Supplementary Textbook Chapters S2 and S3: Special and General Relativity We will consider just a small part of these which introduce Einsteins Special and General Theories of Relativity Young Einstein (better

than oldster disheveled version) Even younger Special Relativity Einsteins Special Theory begins with a proposed axiom that the speed of light in a

vacuum will be measured to be the same by all observers, regardless of how they move relative to the light beams. Ordinary objects in your daily life dont behave that way. So Sounds very anti-intuitive. But if youre Albert Einstein, it seemed obvious. And, experiments proved he was right. A Logical Consequence is

The measured mass of a moving object increases with its speed, until it becomes infinite at the speed of light itself. Which means it would take an infinite force to further accelerate it beyond this speed, which is impossible. Therefore... The speed of light in a vacuum is the ultimate speed limit. No object will be observed to move faster than the speed of light in a vacuum; 299,972 km/sec This is such a fundamental constant (combined with the fact that we can measure time intervals more accurately

than we can measure distance) that we now invert the historical definitions of a meter and a second so that a meter is defined in terms of the speed of light, i.e. 1 kilometer == the distance a photon moves in 1/299,972.458 of a second To Preserve this Fundamental Principle Einstein realized the nature of length and time needed to be relative; that different observers must measure these differently, else the speed of light could not be a universal

constant. Hence, moving objects measure shorter in the direction of their motion, and moving clocks tick slower. Both of these are necessary so that the speed of light (distance covered per unit of time) would stay constant for all differently moving observers The scientific courage of Einstein was in accepting the relativity of length, mass, and time, whereas a more conventional mind would simply reject the notion of universal constancy of the speed of light in order to preserve the

supposed universality of mass, length, and the rate of passage of time In Relativity, even the actual order in which events happen in time, depends on the observers motion General Relativity Our modern theory of gravity General Relativity

Newtons equation for gravity works pretty well at low speeds and weak gravity such as were used to - but Einstein discovered by 1915 that the underlying concept of gravity is actually quite different than Newtonian gravity Mass and/or Energy density warps the geometry of the surrounding space. This warped space has a tension, like a bent steel rod carries tension, and this tension is an energy density which itself warps space. Warped space?! But whats there to warp? Actually space has a physical reality to it, even if it has no mass to it.

Einstein realized this and formulated his General Theory of Relativity our modern theory of gravity. Took 10 years of really REALLY hard work to develop the mathematics (said Albert himself! - tensor calculus applications) Einstein realized there was no difference

between being at rest in a gravity field, and being in a uniformly accelerated coordinate frame

The Equivalence Principle Einstein noted how the acceleration of gravity could be removed if one simply put the falling object into a properly accelerated coordinate frame. He took his intuition to the extreme he felt sure that there could be NO EXPERIMENT POSSIBLE which could distinguish between being at rest in a gravity field vs. being in a suitably accelerated coordinate frame. This is the Equivalence Principle, and predicts the

bending of light (next slide) Note that the difference in gravity across the planet (tidal force), means a coordinate frame appropriate to the whole planet must be rubber and not just accelerate rigidly, but also stretch appropriately in different places with different gravity. A laser beam will appear to bend in an accelerating room (left), simply because the room accelerates upward while the beam crosses it - no new physics there. But by the Equivalence

Principle, a room at rest in a gravity field (right) of equivalent acceleration must show the same bending, despite the fact light, in Newtons physics, was massless and so wasnt expected to bend (E=mc2 says there is a mass equivalent to the energy of light, in Relativity) Coordinate Frames Einstein became fascinated with coordinate frames. Coordinate systems which could be used to describe natural phenomenon, and how the mathematics got simpler if the

right coordinate frame were used. That Nature seems to know about coordinate frames and has Her preferences in given situations. (OK, thats a little anthropomorphic for some people, just so you know). The point is, Einstein had an intuition that the messy movings of things under gravity would look far simpler if we instead thought of them as following simpler paths in a curved space. A curved coordinate frame. Later, as it became clear that empty space wasnt really empty, the reality of a curved space had more than just a semantic meaning, it seemed.

An Inertial Coordinate Frame is one in which the laws of physics are homogenous (the same at all places) and isotropic (no direction is preferred over any other, in the laws). Then laws are as mathematically simple as possible An inertial frame experiences no acceleration. General Relativity is Expressed in Einsteins Field Equations

R = Ricci curvature tensor g = metric tensor T = stress-energy tensor G = Newtonian gravitational constant = the Cosmological Constant:

Even Better When Written on Old Industrial Age Remnants! = the Cosmological Constant: Well encounter the Cosmological Constant again, since, surprisingly, it is not zero, but instead about 10-52 m-2 More precisely, its measured to be 0.69(3H2/(8G)G) where H is the Hubble

parameter = 67.8 km/s-Mpc, and error bars of about 1.5% For now, dont stress on knowing about . When we get to cosmology well learn more (and THATs when youll need to remember just a bit about it for exam purposes) Curved Space! To model changing gravitational fields over large areas, Einstein realized that

the coordinate frames that are natural are not made of straight lines, but curved, and rubber-like. SpaceTime is curved by gravity, and gravity is caused by mass, and Special Relativity also showed that E=mc2, and therefore energy has its own equivalent mass and will also curve space.

