- 3111 - EINSTEIN - Why does energy equal mass? If you extrapolate all the way back, the universe gets smaller and smaller, it gets denser and denser, and warmer and warmer. Finally you get to a point where it's really small, really hot and dense. That's actually the Big Bang theory: that the universe started in such a condition. That's where you really have to stop.
-------------------- 3111 - EINSTEIN - Why does energy equal mass?
- E = mc². In plain English, it tells us that energy is equal to mass multiplied by the speed of light squared, teaching us an enormous amount about the Universe.
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- This one equation tells us how much energy is inherent to a massive particle at rest, and also tells us how much energy is required to create particles (and antiparticles) out of pure energy.
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- It tells us how much energy is released in nuclear reactions, and how much energy comes out of annihilations between matter and antimatter.
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- Why does energy have to equal mass multiplied by the speed of light squared? Why couldn’t it have been any other way?
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- Einstein's equation is amazingly elegant. But is its simplicity real or only apparent? Does E = mc² derive directly from an inherent equivalence between any mass's energy and the square of the speed of light? Or does the equation only exist because its terms are defined in a conveniently particular way?”
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- Energy is a particularly tricky thing to define. There are many examples. There’s potential energy, which is some form of stored energy that can be released.
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- There is gravitational potential energy, like lifting a mass up to a large height, chemical potential energy, where stored energy in molecules like sugars can undergo combustion and be released, or electric potential energy, where built-up charges in a battery or capacitor can be discharged, releasing energy.
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- There’s kinetic energy, or the energy inherent to a moving object due to its motion.
There’s electrical energy, which is the kinetic energy inherent to moving charges and electrical currents.
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- There’s nuclear energy, or the energy released by nuclear transitions to more stable states.
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- And, of course, there are many other types of energy. Energy is one of those things that we all “know it when we see it,” but to a physicist, we want a more universal definition. The best one we have is simply: extracted/extractable energy is a way of quantifying our ability to perform work.
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- Work has a particular definition itself: a force exerted in the same direction that an object is moved, multiplied by the distance the object moves in that direction.
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- Lifting a barbell up to a certain height does work against the force of gravity, raising your gravitational potential energy; releasing that raised barbell converts that gravitational potential energy into kinetic energy; the barbell striking the floor converts that kinetic energy into a combination of heat, mechanical, and sound energy. Energy isn’t created or destroyed in any of these processes, but rather converted from one form into another.
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- The way most people think about E = mc², when they first learn about it, is in terms of what we call “dimensional analysis.” They say, “okay, energy is measured in Joules, and a Joule is a kilogram * meter² per second².
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- So if we want to turn mass into energy, you just need to multiply those kilograms by something that’s a meter² per second², or a (meter/second)², and there’s a fundamental constant that comes with units of meters/second: the speed of light, or c.
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- After all, you can measure any velocity you want in units of meters/second, not just the speed of light. In addition, there’s nothing preventing nature from requiring a proportionality constant, a multiplicative factor like ½, ¾, 2π, etc., to make the equation true.
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- What we can do is imagine that there’s some energy inherent to a particle due to its rest mass, ie: Potential energy, and additional energy that it might have due to its motion, ie: kinetic energy.
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- We can imagine starting a particle off high up in a gravitational field, as though it started off with a large amount of gravitational potential energy, but at rest.
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- When you drop it, that potential energy converts into kinetic energy, while the rest mass energy stays the same. At the moment just prior to impact with the ground, there will be no potential energy left: just kinetic energy and the energy inherent to its rest mass, whatever that may be.
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- How does a particle gains energy when it falls in a gravitational field? There’s some energy inherent to the rest mass of a particle and that gravitational potential energy can be converted into kinetic energy (and vice versa)?
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- Let’s throw in one more idea: that all particles have an antiparticle counterpart, and if ever the two of them collide, they can annihilate away into pure energy.
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- E = mc² tells us the relationship between mass and energy, including how much energy you need to create particle-antiparticle pairs out of nothing, and how much energy you get out when particle-antiparticle pairs annihilate.
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- Instead of having one particle high up in a gravitational field, imagine that we have both a particle and an antiparticle up high in a gravitational field, ready to fall. Let’s set up two different scenarios for what could happen, and explore the consequences of both.
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- It is possible to create matter/antimatter pairs from pure energy, and vice versa.
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- Scenario 1: the particle and antiparticle both fall, and annihilate at the instant they would hit the ground. This is the same situation we just thought about, except doubled. Both the particle and antiparticle start with some amount of rest-mass energy.
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- We don’t need to know the amount, simply that’s whatever that amount is, it’s equal for the particle and the antiparticle, since all particles have identical masses to their antiparticle counterparts.
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- Now, they both fall, converting their gravitational potential energy into kinetic energy, which is in addition to their rest-mass energy. Just as was the case before, the instant before they hit the ground, all of their energy is in just two forms: their rest-mass energy and their kinetic energy.
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- Only, this time, just at the moment of impact, they annihilate, transforming into two photons whose combined energy must equal whatever that rest-mass energy plus that kinetic energy was for both the particle and antiparticle.
