The Derivation of E=mc²

Perhaps the most famous equation of all time is E = mc². The equation is a direct result of the theory of special relativity, but what does it mean and how did Einstein find it? In short, the equation describes how energy and mass are related. Einstein used a brilliant thought experiment to arrive at this equation, which we will briefly review here.

First of all, let us consider a particle of light, also known as a photon. One of the interesting properties of photons is that they have momentum and yet have no mass. This was established in the 1850s by James Clerk Maxwell. However, if we recall our basic physics, we know that momentum is made up of two components: mass and velocity. How can a photon have momentum and yet not have a mass? Einstein’s great insight was that the energy of a photon must be equivalent to a quantity of mass  and hence could be related to the momentum.

Einstein’s thought experiment runs as follows. First, imagine a stationary box floating in deep space. Inside the box, a photon is emitted and travels from the left towards the right. Since the momentum of the system must be conserved, the box must recoils to the left as the photon is emitted. At some later time, the photon collides with the other side of the box, transferring all of its momentum to the box. The total momentum of the system is conserved, so the impact causes the box to stop moving.

Unfortunately, there is a problem. Since no external forces are acting on this system, the centre of mass must stay in the same location. However, the box has moved. How can the movement of the box be reconciled with the centre of mass of the system remaining fixed?

Einstein resolved this apparent contradiction by proposing that there must be a ‘mass equivalent’ to the energy of the photon. In other words, the energy of the photon must be equivalent to a mass moving from left to right in the box. Furthermore, the mass must be large enough so that the system centre of mass remains stationary.

Let us try and think about this experiment mathematically. For the momentum of our photon, we will use Maxwell’s expression for the momentum of an electromagnetic wave having a given energy. If the energy of the photon is E and the speed of light is c, then the momentum of the photon is given by:

                              

 (1.1)

The box, of mass M, will recoil slowly in the opposite direction to the photon with speed v. The momentum of the box is:

                              

 (1.2)

The photon will take a short time, Δt, to reach the other side of the box. In this time, the box will have moved a small distance, Δx. The speed of the box is therefore given by

                              

 (1.3)

By the conservation of momentum, we have

                              

 (1.4)

If the box is of length L, then the time it takes for the photon to reach the other side of the box is given by:

                              

 (1.5)

Substituting into the conservation of momentum equation (1.4) and rearranging:

                              

 (1.6)

Now suppose for the time being that the photon has some mass, which we denote by m. In this case the centre of mass of the whole system can be calculated. If the box has position x₁ and the photon has position x₂, then the centre of mass for the whole system is:

                              

 (1.7)

We require that the centre of mass of the whole system does not change. Therefore, the centre of mass at the start of the experiment must be the same as the end of the experiment. Mathematically:

                              

 (1.8)

The photon starts at the left of the box, i.e. x₂ = 0. So, by rearranging and simplifying the above equation, we get:

                              

 (1.9)

Substituting (1.4) into (1.9) gives:

                              

 (1.10)

Rearranging gives the final equation:

                              

So, let’s think about what this equation means. The equation suggests that a given mass can be converted into energy. But how much energy? Well, suppose we have a kilo of mass. Conversion of this mass into pure energy would result in (1kg * c²) joules of energy. Now note that c² = 8.99 * 10¹⁶ m²s^-2  so that's a WHOLE lot of energy - equivalent to 21.48 megatons of TNT!

In practice, it is not possible to convert all of the mass into energy. However, this equation led directly to the development of nuclear energy and the nuclear bomb - probably the most tangible results of special relativity.

Halley's Comet: Facts About the Most Famous Comet

By Pratyashu Shekhar

An image of Halley's Comet taken in 1986.

Credit: NASA

Halley's Comet is arguably the most famous comet. It is a "periodic" comet and returns to Earth's vicinity about every 75 years, making it possible for a human to see it twice in his or her lifetime. The last time it was here was in 1986, and it is projected to return in 2061.

