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Throughout the years 1907 and 1915, Albert Einstein is going to be working to publish his general theory of relativity, a geometric theory of gravitation, space and time, that will serve as a pillar of modern physics. Einstein wanted to generalize his theory of special relativity to non-inertial situations. This is when there is the presence of acceleration or deceleration.
Indeed, Einstein was not satisfied merely with special relativity, and noticed two holes in his theory as it currently was understood:
Special relativity was based on inertial motions, which are almost never viewed in nature. Everything is in a state of constant acceleration! Special relativity couldn't account for even the simplest kinds of acceleration.
Special relativity did not include gravity. This great new theory of the universe did not describe gravity? This was an embarrassment since gravity is, of course, everywhere. Einstein thought about gravity and pondered a fundamental inconsistency: The speed of light is the ultimate speed limit in the universe right? Light takes 8 minutes and 17 seconds to get from the surface of the Sun to the surface of the Earth right? So how could gravity act instantaneously as it did in the picture of Isaac Newton? This was the contradiction that paved the way for Einstein to think about how Newtonian gravity could be incorporated into special relativity. He began to work on generalizing his theory of relativity to include acceleration and gravitation.
At this point, Einstein began to differentiate his "special" theory of relativity with his "general" theory of relativity. The former was "special" in the sense that it only applied to the special situations where acceleration and gravitation could be completely ignored.
In general relativity, gravity is a phenomenon that results from the curvature of spacetime. This curvature is the result of the presence of a mass. The more mass that is contained in a given volume of space, the greater the curvature of spacetime.
Albert Einstein’s insights for general relativity began with the development of his equivalence principle. The idea hit him one day in 1907: "If a person falls freely, he will not feel his own weight". That very thought propelled him toward a new theory of gravitation. One is momentarily weightless when they fall.
According to this principle, there is an equivalence between:
States of accelerated motion
Being at rest in a field of gravitation.
The laws of physics in an accelerating frame and in a gravitating frame are indistinguishable. They are identical. An example of being at rest in a gravitational field is, me, sitting here, writing about physics. Though I am not accelerating or even moving, I am still under the influence of the planet’s gravitational attraction. Einstein showed that a gravitational field is physically equivalent to a reference frame under acceleration. All masses fall at the same rate under gravity. Inertial mass is the same as gravitational mass.
A person in a free falling elevator experiences weightlessness and cannot tell that they are in free fall.
This equivalence principle also has implications for the idea of an object that is in free fall:
If you are in an elevator and the cord holding you and the rest of the elevator snaps, you begin to fall freely toward the Earth.
You are falling at the same rate as the elevator floor.
You would feel weightless, as if you were floating in air.
Thus, if one is in free fall: the effect of gravitation is canceled by one's acceleration. This would provide the feeling of weightlessness. Gravity can be canceled by acceleration.
In classical mechanics, an object is in freefall, as a consequence of gravitation, or lack thereof. However, according to the equivalence principle, an object in free fall, is falling, since, there is no force being exerted on it or underneath it. That is what freefall is, in general relativity: there is no force acting on it. The only force acting on an object in free fall is gravity. Also, in classical mechanics, objects in free fall, can accelerate, with respect to each other, however, inertially moving objects cannot. For the equivalence principle, one who is in a free falling elevator, cannot tell that they are in free fall.
One who is dropping a ball in an accelerated rocket that is moving, will witness an equivalent effect to one who is stationary, tied to the ground by the Earth's gravitation. He would not be able to tell whether he was in free fall, or, being held to the Earth by it's gravity. He would witness the equivalent effect.
Thus, with Albert Einstein’s theory of general relativity, will come the proposal, that spacetime, is curved. In 1915, Albert Einstein will publish his field equations which will relate the curvature of spacetime to the momentum and energy of the mass present in that space. In the words of John Archibald Wheeler; “Spacetime tells matter how to move, and matter tells spacetime how to curve.”
Spacetime, is a unified relationship between space and time, as a 4-dimensional continuum.
Marcel Grossman was an old classmate of Albert Einstien. He was also the one to emphasize to Einstein: Riemannian geometry. This is the language that general relativity was going to be formulated in. Riemann's contributions to differential geometry, laid the foundations of general relativity. It was his 1854 lecture, that would prove to be critical in Einstein's proposal to how spacetime curved. It was a necessary step.