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- For a photon, however, which has no mass, the energy is simply given by its momentum, p, multiplied by the speed of light: E = pc. Whatever the energy of both particles was before they hit the ground, the energy of those photons must equal that same total value.
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- How do photons gain energy when they fall in a gravitational field?
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- Scenario 2: the particle and antiparticle both annihilate into pure energy, and then fall the rest of the way down to the ground as photons, with zero rest mass. Now, let’s imagine an almost identical scenario.
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- We start with the same particle and antiparticle, high up in a gravitational field. Only, this time, when we “release” them and allow them to fall, they annihilate into photons immediately: the entirety of their rest-mass energy gets turned into the energy of those photons.
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- The total energy of those photons, where each one has an energy of E = pc, must equal the combined rest-mass energy of the particle and antiparticle in question.
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- We measure photons energies when they reach the ground. By the conservation of energy, they must have a total energy that equals the energy of the photons from the previous scenario.
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- This proves that photons must gain energy as they fall in a gravitational field, leading to what we know as a gravitational blueshift, but it also leads to something spectacular: the notion that E = mc² is what a particle’s (or antiparticle’s) rest mass has to be.
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- The physics of gravitational redshift/blueshift is a core feature of General Relativity.
When a quantum of radiation leaves a gravitational field, its frequency must be redshifted
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- There’s only one definition of energy we can use that universally applies to all particles, massive and massless, alike, that enables scenario #1 and scenario #2 to give us identical answers: E = √(m²c4 + p²c²).
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------------------------ Think about what happens here under a variety of conditions:
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- If you are a massive particle at rest, with no momentum, your energy is just √(m²c4), which becomes E = mc².
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- If you’re a massless particle, you must be in motion, and your rest mass is zero, so your energy is just √(p²c²), or E = pc.
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- If you’re a massive particle and you’re moving slow compared to the speed of light, then you can approximate your momentum by p = mv, and so your energy becomes: E = √(m²c4 + m²v²c²).
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- You can rewrite this as E = mc² · √(1 + v²/c²), so long as v is small compared to the speed of light.
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- Perform what’s known, mathematically, as a “Taylor series expansion“, where the second term in parentheses is small compared to the “1” that makes up the first term.
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- If you do, you’ll get that E = mc² · [1 + ½(v²/c²) + ...], where if you multiply through for the first two terms, you get E = mc² + ½mv²: the rest mass plus the non-relativistic formula for kinetic energy.
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- Another way to derive that E = mc² for a massive particle at rest, it would be to consider a photon, which always carries energy and momentum, traveling in a stationary box with a mirror on the end that it’s traveling towards.
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- When the photon strikes the mirror, it temporarily gets absorbed, and the box with the absorbed photon has to gain a little bit of energy and start moving in the direction that the photon was moving which is the only way to conserve both energy and momentum.
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- When the photon gets re-emitted, it’s moving in the opposite direction, and so the box having lost a little mass from re-emitting that photon has to move forward a little more quickly in order to conserve energy and momentum.
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- By considering these three steps the total energy and the total momentum must be equivalent. If you solve those equations, there’s only one definition of rest-mass energy that works out: E = mc².
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- In our Universe, energy is conserved, momentum is conserved, and General Relativity is our theory of gravitation.
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- This all started with the whole universe packed together in an infinitely small point, then it exploded, and the entire mass that made up the universe was sent out into space.
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- An astrophysicist would tell you that everything about that statement is wrong. That's not at all how we should think about the Big Bang.
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- What does "Big Bang" really mean? The Big Bang theory is that about 14 billion years ago the universe was in a state that was much warmer and much denser, and that it expanded.
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- Since then space has continued to expand and has become colder. When the universe was about 10^-32 seconds old. That's 0.0000000000000000000000000000000001 seconds.
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- This was not an explosion. Here is how the idea came about. In the early 1920s, mathematician Alexander Friedmann discovered that Einstein's general theory of relativity provides for an expanding universe.
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- The Belgian priest Georges Lemaître came to the same conclusion. Shortly afterwards, Edwin Hubble showed that galaxies are actually moving apart.
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- The galaxies are moving away from us. The light from them is “red-shifted“, meaning the waves have become longer and shifted towards the red end of the light spectrum. Not only that, galaxies are disappearing from us faster and faster.
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- Someday, almost all the galaxies we can currently observe in telescopes will be out of view. Eventually the stars will go out and observers will look out into an eternally dark and lonely sky. Fortunately, that's an extremely long way off.
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- We can also play the story the opposite way. The galaxies are moving apart and they have been closer before. If you take the entire observable universe and rewind all the way back, everything fit into a very, very small area.
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- Then we come to the point in time of the Big Bang. What happened? It's easy to think that the Big Bang was an explosion, in which substances were thrown out, like pieces of wood flying off after a hand grenade goes off.
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- But the Big Bang, it's not the substance that travels out. The universe itself expands, space itself expands.
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- An explosion where the mass explodes in all directions is not an accurate picture of the Big Bang.
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- The second myth is that the universe is expanding into something. It is not the galaxies that are moving apart, but space in between them that's expanding.