The comet is named after English astronomer Edmond Halley, who examined reports of a comet approaching Earth in 1531, 1607 and 1682. He concluded that these three comets were actually the same comet returning over and over again, and predicted the comet would come again in 1758.

Halley didn't live to see the comet's return, but his discovery led to the comet being named after him. (The traditional pronunciation of the name usually rhymes with valley.) Halley's calculations showed that at least some comets orbit the sun.

Further, the first Halley's Comet of the space age – in 1986 – saw several spacecraft approach its vicinity to sample its composition. High-powered telescopes also observed the telescope as it swung by Earth.

Halley's in history

The first known observation of Halley's took place in 239 B.C., according to the European Space Agency. Chinese astronomers recorded its passage in the Shih Chi and Wen Hsien Thung Khao chronicles.

When Halley's returned in 164 B.C. and 87 B.C., it probably was noted in Babylonian records now housed at the British Museum in London. "These texts have important bearing on the orbital motion of the comet in the ancient past," noted a Nature research paper about the tablets.

This portion of the Bayeux Tapestry shows Halley's Comet during its appearance in 1066. Credit: Public domain

Halley's most famous appearance occurred shortly before the 1066 invasion of England by William the Conquerer. It is said that William felt the comet heralded his success. In any case, the comet was put on the Bayeux Tapestry – which chronicles the invasion – in William's honor.

Another appearance of the comet in 1301 possibly inspired Italian painter Giotto's rendering of the Star of Bethlehem in "The Adoration of the Magi," according to the Britannica encyclopedia.

Astronomers in these times, however, saw each appearance of Halley's Comet as an isolated event. Comets were often foreseen as a sign of great disaster or change.

Even when Shakespeare wrote his play "Julius Caesar" around 1600, just 105 years before Edmond Halley calculated that the comet returns over and over again, one famous phrase spoke of comets as heralds: "When beggars die there are no comets seen; The heavens themselves blaze forth the death of princes."

Discovery of Halley's recurrence

Astronomy began changing swiftly around the time of Shakespeare, however. Many astronomers of his time held that Earth was the center of the solar system, but Nicolaus Copernicus – who died about 20 years before Shakespeare's birth – published findings showing that the center was actually the sun.

It took several generations for Copernicus' calculations to take hold in the astronomy community, but when they did, they provided a powerful model for how objects move around the solar system and the universe.

Edmond Halley

Edmond Halley published "A Synopsis of the Astronomy of Comets" in 1705, cataloguing what he had found from searching historical records of 24 comets appearing near Earth between 1337 to 1698. Three of those observations appeared to be very similar in terms of orbit and other parameters, leading Halley to propose that one comet might be visiting Earth again and again.

The comet appeared in 1531, 1607 and 1682. Halley suggested the same comet could return to Earth in 1758. Halley did not live long enough to see its return – he died in 1742 – but his discovery inspired others to name the comet after him.

On each successive journey to the inner solar system, astronomers on Earth turned their telescopes skyward to watch Halley's approach.

The comet's pass in 1910 was particularly spectacular, as the comet flew by about 13.9 million miles (22.4 million kilometers) from Earth, which is about 1/15 the distance between Earth and the Sun. On that occasion, Halley's was captured on camera for the first time.

According to biographer Albert Bigelow Paine, the writer Mark Twain said in 1909, "I came in with Halley's Comet in 1835. It is coming again next year, and I expect to go out with it." Twain died on April 21, 1910, one day after perihelion, when the comet emerged from the far side of the sun.

This photo of Halley's comet was taken by the Russian Vega 2 spacecraft, one of two Soviet probes (Vega 1 was the other) to rendezvous with the comet during its 1986 trip through the solar system in March 1986. The closest approach of Vega 1 to Halley was 8890 km while Vega 2 had a close encounter at 8030 km. Credit: ESA

Halley's in the Space Age

When Halley's Comet came by Earth in 1986, it was the first time we could send spacecraft up to look at it.

