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👉 This channel was created in collaboration with https://www.youtube.com/@presura
0:00 - Introduction to Energy and Einstein's Formula
0:41 - Origins of Relativity in Physics
1:23 - Four-Vectors and Mass Variation
3:42 - Relativistic Mass and Verification
6:38 - Energy Equivalence and Inertia
7:51 - Electromagnetic Field and Mass
12:35 - Experimental Proof and Applications
14:50 - Closing Remarks
👉 This channel was created in collaboration with https://www.youtube.com/@presura
0:00 - Introduction to Energy and Einstein's Formula
0:41 - Origins of Relativity in Physics
1:23 - Four-Vectors and Mass Variation
3:42 - Relativistic Mass and Verification
6:38 - Energy Equivalence and Inertia
7:51 - Electromagnetic Field and Mass
12:35 - Experimental Proof and Applications
14:50 - Closing Remarks
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TechTranscript
00:00This book contains an amount of energy equivalent to tens of millions of billions of joules,
00:06enough to power the world economy for almost a quarter of an hour.
00:10If I sold this energy, I would make almost a billion euros. I would be rich.
00:14How do I know how much energy there is?
00:16That's what Einstein's famous equation tells us, E equals MC squared.
00:21Let's talk about what this famous formula means, how it was verified, and what its applications are.
00:27Fasten your seatbelts because you're going to see some formulas, Maestro Banda.
00:41Many discoveries in physics were made in an armchair.
00:44It's fascinating how by thinking and thinking deeply, physicists reached some conclusions that were later just validated by experiment.
00:53One example is mass changing with speed.
00:55The fact that the inertial mass of a body increases with its speed.
01:00In fact, in another video, I used this dependence to show why bodies cannot accelerate to the speed of light.
01:06We saw that as the speed of a body approaches the speed of light, its mass tends to infinity,
01:12making it impossible to accelerate the body beyond the speed of light.
01:15But where does this dependence come from?
01:18Here, we can intuit it from his formulas for the structure of spacetime.
01:23Without going into details which can be found in the book Physica Povestita, I want to present to you the
01:28basic idea.
01:29Here is how a vector is written in spacetime.
01:32You can see here the four coordinates.
01:34c, t, t is time, c is the speed of light, and the positions x, y, and z.
01:39It is a four-dimensional vector, basically a generalization of the three-dimensional position vector.
01:45The variation of a position vector over time defines a velocity vector.
01:50Einstein then generalized the usual three-dimensional velocity to a four-dimensional velocity vector in spacetime,
01:57for a clock moving along a world line.
01:59Here is the result.
02:01You see that the position vector, now four-dimensional, is differentiated not with respect to time t,
02:06but with respect to the proper time of the clock, tau, which is related to time t by the gamma
02:11factor given by time dilation.
02:13Why?
02:13Because the proper time tau is physical, it is what you see on the screen, the elapsed time,
02:18and that's why it doesn't depend on the inertial reference frame.
02:22We say that it is relativistically invariant, the four-dimensional quantity that is obtained
02:27and that we see here will then retain the characteristics of a four-dimensional vector.
02:33In the formula here, I multiplied both the numerator and the denominator by dt,
02:37so that on the right side, we are left with the velocities vx, vi, and v's,
02:42as we usually know them from three-dimensional space.
02:46We see that the gamma factor is in front.
02:48In the end here, we also multiplied the numerator by the rest mass m0 of the clock,
02:53which is also a relativistic invariant quantity, meaning it is the same in any inertial reference frame.
03:00Let's examine the result closely.
03:02We see here that m appears instead of m0, m being m0 multiplied by gamma.
03:08The quantity m is called the relativistic mass of the body because it depends on speed through gamma.
03:14So, what does this relationship tell us?
03:16First of all, the momentum of the particle has become m times v and not m0 times v.
03:21In other words, m justifies its name as relativistic mass.
03:25The mass m of a body, that is the relativistic mass, will then vary with speed according to this relation.
03:31Here is Einstein's first prediction.
03:33Looking again at the result, in the first term, we also see a quantity m times c squared,
03:38which has the dimension of energy.
03:40This is the second prediction.
03:42What is this term m times c squared, which has the unit of energy?
03:46This is what Einstein asked himself.
03:48To answer, let's go back to relativistic mass.
03:51Does the mass m of a body really vary with its speed according to this formula?
03:56Especially since, as you can see from the formula, the mass of the electron should tend to infinity as its
04:01speed approaches the speed of light.
04:03The answer is yes.
04:05And here is the result of a measurement made by the physicist Alfred Bucherer in 1908 for the motion of
04:11an electron.
04:12On the horizontal axis is the speed, on the vertical axis the mass.
04:16We can see how, indeed, the mass of the electron increases with speed as its speed approaches the speed of
04:21light.
