A Note on E = mc^2

May 30, 2022
Note that for E = mc^2 we can write for a speed = v = c: E = mvc. Now the mystery becomes why is Energy equal to the momentum of a particle (with speed c replacing v) times c. This rhythms with E^2= p^2c^2. Indeed E^2= m^2v^2c^2 follows from this. There must be an operation on the mass circle of a particle and the velocity circle of a photon.

Using E = hf we can say: c =hf/p for a spin 1 particle. Thus c = hf/2p for a spin 1/2 particle moving at the speed of light.
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Energy = mass x speed of light squared.

Energy is not equal to the momentum alone.

Energy = the mass of a particle, not its momentum.

Isaac Newton's First Law of Motion states, "A body at rest will remain at rest, and a body in motion will remain in motion unless it is acted upon by an external force." What, then, happens to a body when an external force is applied to it? That situation is described by Newton's Second Law of Motion.

According to NASA, this law states, "Force is equal to the change in momentum per change in time. For a constant mass, force equals mass times acceleration." This is written in mathematical form as F = ma

is force, m is mass and a is acceleration. The math behind this is quite simple. If you double the force, you double the acceleration, but if you double the mass, you cut the acceleration in half.

Newton's second law of motion states that F = ma, or net force is equal to mass times acceleration. A larger net force acting on an object causes a larger acceleration, and objects with larger mass require more force to accelerate. Both the net force acting on an object and the object's mass determine how the object will accelerate. Created by Sal Khan.

In physics, you perform work when you apply force to an object and move it over a distance. No work happens if the object does not move, no matter how much force you apply. When you perform work, it generates kinetic energy. The mass and velocity of an object impact how much kinetic energy it has. Equating work and kinetic energy allows you to determine velocity from force and distance. You cannot use force and distance alone, however; since kinetic energy relies on mass, you must determine the mass of the moving object as well.

Einstein’s theory of gravity, the general theory of relativity, differs in many ways from Newton’s earlier theory of gravity.

One of the most important is that Einstein’s theory incorporates the cosmic speed limit: the speed of light.

Newton had assumed that gravity is felt everywhere in the Universe instantaneously, in other words that it travels at infinite speed.

Newton would therefore have predicted that, if the Sun vanished at this moment, the Earth would notice the lack of gravitational pull immediately and fly off out of the Solar System.

Einstein recognized that since nothing, not even gravity, can travel faster than the speed of light, the Earth would not notice the absence of gravity for 8.5 minutes, the time it takes gravity to travel (at 300,000km/s or 6.7 million mph) from the Sun to the Earth.

Another difference between Newton and Einstein is that Einstein’s theory recognises that the source of gravity is not mass, as Newton believed, but energy, one form of which is mass.

This means that all forms of energy have gravity: sound energy, heat energy and so on.

Crucially, gravity itself is a form of energy, so gravity creates more gravity.

What this means is that close to the Sun where solar gravity is at its most powerful, gravity is slightly stronger than Newton predicted.

In Newtonian gravity, a planet can only follow an elliptical orbit, but Mercury’s orbit continually shifts so that it traces out a pattern like a rosette.

The prediction of this ‘precession’ of the perihelion of Mercury was one of the key triumphs of Einstein’s theory of gravity.

See: https://www.skyatnightmagazine.com/space-science/newton-einstein-gravity

See: https://www.nbcnews.com/mach/scienc...-about-gravity-now-scientists-are-ncna1038671

See: https://www.popularmechanics.com/sp...y-contradicts-newton-einstein-theory-gravity/

A new study brings them closer to the answer.

The study, published Aug. 16 in the journal Science, shows that gravity works just as Einstein predicted even at the very edge of a black hole — in this case Sagittarius A*, the supermassive black hole at the center of our Milky Way galaxy. But the study is just the opening salvo in a far-ranging effort to find the point where Einstein’s model falls apart.

"We now have the technological capacity to test gravitational theories in ways we've never been able to before,” study co-author Jessica Lu, an astrophysicist at the University of California, Berkeley, said. “Einstein's theory of gravity is definitely in our crosshairs."

That means we may be closer to the day when Einstein’s relativity is supplanted by some as-yet-undescribed new theory of gravity.