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- A few galaxies are blue-shifting, meaning they're moving towards us. This applies to some nearby galaxies. But over large distances, this effect is eclipsed by Hubble-Lemaître's law, which states how fast galaxies are moving away in proportion to distance. In fact, the distance increases faster than light between points that are extremely far apart.
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- The universe doesn't expand into anything. The universe has an edge.
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- The observable universe is a bubble surrounding us that is 93 billion light-years in diameter. The more distant something is that we look at, the farther back in time we're seeing. We can't observe or measure anything farther away than the distance light has managed to travel towards us since the Big Bang.
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- Since the universe has been expanding, the observable universe is counter intuitively larger than 14 billion light-years. Scientists calculate that the universe outside our bubble is much, much larger than that, perhaps infinite in size.
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- The universe can be "flat," it appears not to be curved. That would mean that two light rays would remain parallel and never meet. If you tried to travel to the end of the universe, you would never reach it. The universe goes on infinitely.
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- If the universe has “positive curvature“, it could in theory be finite. But then it would be like a kind of strange sphere. If you traveled to the "end" you would end up in the same place you started, no matter which direction you took. It's a bit like being able to travel around the world and ending up back where you started.
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- In either case, the universe can expand without having to expand into anything. An infinite universe that's getting bigger is still infinite. A "spherical universe" has no edge.
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- The third myth is that the Big Bang had a center. If we imagine the Big Bang as an explosion, it's easy to think that it exploded outwards, from a center. That's how explosions work.
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- But that wasn't the case with the Big Bang. Almost all galaxies are moving away from us, in all directions. It seems like the Earth was the center of the beginning of the universe. But it wasn't.
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- All other observers would see the same thing from their home galaxy. The universe is expanding everywhere at the same time. The Big Bang didn't happen in any particular place. It happened everywhere.
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- The forth myth is that the whole universe was gathered in a tiny little point. It's true that our entire observable universe was gathered incredibly tightly together in very little space at the beginning of the Big Bang.
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- But how can the universe be infinite, and at the same time have been so small? You might read that the universe was smaller than an atom at first and then the size of a football. But that analogy insinuates that space had boundaries in the beginning, and an edge.
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- There's nothing that says that the universe wasn't already infinite at the Big Bang. It was just smaller in the sense that what was then a meter, has now expanded into enormous distances of many billions of light years.
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- When you talk about how big the universe was at certain times, it refers to our observable universe. The whole observable universe comes from a tiny little area that you can call a point. But the point next to it has also expanded, and the next point as well. It's just that it's so far away from us that we can't observe it.
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- The fifth myth is that the universe was infinitely small, hot and dense. The universe began as a “singularity“. It was infinitely small am infinitely hot.
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- Singularities are an expression for mathematics that breaks down and can't be described with ordinary physics. The universe today is a little bigger than it was yesterday. And it's even a little bigger still than it was a million years ago. The Big Bang theory involves extrapolating this back in time. Then you need a theory for that: and that's the general theory of relativity.
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- If you run the general relativity theory all the way back you reach a point of infinitely high density and heat, where the size is zero. That's pure mathematical extrapolation beyond what the theory actually allows. You then come to a point where the energy density and temperatures are so high that we no longer have physical theories to describe them.
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- In order to describe such an extreme condition you need a theory that combines gravity and quantum theory. No one has been able to formulate it yet. The expectation is precisely that a “quantum gravity theory” wouldn't lead to the conclusion that everything goes back to one point.
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- So what happened at this time, the earliest point in the history of the universe, is still hidden from us. That is a review I will never get to write.
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- 3050 -
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- 3047 - EINSTEIN - are his theories real life? One of the most mysterious components in the entire Universe is “dark energy“, which wasn’t supposed to exist. We had assumed that the Universe was a balancing act, with the expansion of the Universe and the gravitational effects of everything within it fighting against one another.
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- 2687 - EINSTEIN - his cosmological constant? - In 1917 Albert Einstein completed his equations for General Relativity and a new theory for gravity. He recognized that a problem with his equations required the contraction or the expansion of the Universe but not the static condition that everyone thought existed at the time.
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- 2596 - EINSTEIN - has theories of the Universe? The basic principles of general relativity can be stated quite simply: The presence of matter distorts the fabric of space and time, and objects travel on the shortest path in that distorted space-time universe. Getting to this conclusion is another story.
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- 2556 - Einstein’s legacy 100 years later.
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- 2483 - Testing Einstein’ theories.
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- 2284 - Einstein’s theory of gravity.
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- 2234 - What did Einstein say about the Universe?
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- 2216 - Einstein’s math and the theory of he gravity lens.
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- 1582 - Using the Pythagorean theory to derive the theory of relativity.
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- 1142 - Can Einstein’s equations pass the tests? His equations are alone in unifying space, time, mass, energy, motion and light.
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- 929 - Einstein’s Legacy. If you can link the equations of General Relativity and Quantum Mechanics it would be a supertechifragilisticexpialedocious breakthrough in physics.
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- 395 - deriving E=mc^2 using a teeter totter
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- 409 - Einstein is right again. Measurements with the Gravity Probe satellite.
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- March 28, 2021 - EINSTEIN - Why does energy equal mass? 3111
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