Relativity Dr Walker SpaceTime Prior to Einstein physicists

Relativity Dr. Walker

Space-Time • Prior to Einstein, physicists thought of the space and the universe as an infinite expanse where all things exist • Einstein theorized space and time exist in the known universe and it is intertwined • This is known as space-time

Special Theory of Relativity • Einstein’s special theory of relativity describes how time is affected by motion in space based on – Variations in velocity (including near light speed) – The relationship between energy and mass • This in particular had not been discussed

Space-Time Travel • The universe has a space and time component. – When you stand still, time still moves. You travel “forward” through time. – When you move normally, you travel through space AND time. Most of the travel is still through time

Space-Time Travel • Based on the previous, a photon travelling at the speed of light doesn’t travel through time – The photon’s movement is only through space. Time doesn’t move. – Time doesn’t move for light! – If you moved at the speed of light, time would stand still

Postulates of Special Relativity • The first postulate (fundamental assumption) of special relativity states that all the laws of nature are the same in all uniformly moving frames of reference – We can only measure our speed relative to other objects within our own frame of reference – For example, think about measuring velocity on Earth • The Earth itself is moving through space, rotating in place and revolving around the sun at fairly high velocities

Postulates of Special Relativity • Which one is moving? Are they both moving? • Each ship passenger only observes the relative motion of the other ship

Postulates of Special Relativity • The second postulate of special relativity states that the speed of light in empty space will always have the same value regardless of the motion of the source or the motion of the observer – The constancy of the speed of light is the connection between space and time – Velocity (speed of light) = distance (space) / time =

Time Dilation • The stretching of time based on the relationship between an observer and his observations is known as time dilation • to = time experienced by traveler, t = “real time” • When people are travelling at “normal” speeds, the denominator is essentially 1, and time is normal • The faster an object moves, the slower time passes

Time Dilation Example • If a traveler is moving at 90% of the speed of light, how much time will they experience when the average person experiences 60 s?

Time Dilation Example • If a traveler is moving at 90% of the speed of light, how much time will they experience when the average person experiences 60 s? • • 60 s = to / sqrt (1 – (. 9/1)2) 60 s = to / sqrt (1 – 0. 81) 60 s = to / 0. 436 to = 26. 2 s – the traveler experiences 26. 2 s

More Info About Time Dilation • https: //www. youtube. com/watch? v=yu. D 34 t. E p. RFw

Light Clock Example • A stationary light clock is shown here. Light bounces between parallel mirrors and “ticks off” equal intervals of time.

Light Clock Example • Suppose we view the light clock as it whizzes past us in a high-speed spaceship. • We see the light flash bouncing up and down along a longer diagonal path.

Light Clock Example • Remember the second postulate of relativity: The speed will be measured by any observer as c. • Since the speed of light will not increase, we must measure more time between bounces! • From the outside, one tick of the light clock takes longer than it takes for occupants of the spaceship. • The spaceship’s clock has slowed down. • However, for occupants of the spaceship, it has not slowed.

Twin Paradox • Another example of time dilation is known as the twin paradox • A pair of identical twins, one an astronaut who takes a high-speed round-trip journey while the other stays home on Earth. • When the traveling twin returns, he is younger than the stay-at-home twin. How much younger depends on the relative speeds involved.

Twin Paradox • How does one twin age differently? !? – If the traveling twin maintains a speed of 50% the speed of light for one year (according to clocks aboard the spaceship), 1. 15 years will have elapsed on Earth. – If the traveling twin maintains a speed of 87% the speed of light for a year, then 2 years will have elapsed on Earth. – At 99. 5% the speed of light, 10 Earth years would pass in one spaceship year. At this speed, the traveling twin would age a single year while the stay-at-home twin ages 10 years.

Twin Paradox – Time Dilation Example • https: //www. youtube. com/watch? v=i. IEe. Si. T 3 S I 4

Additional Time Dilation Example • Picture two people – One is standing watching a clock – The other is watching a clock while away from it at the speed of light…

Additional Time Dilation Example • Picture two people – One is standing watching a clock • The clock moves normally for the person standing still – The other is watching a clock while away from it at the speed of light… • The light “signal” from the clock never reaches the 2 nd observer, so the clock appears to stand still

Additional Time Dilation Example • Picture two people – One is standing watching a clock • The clock moves normally for the person standing still – The other is watching a clock while away from it at the speed of light… • The light “signal” from the clock never reaches the 2 nd observer, so the clock appears to stand still • If this person moves back towards the clock at the speed of light, more signals are received from the clock in relative time, so it looks like the clock has sped up – In reality, the clock’s “time” is faster because your time is slower, since you’re moving at close to light speed

Time Dilation and Space Travel Applications • A light year is the distance a photon of light will travel in year, which is equal to 5. 88 trillion miles – The closest star to Earth after the sun, is Alpha Centauri, which is 4 light years away – This means it would take someone 4 years travelling at light speed to get to Alpha Centauri – The fastest space shuttle travels at 17, 500 miles per hour • It would take 3. 36 x 108 hours, or roughly 38, 000 years at that speed to make it to the nearest star – YIKES!!

