source (e.g. water)

• The changing Magnetic field induces an AC current in nearby wires

• The current can then be sent to an "outlet" (or applied directly as shown here)
This is precisely how "induction stoves" work

• The oscillating magnetic field wirelessly induces a current in the pot

• The current flowing through the pot results in resistive heating

The coolest thing about induction stoves is that the stovetop itself doesn't get hot!
Read 4 tweets
〈 Berger |🎄| Dillon 〉 Profile picture
Jun 13, 2019
How was the Higgs Boson discovered? (1/n)

👉The main process which led to its discovery is its decay into 2 photons: ℎ → γ + γ
👉This process is occurs at "loop level" mediated by the top and bottom quarks 𝑡 & 𝑏
👉The higgs lifetime is 1.56×10⁻²² s
Step (1): Smash Protons

👉The LHC smashes together protons
👉When two protons collide, it produces a "shower" of its constituent quarks 𝑞 and gluons 𝐺
👉The 𝐺 can then decay into pairs of t̅t (top, anti top)
Step (2): Higgs Production

👉These t̅t coming from the Gluon decay, can then interact with one another
👉These t̅t can then "annihilate" to produce a higgs particle, via the process shown below.
Read 6 tweets

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