Heres a YouTube on understanding GR Its called Einsteins Field Equations for Beginners! Hey, its only 2 hrs long, and you have lots of free time, so have at it! (and, dont believe the title) Orbits, Einstein-style

An Object Follows an Unforced Path Through a Curved Space, and hence, it Curves In 3-D Putting General Relativity (GR) to the Test These realizations allowed Einstein to deduce

that there were testable predictions to this radical new understanding of gravity This is the hallmark of a true Scientist! Testable predictions which were specific, and uniquely not predicted by Newtonian gravity Because he gave testable predictions, his radical new ideas were not dismissed as the rantings of a lowly patent clerk disconnected from Reality. Now go read the typical grandiose flowery

claims of New Age people, who smugly provide no objective tests, and note the contrast! The additional 43/century precession of Mercurys orbit was already observed by 1916 but unexplained until Einstein showed it was a prediction of GR Light, too, will bend as it moves through space curved by nearby mass. Note how the star appears farther

from the sun than it actually is, by the bending The Bending of Starlight Confirmed at 1919 and 1922 Solar Eclipses Glass Photographic Plate of 1922 Eclipse with Stars Circled and Measured Black Holes hugely bend nearby space. Youd see one coming! (which is good)

Later came new tests

Black holes in galaxy cores, now too in globular clusters, and binary stars discovered Gravity Probe B and frame dragging confirmed Gravitational Waves (from the Binary Pulsar) indirectly detected at proper level All passed with flying colors. However, GR is a classical theory it does not take account of the quantum nature of the very tiny. We still do not have a verified quantum

theory which includes gravity. One theory getting a lot of attention for many years now is String Theory, in which the fundamental physical units are loops of energy called string rather than 0-dimensional elementary particles. This has the benefit of unifying gravity with the rest of quantum fields. However, it makes no testable predictions and also requires 11 dimensions to space/time, vs. the 4 actually observed. Some are excited by it because of its beauty, but others are not convinced its all that beautiful and await something more testable and obvious in its beauty. An interesting extrapolation of Einsteinian conceptions of physics is the

ideas of physicist Lee Smolin and colleagues, who consider fundamental physical ideas as relational, rather than involving absolute space and time. The mathematics is extremely difficult and unfortunately his ideas are not yet testable, but for the philosophically and math-minded, worth reading (e.g. here) Physics isnt done yet. Were still working at it An Einstein Ring, as light from a background galaxy is focused into a ring by the orange foreground elliptical galaxy

Just as accelerating a charge will create waves in the electromagnetic field, accelerating a mass will create waves in the gravitational field. But since gravity as a force is ~10-37 times weaker

than EM, the waves are VERY hard to detect LIGO is two identical interferometers thousands of miles apart using beam splitters to detect the slight contraction along one arm and expansion along the other arm as a gravity wave passes through Finally in 2016, we succeeded in direct detection of gravity waves. LIGO the Laser Interferometry Gravitational

Observatory observed waves produced by the in-spiraling merger of 2 black holes 10s of times more massive than the sun. To put the y-axis numbers in perspective, the 4km long arms expanded and contracted by only 1/250th of the diameter of a proton and we detected it (!) Another Prediction of Relativity, is that light must lose energy climbing out of a gravity field. This energy loss shows it self by the frequency and wavelength of the photons as observed by

someone beyond that gravity field. This is called Gravitational Redshift a new kind of redshift different than the Doppler Effect. This is what makes black holes BLACK! Photons are infinitely redshifted, losing all their energy trying to escape. S2: Key Ideas Relativity Special Relativity follows from: All observers measure the speed of light in a vacuum as being exactly the same. For this to remain true, Einstein realized measured size and

mass the pace of clocks depended on the relative velocity between observed and observer Moving masses are more massive. Length of moving objects contract along direction of motion E=mc2 Energy has a mass equivalent General Relativity = our modern theory of gravity, follows from The Equivalence Principle: The effects of gravity are exactly equivalent to what would be experienced in an accelerated coordinate frame.

Presence of mass (or energy) warps the geometry of space so that objects moving through that space follow curved paths. Slower speed-> more curve. Faster speed->less curve. Speed of light -> very little curve, so we didnt notice till technology advanced Gravitational Redshift: Light loses energy and redshifted climbing out of a gravity field. Observationally confirmed