04:25Now we can actually see why an electron, or any other particle with non-zero rest mass, cannot reach the
04:31speed of light.
04:32Its mass becomes infinite.
04:34That's true, no matter how hard we push the electron.
04:37In fact, the more energy we pump in to increase the electron's speed, the more its mass will increase, and
04:43it will then be harder to accelerate.
04:46That's because the acceleration of a body, as we remember from Newton, is the ratio between force and mass.
04:51If the mass approaches infinity, the ratio becomes zero.
04:54So, the acceleration will remain almost zero, no matter how great the force used is.
05:00That means it won't accelerate anymore.
05:02The question then arises, if the body hardly accelerates anymore, where does all that energy used to push it go?
05:09In the theory of relativity, you can calculate the energy delta E pumped into a body to bring it from
05:16rest to a certain speed.
05:18For this, first dm is calculated, meaning by how much the relativistic mass m increases when the speed increases by
05:26dv.
05:27This is the result.
05:29Then, the previous relation is used to calculate the mechanical work L done by the force over a distance delta
05:35x, meaning the energy consumed.
05:38As you can see, a remarkable result is obtained.
05:41The change in the energy pumped in, delta E, is proportional to the variation in mass delta m.
05:47The proportionality constant is the speed of light c squared.
05:52Looking at this form among other equations, Einstein had a brilliant suggestion.
05:57This is what he said.
05:58What if we assume that all the energy that was ever pumped into the body is found in its relativistic
06:03mass according to the relationship here?
06:06Basically, the delta that appeared in the previous equation was dropped.
06:10This relationship between a body's energy and its relativistic mass makes sense.
06:15We saw that it was also present in the four-dimensional four-vector of velocity in spacetime.
06:21Here's the interpretation of Einstein's famous equation, E equals mc squared.
06:25The inertial mass m of a body tells us, according to Einstein's assumption, how much energy has ever been pumped
06:32into that body.
06:33It is also all the energy we can extract from that body, as we will see in the next episode.
06:38It's interesting how, at low speeds, the energy mc squared actually turns out to be the sum of the rest
06:44energy m0c squared and the kinetic energy m0v squared divided by 2.
06:50In other words, kinetic energy is where that energy ended up when the body was accelerated.
06:57Einstein's relation is called the principle of equivalence between mass and energy.
07:02The word equivalent is very dangerous in physics.
07:05When two things are mathematically equivalent, as is the case here, it means their physical interpretation is the same.
07:12Simply put, it doesn't mean that mass is the same thing as energy.
07:16It's also important to mention that we're talking about inertial mass, not gravitational mass, meaning we're referring to mass in
07:23kilograms measured as resistance to attempts to change the object's velocity.
07:28This is the definition of inertia.
07:30When an object has high inertia, it's hard to stop, meaning it's hard to change its velocity.
07:35We're not talking here about gravitational mass, meaning the ability of objects to attract each other gravitationally.
07:46Interestingly, an approximation of Einstein's equation can be found in the classical case as well.
07:51Let me show you a calculation that I also reproduced in Physica Povestita, and that fascinates me every time.
07:57Look, let's take an electron. It has an electric charge, so there's an electric field around it.
08:03But we know that any electromagnetic field stores energy in it, which is distributed in space depending on the values
08:09of the electric and magnetic fields.
08:10The higher the values, the more energy is present there.
08:14Since the electric field is strong near the charge, it's no wonder that a lot of electromagnetic energy will be
08:20stored there.
08:21That's shown on the left of this figure.
08:24Here, at the top, is also the formula for the energy density of the electromagnetic field, which says the same
08:30thing.
08:31The first term is the electric energy, and the second one is the magnetic energy.
08:35The formula below, the pointing vector, is even more beautiful.
08:39It tells us that the energy of the electromagnetic field doesn't stay in one place.
08:44It flows through space in the direction given by the cross product of the electric and magnetic fields.
08:49This is extremely useful when the electron is moving, as you can see in this figure.
08:54Here you see a positive electric charge moving from left to right.
08:58I've also shown the electric and magnetic fields in the figure.
09:01The pointing vector, S, which tells us how the energy of the electromagnetic field flows, is oriented to the right,
09:08just like the electric charge is moving.
09:12Logically, as it moves, the energy of the electromagnetic field follows the motion of the electric charge, so that the
09:18energy of the electromagnetic field is always high near the electron.
09:22This is what the situation looks like for a moving electron.
09:25On the left, the electron is stationary. The electromagnetic field has energy around it.
09:31On the right, the electron is moving. The electromagnetic field redistributes its energy in such a way as to follow
09:37the motion of the electron.
09:39Moreover, the electromagnetic field changes. For example, an additional magnetic field also appears.
09:45As the figure suggests, and as the calculations show, when the electron is in motion, the electromagnetic field stores more
09:52energy.