“Newton had a great time for a long time with his description [of gravity], and then at some point it was clear that that description was fraying at the edges, and then Einstein offered a more complete version,” said Andrea Ghez, an astrophysicist at UCLA and a co-leader of the new research. “And so today, we're at that point again where we understand there has to be something that is more comprehensive that allows us to describe gravity in the context of black holes.”

In Newton’s view, all objects — from his not-so-apocryphal apple to planets and stars — exert a force that attracts other objects. That universal law of gravitation worked pretty well for predicting the motion of planets as well as objects on Earth — and it's still used, for example, when making the calculations for a rocket launch.

But Newton's view of gravity didn't work for some things, like Mercury’s peculiar orbit around the sun. The orbits of planets shift over time, and Mercury’s orbit shifted faster than Newton predicted.

Einstein offered a different view of gravity, one that made sense of Mercury. Instead of exerting an attractive force, he reasoned that each object curves the fabric of space and time around them, forming a sort of well that other objects — and even beams of light — fall into. Think of the sun as a bowling ball on a mattress. It creates a depression that draws the planets close.

This new model solved the Mercury problem. It showed that the sun so curves space that it distorts the orbits of nearby bodies, including Mercury. In Einstein’s view, Mercury might look like a marble forever circling the bottom of a drain.

Einstein’s theory has been confirmed by more than a century of experiments, starting with one involving a 1919 solar eclipse in which the path of light from distant stars was shifted by the sun’s intense gravitation — by just the amount Einstein had predicted.

But Ghez and her colleagues wanted to subject Einstein to a more rigorous test. So they watched what happened when light from the star S0-2 passed Sagittarius A*, which is four million times more massive than the sun.

For the new research, Lu, Ghez and their collaborators used a trio of giant telescopes in Hawaii to watch as a bluish star named S0-2 made its closest approach to Sagittarius A* in its 16-year orbit around the black hole.

If Einstein was right, the black hole would warp space and time in a way that extended the wavelength of light from S0-2. In short, the waves would stretch out as the intense gravity from the black hole drained their energy, causing the starlight's color to shift from blue to red. If the star continued to glow blue, it would give credence to Newton's model of gravity, which doesn't account for the curvature of space and time. If it turned a different color, it would have hinted at some other model of gravity altogether.

Just as Einstein would have predicted, the star glowed red.

“You might say, ‘Who cares?’ But in fact, no one has looked there,” Ghez, from UCLA, said. “So we've been able to take a big step forward in terms of exploring a regime that's not been explored before … You know there's a cliff, and you want to get close to that cliff, but you don't know where the drop-off is.”

Scientists know that at some point in a black hole, Einstein's theory stops working. “The curvature of spacetime is so extreme that Einstein's general relativity fails," said Kip Thorne, a Nobel Prize-winning theoretical physicist at the California Institute of Technology, who wasn't involved in the new research. "We don't understand how it works when the thing you're dealing with is extreme.”

This experiment brings scientists a little closer to understanding.

"It's definitely exciting," said Zoltan Haiman, a Columbia University astrophysicist who wasn't involved in the new research. "It's pushing the envelope. This is how we get to some place where we discover [Einstein's] theory no longer works."

Haimain said he was "in awe" of the work researchers had done, likening tracking S0-2 from an observatory on Earth to studying a tree in Paris from a balcony in New York City.

"This test is just the beginning," Lu said. Researchers plan to use a new generation of high-powered instruments to conduct more tests of gravity around black holes. For example, they'll keep an eye on SO-2, to see if its orbit proceeds as Einstein would have expected, or if it takes a different path around Sagittarius A*, suggesting an alternate model of gravity.

In the next 10 years, Lu said, "we should be able to push Einstein's theory of gravity to its limits and hopefully start to see cracks."

What would that mean for science?

“It’s very hard to predict how new discoveries in fundamental physics will impact our day-to-day lives,” Lu said. “But a new theory of gravity might help us understand how our own universe was born, and how we got to where we are today 13½ billion years later.”