That was a fortunate occurrence, as the comet ended up being underwhelming in observations from Earth. When the comet made its closest approach to the sun, it was on the opposite side of that star from the Earth – making it a faint and distant object, some 39 million miles away from Earth.

Several spacecraft successfully made the journey to the comet. This fleet of spaceships is sometimes dubbed the "Halley Armada." Two joint Soviet/French probes (Vega 1 and 2) flew nearby, with one of them capturing pictures of the heart or nucleus of the comet for the first time.

The European Space Agency's Giotto got even closer to the nucleus, beaming back spectacular images to Earth. Japan sent two probes of its own (Sakigake and Suisei) that also obtained information on Halley.

Additionally, NASA's International Cometary Explorer (already in orbit since 1978) captured pictures of Halley from 17.3 million miles (28 million kilometers away.)

"It was inevitable that this most famous of all comets would receive unprecedented attention, but the actual magnitude of the effort has surprised even most of those involved in it," NASA noted in an account of the event.

Sadly, the astronauts aboard Challenger's STS-51L mission were also scheduled to look at the telescope when they arrived in the orbit, but they never got the chance. The shuttle exploded about two minutes after launch on Jan. 28, 1986, due to a rocket malfunction.

It will be many decades until Halley's gets close to Earth again, but in the meantime you can see its remnants every year. The Orionid meteor shower, which is spawned by Halley's fragments, occurs annually in October. Halley's also produced a shower in May, called the Eta Aquarids.

When Halley's sweeps by Earth in 2061, the comet will be on the same side of the sun as Earth and will be much brighter than in 1986.

One astronomer predicted it could be as bright as apparent magnitude -0.3. This is relatively bright, but well below that of the brightest star in Earth's sky: Sirius, at magnitude -1.4 as seen from Earth.

 Minkowski diagram

The Minkowski diagram, also known as a spacetime diagram, was developed in 1908 by Hermann Minkowski and provides an illustration of the properties of space and time in the special theory of relativity. It allows a qualitative understanding of the corresponding phenomena like time dilation and length contraction without mathematical equations.

The term Minkowski diagram is used in both a generic and particular sense. In general, a Minkowski diagram is a graphic depiction of a portion of Minkowski space, often where space has been curtailed to a single dimension. These two-dimensional diagrams portray worldlines as curves in a plane that correspond to motion along the spatial axis. The vertical axis is usually temporal, and the units of measurement are taken such that the light cone at an event consists of the lines of slope plus or minus one through that event.

A particular Minkowski diagram illustrates the result of a Lorentz transformation. The horizontal corresponds to the usual notion of simultaneous events, for a stationary observer at the origin. The Lorentz transformation relates two inertial frames of reference, where an observer makes a change of velocity at the event (0, 0). The new time axis of the observer forms an angle α with the previous time axis, with α < π/4. After the Lorentz transformation the new simultaneous events lie on a line inclined by α to the previous line of simultaneity. Whatever the magnitude of α, the line t = x forms the universal bisector.

How Does The Sun Produce Energy?

There is a reason life that Earth is the only place in the Solar System where life is known to be able to live and thrive. Granted, scientists believe that there may be microbial or even aquatic life forms living beneath the icy surfaces of Europa and Enceladus, or in the methane lakes on Titan. But for the time being, Earth remains the only place that we know of that has all the right conditions for life to exist.

One of the reasons for this is because the Earth lies within our Sun’s Habitable Zone (aka. “Goldilocks Zone”). This means that it is in right spot (neither too close nor too far) to receive the Sun’s abundant energy, which includes the light and heat that is essential for chemical reactions. But how exactly does our Sun go about producing this energy? What steps are involved, and how does it get to us here on planet Earth?