Time Dilation and Space Travel Applications • The space shuttle does not travel at close enough to light speed to provide significant time dilation • An astronaut travelling at a significant portion of light speed would experience less time due to time dilation

Time Dilation and Space Travel Applications • Travelling to Procyon (11. 4 light years) – Astronauts travelling at 99% of light speed would make a round trip in 23 years time – Time dilation would make the astronauts age only 3 years during that trip – The people of Earth age 23 years • This type of travel would require astronauts to leave behind most of what they know and return to a “future” time • Astronauts could “jump” into future with near light speed travel – not into the past

Length Contraction • The observable shortening of objects moving at speeds approaching the speed of light is length contraction. • The amount of contraction is related to the amount of time dilation. For everyday speeds, the amount of contraction is much too small to be measured.

Length Contraction • Think of a meter stick…. – At 87% of c, it would appear to you to be 0. 5 meter long. – At 99. 5% of c, it would appear to you to be 0. 1 meter long. – As relative speed gets closer and closer to the speed of light, the measured lengths of objects contract closer and closer to zero. – The width of a stick, perpendicular to the direction of travel, doesn’t change.

Length Contraction • People travelling at near light speed notice no difference in themselves • They would see objects travelling at normal speeds contracted based on their frame of reference, which is travelling at near light speed – The effects of relativity are always attributed to “the other guy”

Length Contraction • • V is the speed of the object relative to the observer c is the speed of light L is the length of the moving object as measured by the observer L 0 is the measured length of the object at rest – Similar to the time dilation equation

Length Contraction Example • A space shuttle that is 184 feet long is retrofitted with an engine that allows it to travel at 80% of the speed of light. When travelling at this speed, what is the length of the shuttle when accounting for length contraction?

Length Contraction Example • A space shuttle that is 184 feet long is retrofitted with an engine that allows it to travel at 80% of the speed of light. When travelling at this speed, what is the length of the shuttle when accounting for length contraction? • L = 184 ft (sq rt (1 – 0. 82/12)) • L = 184 (sq rt 0. 36) • L = 110. 4 ft – The shuttle is 110. 4 ft long when travelling at 80% of light speed

Length Contraction • Based on the equation for length contraction, an object travelling at light speed would have a length = 0. – This is one way that light speed is defined as an upper limit. An object can not have a length < 0.

Back To Kinematics • Based on previous topics, we know F = ma or a = F/m – More force means more acceleration – More acceleration means more speed – Based on what we’ve discussed previously, we know the speed of light is the physical speed limit of the universe https: //www. physicsforums. com/insights/wp-content/uploads/2015/07/Speed-Limit-Light. jpg

Back To Kinematics • • Based on F = ma Acceleration is velocity/time F = m. DV/Dt Rearrange this and you get the expressions for momentum – p = m. DV = FDt • The speed of the universe is finite, yet momentum can increase without limit – The math doesn’t say this – how does THAT work?

Relativistic Momentum • Similar to time dilation and length contraction at near light speed, relativistic momentum is a correction at near light speed • Notice that at typical speeds, the denominator is effectively 1, and the standard kinematic definition for momentum holds

Relativistic Momentum • At near light speed, the momentum of particles increases more than their speed • This is proven in experiments with subatomic particles that are accelerated to near light speed – When electrons are directed into a magnetic field, the particles are deflected – They are not directed as much as p = mv would predict, since they have more momentum are very high speed • Particles are harder to deflect with more momentum and inertia

Mass-Energy Equivalence • The most unusual insight of Einstein’s special theory of relativity is his conclusion that mass is simply a form of energy. • A piece of matter has an “energy of being” called its rest energy. • Einstein concluded that it takes energy to make mass and that energy is released when mass disappears. • Rest mass is, in effect, a kind of potential energy.

Mass-Energy Equivalence • Einstein determined c 2 to be a conversion factor between mass and energy – Mass and energy are two sides of the same coin – This only occurs practically during nuclear reactions • c 2 = E/m, which rearranges to E = mc 2

Mass-Energy Equivalence • In one second, 4. 5 million tons of rest mass in the sun is converted to radiant energy – This is converted directly from mass to energy – The sun has enough mass for this to continue for another 5 billion years

Mass-Energy Conversions • Other examples of conversions of mass to energy • For example, when we strike a match, a chemical reaction occurs and heat is released. • The molecules containing phosphorus in a match head rearrange themselves and combine with oxygen to form new molecules. • These molecules have very slightly less mass than the separate phosphorus- and oxygen-containing molecules by about one part in a billion. • For all chemical reactions that give off energy, there is a corresponding decrease in mass. • The example above involves the conversion of a very small amount of mass to energy

Correspondence Principle • In science, we know that accepted theories may be incomplete – The relationship between relativity and Newtonian mechanics is an example • The correspondence principle states that new theory and old theory must overlap and agree in the area where the results of the old theory have been fully verified – Our expressions for time dilation and length contraction follow accepted theory for standard conditions

General Relativity • The special theory of relativity primarily deals with motion in uniformly moving frames of reference • The general theory of relativity explains that gravity causes curvature in space-time and time to slow down

Artificial Gravity • The sensation of gravity exists based on universal gravitation between objects – The objects accelerate towards one another • If a spaceship accelerates, objects would move as if they are falling due to gravity – The objects can’t tell the difference between gravity and acceleration – because they are both a form of acceleration!