Space is a big place, and even Einsteins sometimes meet their limit. One of the most well-known of these limits is a black hole’s center, or singularity, where Einstein’s famous theory appears to break down completely. Now, a new study from scientists at South Korea’s Sejong University suggests that another limit to Newton and Einstein’s conception of gravity can be found in the orbital motions of long-period, widely separated, binary stars—also known simply as “wide binaries.” The results of this study were published in The Astrophysical Journal.

The main difference between Einstein and Newton gravity is that Einstein described gravity as a curvature in a 4D space-time fabric proportional to the masses of objects, whereas Newton described gravity as a force between two objects based on their masses. Gravity is a fundamental interaction (a force of attraction) that causes mutual attraction between all things with mass or energy. Gravity is a universal force, meaning it acts on all objects with mass, no matter how large or small the object is. However, compared to other fundamental forces, gravity is a relatively weak force.
May 30, 2022
Most of that is irrelevant. The mystery is: why is a particles energy equal to its momentum if going at the speed of light times lightspeed?
Energy = mass x the speed of light squared.

Energy = mass

Momentum can be defined as "mass in motion." All objects have mass; so if an object is moving, then it has momentum - it has its mass in motion. The amount of momentum that an object has is dependent upon two variables: how much stuff is moving and how fast the stuff is moving. Momentum depends upon the variables mass and velocity. In terms of an equation, the momentum of an object is equal to the mass of the object times the velocity of the object.

Momentum = mass • velocity
In physics, the symbol for the quantity momentum is the lower case p. Thus, the above equation can be rewritten as

p = m • v

where m is the mass and v is the velocity. The equation illustrates that momentum is directly proportional to an object's mass and directly proportional to the object's velocity.

The units for momentum would be mass units times velocity units. The standard metric unit of momentum is the kg•m/s. While the kg•m/s is the standard metric unit of momentum, there are a variety of other units that are acceptable (though not conventional) units of momentum. Examples include kg•mi/hr, kg•km/hr, and g•cm/s. In each of these examples, a mass unit is multiplied by a velocity unit to provide a momentum unit. This is consistent with the equation for momentum.

Mass, in physics, is defined as the measure of the amount of matter present in an object or a system. Mass is an intrinsic characteristic of an object and is independent of its location or the forces acting upon it. The standard unit of mass in the International System of Units (SI) is the kilogram (kg). The greater the mass of an object, the more matter it possesses. To illustrate the concept, let’s consider a bowling ball and a football. The bowling ball has more mass because it contains more matter.

It is important to note that mass is distinct from weight. Weight is the force experienced by an object due to the gravitational pull of the Earth. Mass remains constant regardless of the gravitational field, whereas weight varies depending on the strength of gravity. Mass is a scalar quantity, meaning it only has magnitude and no direction.

Mass plays a fundamental role in various areas of physics as it is a key factor in understanding concepts like energy, inertia, and conservation laws in physics.

Mass is the measure of the inertia of a body. Inertia is the object’s resistance to changes in its state of motion and is a fundamental property of matter. The greater the mass of an object, the greater its inertia.

In simpler terms, mass determines how difficult it is to change the motion of an object. An object with a higher mass requires more force to accelerate or decelerate than an object with a lower mass. This is because objects with more mass have more matter within them, resulting in stronger intermolecular forces and higher resistance to changes in motion.

For example, consider a heavy stone and a lightweight feather. When you try to push or stop the stone, it offers significant resistance due to its higher mass. On the other hand, the feather can be easily moved or stopped as it has a much lower mass. This difference in inertia is directly related to the mass of the objects.

Therefore, mass is crucial in determining how objects respond to external forces. It influences the acceleration, deceleration, and overall motion of an object.

E = mc2 as part of a wider relationship between energy, mass, and momentum. We start by defning energy and momentum in the everyday sense. We then build on the stretching‐triangle picture of spacetime vectors to see how energy, mass, and momentum have a deep relationship that is not obvious at everyday low speeds. When momentum is zero (a mass is at rest) this energy‐momentum relation simplifes to E = mc2, which implies that mass at rest quietly stores tremendous amounts of energy. The energy momentum relation also implies that traveling near the speed of light (e.g., to take advantage of time dilation for interstellar journeys) will require tremendous amounts of energy. We look at the simplifed form of the energy‐momentum relation when the mass is zero. Tis gives us insight into the behavior and energy of massless particles such as the photon.