Marcel Grossman
Bernhard Riemann
Riemann was shy and often suffered from nervous breakdowns. He also suffered from poverty and had tuberculosis. He would often retreat into the world of mathematics. In fact he read the 859-paged "Theory of Numbers" in 6 days.
Thus, spacetime's curvature is determined by: the energy/momentum of the present matter/radiation.
As with special relativity, the key to Einstein's thinking is unification through symmetry. The way this worked in general relativity took a few steps:
Einstein had the picture: gravity was caused by the warping of space and time.
Einstein recognized a fundamental disagreement between Newtonian gravity, where gravity traveled instantaneously, with special relativity, where nothing can travel faster than the speed of light.
Einstein than put forth his equivalence principle: accelerating and gravitational frames obey the same laws of physics.
Lastly, will come general covariance: the symmetry behind gravity and the coordinate transformations of spacetime. This is a generalization of Lorentz covariance, which is applicable for all possible transformations and accelerations, not just inertial ones. This led to his principle of general covariance.
Paul Ehrenfest
Einstein needed a field theory of gravity. By 1912, he knew he needed a new kind of geometry to describe space and time. Ehrenfest's paradox would send Albert Einstein on the road to understanding the geometry of curved spaces. We begin with a spinning disk moving like a merry-go-round. Before this disk begins to spin, its circumference is π times the diameter. Once the disk begins to spin, the outer rim will travel faster than the interior. According to relativity, the outer rim should shrink more than the interior of the disk. This would distort its shape: the circumference has been made shorter than π times the diameter. The result is that space is no longer flat, however, it is curved. This non-Euclidean idea was crucial for Einstein's development of general relativity. From this picture we can draw an analogy:
If one places a bowling ball in the center of a trampoline, the ball will create a depression and sink into the trampoline. Then, one puts a marble some arbitrary distance from the bowling ball, also on the trampoline. This marble will follow the curve in the trampoline that is laid out by the bowling ball. It does not move in a straight line. This scenario would have two different interpretations: one for the Newtonian and for Einsteinian.
To Isaac Newton: There is a mysterious invisible force that emanates from the bowling ball, forcing the marble to change its path. This invisible force pulls the marble toward the bowling ball.
To Albert Einstein: There is no "force" per se, acting on the marble. It is just the depression that the bowling ball has created on the trampoline. This depression will dictate the motion of the marble and will guide it along its new path that pushes it in a circular motion around the bowling ball.
All you have to do is replace the trampoline with the fabric of 4-dimensional spacetime, the bowling ball with the Sun and the marble with the Earth! There is no gravitational pull or mysterious force, as in Newton's picture of gravity, however, the Earth is following curves in spacetime. The spacetime that is warped by the presence of the Sun's mass, pushes the Earth along it's orbit!
Red - Newtonian orbit
Blue - Einsteinian orbit
General relativity also predicts that orbits, act in a manner different than in Newtonian physics. In general relativity: orbits precess. In Newtonian physics, the orbit of a planet around a star will trace out a perfect ellipse, while the star is located at the focus of this elliptical path traced out by the orbit. However, for Albert Einstein, not only will the path of a planet's orbit trace out an ellipse, however, the path itself will be rotating around the star. This is known as the anomalous perihelion shift. This phenomenon was first observed for the planet Mercury, in 1859 and it was later proven by radio telescope measurements. The behavior of the orbit of Mercury was an astronomic mystery for some time. Einstein was able to explain that the behavior of this orbit was a consequence of Mercury following a path in spacetime traced out by the Sun's gravity. General relativity is able to accurately describe this phenomenon.
I would like to discuss gravitational time dilation. This is the phenomenon where clocks run slower in regions of spacetime under the influence of stronger gravitational attraction. This will be an actual difference in elapsed time, between two observers, situated at varying distances from a source of gravitation. A clock that is farther from a source of gravitation, will tick faster.
Gravitational time dilation is a consequence of general relativity. According to this postulate, the closer a clock is to a source of gravitation, the slower it will move. The farther you are from a source of gravitation, the faster a clock will move. Clocks at higher gravitational potential, or, farther from a source of gravitation run more quickly, than clocks at lower gravitational potential or those clocks that are closer to massive objects.