09:53That's why, when we set the electron in motion, we have to pump additional energy into the electromagnetic field.
10:00By pushing the electron, we will feel that it is heavier, as if the electron has a greater inertial mass,
10:07with a component called electromagnetic mass, which is due exclusively to the electromagnetic field.
10:12As I mentioned, this calculation is in my book, Physica Povestita.
10:16Here is the figure I used for the calculation of the electromagnetic field energy of a moving electron.
10:23I used a model in which the electric charge of the electron is distributed uniformly over the surface of a
10:29sphere.
10:29The calculation, which is not difficult, it's high school level,
10:33integrates the energy stored around a moving electric charge at non-relativistic speeds, the upper part,
10:39and the energy stored in the magnetic field, the lower part.
10:42The result of the calculation is here.
10:44On the left is all the energy stored in the electromagnetic field of the moving charge.
10:49On the right is the calculated value for what is called the mass of the electromagnetic field.
10:55What do we see in the formula?
10:57On the left side, we see how the electromagnetic field has an energy that does not depend on the speed
11:02of the electron,
11:03in a formula very similar to Einstein's.
11:06The only difference is the factor of 3 over 4.
11:10On the right side, there is a term that is similar to kinetic energy,
11:13because it depends on the square of the electron's velocity.
11:17The fact that a simple classical calculation comes so close to the relativistic one shouldn't surprise us.
11:24Relativity is in everything, and the consistency of mathematics assures us that we will find hints of the theory of
11:30relativity
11:30in the most unusual places, such as the kinetic energy of the electron, or its electromagnetic field energy.
11:38Moreover, we don't expect to get an exact result.
11:40The model doesn't take into account the energies that arise between the charges distributed on the surface of the sphere.
11:46It's just a model, but the result tells us that we can associate a mass, a rest energy, and even
11:51a kinetic energy with an electromagnetic field.
11:55The observation is crucial, because when we push an electron and measure its mass,
12:00it actually means we're also measuring the mass of its electromagnetic field.
12:04Basically, we can never say what the mass of the electron alone is, and what the mass of the electromagnetic
12:09field is.
12:10We always measure both of them together, because we can't separate them, like we did for example with a snail
12:16and its shell.
12:17This remains a fundamental problem, even in quantum field theories,
12:21where the contribution of the electromagnetic field is taken into account from the very beginning in the mass of the
12:26electron.
12:26It's just that an accurate calculation always depends on the model used, like in our case.
12:35How can we understand Einstein's equation?
12:37Here's an example. In the process of fusion, when two components bind to each other, energy is released.
12:44We can visualize this by looking at two magnets attracting each other.
12:47When they get closer, their kinetic energy increases, and when they collide,
12:52the kinetic energy is transformed into thermal energy, which then dissipates.
12:57In other words, in the end, the bound components will have less energy, because they have lost some of it.
13:02However, energy is proportional to mass, as Einstein tells us.
13:07It follows from this that when several components come together, they will not only have less energy,
13:12but also a total mass that is smaller than the sum of the components.
13:16Let's take the case of a nucleus made up of a proton and a neutron that have fused together in
13:21that nucleus.
13:22The previous relationship then tells us that the mass of the nucleus must be less than the sum of the
13:28masses of the protons and neutrons that make it up,
13:31if they were weighed separately, because the protons and neutrons lost energy when they fused into the nucleus.
13:37A first direct experimental verification of this phenomenon came in 1932,
13:43when physicists John Cockcroft and Amst Walton, both Nobel Prize laureates,
13:47in 1951, they managed to show that the mass of the lithium nucleus is indeed smaller than the sum of
13:55the protons and neutrons' masses.
13:57However, they couldn't measure the energy lost when the nucleus formed, which they should have compared to the mass difference.
14:04Only in the last few decades have such experiments been possible,
14:08where the lost energy was measured from the radiation emitted during the formation process
14:12and was then compared with the mass difference using Einstein's equation.
14:17The best experiments suggest an agreement with Einstein's formula of up to 99.9999%.
14:24Meanwhile, the equivalence between energy and mass has found direct applications.
14:29Some are useful. Nuclear reactors. Others are frightening. The atomic bomb.
14:35It's no wonder if we think that in a single kilogram there is enough energy to supply humanity for more
14:41than an hour.
14:42About how much energy is actually in matter and especially how it can be extracted,
14:47without resorting to antimatter, we will talk next time.
14:50Until then, please subscribe if you haven't already by clicking this subscribe button right now.
14:56I thank those who support us on Patreon and here by becoming channel members.
15:00As I mentioned, channel members can watch videos without ads.
15:03I am Christian Pressura, you are cool, and I wish you all the best. Goodbye.
15:07Bye.
15:08Bye.
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