The simple answer is that the Sun, like all stars, is able to create energy because it is essentially a massive fusion reaction. Scientists believe that this began when a huge cloud of gas and particles (i.e. a nebula) collapsed under the force of its own gravity – which is known as Nebula Theory

. This not only created the big ball of light at the center of our Solar System, it also triggered a process whereby hydrogen, collected in the center, began fusing to create solar energy.

Technically known as nuclear fusion, this process releases an incredible amount of energy in the form of light and heat. But getting that energy from the center of our Sun all the way out to planet Earth and beyond involves a couple of crucial steps. In the end, it all comes down to the Sun’s layers, and the role each of them plays in making sure that solar energy gets to where it can help create and sustain life.

DARKNESS

What  is  dark  matter ?

 All the ordinary matter we can find accounts for only about 4 percent of the universe. We know this by calculating how much mass would be needed to hold galaxies together and cause them to move about the way they do when they gather in large clusters. Another way to weigh the unseen matter is to look at how gravity bends the light from distant objects. Every measure tells astronomers that most of the universe is invisible.
It's tempting to say that the universe must be full of dark clouds of dust or dead stars and be done with it, but there are persuasive arguments that this is not the case. First, although there are ways to spot even the darkest forms of matter, almost every attempt to find missing clouds and stars has failed. Second, and more convincing, cosmologists can make very precise calculations of the nuclear reactions that occurred right after the Big Bang and compare the expected results with the actual composition of the universe. Those calculations show that the total amount of ordinary matter, composed of familiar protons and neutrons, is much less than the total mass of the universe. Whatever the rest is, it isn't like the stuff of which we're made.
The quest to find the missing universe is one of the key efforts that has brought cosmologists and particle physicists together. The leading dark-matter candidates are neutrinos or two other kinds of particles: neutralinos and axions, predicted by some physics theories but never detected. All three of these particles are thought to be electrically neutral, thus unable to absorb or reflect light, yet stable enough to have survived from the earliest moments after the Big Bang. 

What  is  dark  energy ?

Two recent discoveries from cosmology prove that ordinary matter and dark matter are still not enough to explain the structure of the universe. There's a third component out there, and it's not matter but some form of dark energy.

The first line of evidence for this mystery component comes from measurements of the geometry of the universe. Einstein theorized that all matter alters the shape of space and time around it. Therefore, the overall shape of the universe is governed by the total mass and energy within it. Recent studies of radiation left over from the Big Bang show that the universe has the simplest shape—it's flat. That, in turn, reveals the total mass density of the universe. But after adding up all the potential sources of dark matter and ordinary matter, astronomers still come up two-thirds short.

The second line of evidence suggests that the mystery component must be energy. Observations of distant supernovas show that the rate of expansion of the universe isn't slowing as scientists had once assumed; in fact, the pace of the expansion is increasing. This cosmic acceleration is difficult to explain unless a pervasive repulsive force constantly pushes outward on the fabric of space and time.

Why dark energy produces a repulsive force field is a bit complicated. Quantum theory says virtual particles can pop into existence for the briefest of moments before returning to nothingness. That means the vacuum of space is not a true void. Rather, space is filled with low-grade energy created when virtual particles and their antimatter partners momentarily pop into and out of existence, leaving behind a very small field called vacuum energy.

That energy should produce a kind of negative pressure, or repulsion, thereby explaining why the universe's expansion is accelerating. Consider a simple analogy: If you pull back on a sealed plunger in an empty, airtight vessel, you'll create a near vacuum. At first, the plunger will offer little resistance, but the farther you pull, the greater the vacuum and the more the plunger will pull back against you. Although vacuum energy in outer space was pumped into it by the weird rules of quantum mechanics, not by someone pulling on a plunger, this example illustrates how repulsion can be created by a negative pressure.

Scientists have long established that the speed of light just shy at 300000km/s or around 186000 miles per second. While it might seem logically that the existence of speed of light means that there must be a way to determine the speed of darkness, this isn't necessarily true.While it may be possible to determine the speed of darkness, this determination is heavily dependent on just how you define the dark.

Scientific Destruction