Another consequence of general relativity, that is related to gravitational time dilation is gravitational redshift. This is the phenomenon, where, an object moving upward from a source of gravitation, would have it's emitted electromagnetic radiation be reduced in frequency, moving more towards the red end of the visible light portion of the spectrum of electromagnetic radiation, hence, "redshifted". This has been measured both in the laboratory and in astronomical observations. As I mentioned before, gravitational redshift, is a consequence of gravitational time dilation. Remember: as you move away from the source of gravitation, clocks tick faster. Thus, the time that would be required for each wave oscillation, depends on your distance from the source of gravitation, thus changing the frequency.
This picture is a highly dramatized example of gravitational redshift. Einstein argued that, an object moving upward from a source of gravitation, would have it's emitted electromagnetic radiation be reduced in frequency, moving more towards the red end of the visible light portion of the spectrum of electromagnetic radiation, hence, "redshifted".
Accelerating objects that generate changes in the curvature of spacetime, that propagate outward, at the speed of light, in a wave-like manner, are known as gravitational waves. Gravitational waves are ripples in spacetime, that travel at the speed of light. These can be thought of as ripples in the fabric of spacetime. Gravitational waves can move in regions of space that electromagnetic waves cannot, and, could theoretically travel at any frequency. Gravitational waves are constantly passing the Earth, however, even the strongest ones have a minuscule effect.
Gravitational waves were first proposed by Henri Poincare in 1905.
Gravitational waves are ripples in spacetime that propagate at the speed of light: a consequence of general relativity. They are a direct consequence of Einstien's theory. This is different from Newtonian gravity. For Isaac Newton, the gravitational force was instantaneous and propagated at an infinite speed. Newton's law of universal gravitation does not account for the existence of gravitational waves. Instead, in Newton's theory, forces propagate at infinite speed. Of course, in relativity, this is not possible, because, nothing can move faster than the speed of light, even gravity.
A consequence of the equivalence principle is that a beam of light will bend in the presence of some gravitational attraction. Light is deflected in a gravitational field. Einstein knew that this effect would be calculable. Einstein was curious as to if the gravitation from the Sun could bend light from other stars. His idea was that perhaps, during a solar eclipse, he could measure this distortion, as the other stars' light would become visible. Indeed, the equivalence principle allowed Einstein to calculate how a beam of light would move under some gravitational influence.
In other words, gravitational lensing is the phenomenon where light bends in a gravitational field. This is when some source of gravitation, such as a galaxy, that exists in between a source of light and an observer, is actually massive enough to distort the light rays from the source, thus, making it appear distorted, to the observer. An example of gravitational lensing in action is the “Einstein cross”. This is a quasar that has had it’s light distorted by a gravitational lens, thus, making it appear in the shape of a cross. It is four images of the same quasar. Einstein predicted that a beam of light could bend in a field of intense gravitation.
This is a distribution of matter, in between a distant observer and a source of light, that is capable of bending light. There is a prediction in classical physics that light can bend in a source of gravitation. However, it is only half that predicted by general relativity. Einstein published a famous article on this phenomenon in 1936.
For a gravitational lens, there is not a focal point, however, a focal line. When the source, lens and observer are perfectly aligned, the light source will appear as a ring around the massive object. This is known as an Einstein ring. Any misalignment will result in an arc segment being observed. These arcs will be scattered along the lens. Gravitational lensing acts equally on all kinds of electromagnetic radiation, not just light.
There are 3 kinds of gravitational lensing:
1) Strong gravitational lensing is when the lensing effect is strong enough to produce multiple images of the same astronomical object. These are easily visible distortions. Strong lenses have been observed in x-ray and radio wave regimes.
2) Weak gravitational lensing is when the deflection of light cannot be detected by observing only one galaxy, however, by averaging over a large number of galaxies. This is an intrinsically statistical measurement. Weak lensing has been used to study the cosmic microwave background. It is also used to study galaxy surveys.
3) Gravitational microlensing is when the amount of light received from a light source changes with time. However, no distortion in shape can be seen. This allows astronomers to observe objects that emit little or no light.
An example of light being distorted in a gravitational field.
"Einstein cross"
Another consequence of general relativity that I would like to talk about is frame-dragging. This is literally a dragging effect on spacetime caused by the rotation of a mass. Indeed, this effect is related to the rotation of masses.
An example of frame dragging: when spacetime is dragged by the gravitational presence of a massive body.
Lastly, general relativity, can describe the expansion of the universe, to a degree. It allows for space to be expanding, and for objects to be moving away from each other, faster than the speed of light. The expansion of the universe cannot be modeled with special relativity: it is completely general relativistic.
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