“Gravity” In Space a. Everything inside is weightless when the spaceship isn’t accelerating. b. When the spaceship accelerates, an occupant inside feels “gravity. ”

“Gravity” In Space To an observer inside the accelerating ship, a lead ball and a wooden ball accelerate downward together when released, just as they would if pulled by gravity.

General Relativity A ball is thrown sideways in an accelerating spaceship in the absence of gravity. a. An outside observer sees the ball travel in a straight line. b. To an inside observer, the ball follows a parabolic path as if in a gravitational field.

General Relativity A light ray enters the spaceship horizontally through a side window. a. Light appears, to an outside observer, to be traveling horizontally in a straight line. b. To an inside observer, the light appears to bend.

General Relativity • Based on this principle, Einstein reasoned that since acceleration (a space-time effect) can mimic gravity (a force), gravity is not a separate force after all, but a form of space-time • From this bold idea he derived the mathematics of gravity as being a result of curved space-time. • He predicted light would bend around larger objects, like stars, because of the curved space time around large objects – Proven right by Eddington

Space-Time • Space-time has four dimensions—three space dimensions (length, width, and height) and one time dimension (past to future). • Einstein perceived a gravitational field as a geometrical warping of four-dimensional space-time. • Four-dimensional geometry is altogether different from the three-dimensional geometry introduced by Euclid centuries earlier. • Euclidean geometry (the geometry YOU learn) is no longer valid when applied to objects in the presence of strong gravitational fields.

Not Your Normal Geometry • Euclidian geometry deals with figures on a flat surface – C = pd – 3 angles of triangle = 180 o – Shortest distance between two points = straight line • Valid in flat space, but if shapes are drawn on a curved surface, the rules no longer apply

Non-Euclidian Geometry The sum of the angles of a triangle is not always 180°. a. On a flat surface, the sum is 180°. b. On a spherical surface, the sum is greater than 180°. c. On a saddle-shaped surface, the sum is less than 180°.

Non-Euclidian Geometry The geometry of Earth’s two-dimensional curved surface differs from the Euclidean geometry of a flat plane. a. The sum of the angles for an equilateral triangle (the one here has the sides equal ¼ Earth’s circumference) is greater than 180°. b. Earth’s circumference is only twice its diameter instead of 3. 14 times its diameter.

It’s Not A Straight Line After All • The lines forming triangles on curved surfaces are not “straight” from the three-dimensional view • They are the “straightest” or shortest distances between two points if we are confined to the curved surface. • These lines of shortest distance are called geodesics.

Geodesics and Light • Light beams follow geodesic paths • Three experimenters on Earth, Venus, and Mars measure the angles of a triangle formed by light beams traveling between them. • The light beams bend when passing the sun, resulting in the sum of the three angles being larger than 180°. • So the three-dimensional space around the sun is positively curved. • The planets that orbit the sun travel along fourdimensional geodesics in this positively curved spacetime. • Freely falling objects, satellites, and light rays all travel along geodesics in four-dimensional space-time.

Geodesics and Light The light rays joining the three planets form a triangle. Since the sun’s gravity bends the light rays, the sum of the angles of the resulting triangle is greater than 180°.

Warping Space-time • Instead of visualizing gravitational forces between masses, we shouldn’t of the idea of gravitational force – Think of masses responding in their motion to the curvature or warping of the space-time they inhabit. • General relativity tells us that the bumps, depressions, and warpings of geometrical space-time are gravity.

Warping Space-Time • Consider a simplified analogy in two dimensions: a heavy ball resting on the middle of a waterbed. • The more massive the ball, the more it dents or warps the two-dimensional surface. • A marble rolled across such a surface may trace an oval curve and orbit the ball. • The planets that orbit the sun similarly travel along four-dimensional geodesics in the warped space-time about the sun.

Warping Space-Time Space-time near a star is curved in a way similar to the surface of a waterbed when a heavy ball rests on it.

Relativistic Phenomena • Deflection of Starlight – Einstein predicted that starlight passing close to the sun would be deflected by an angle of 1. 75 seconds of arc • This can only be observed during a solar eclipse • Photography of the eclipse allows astronomers to determine the bending of starlight, first observed by Campbell and Eddington independently

Relativistic Phenomena • Gravitational Waves – Every object has mass, and therefore makes a bump or depression in the surrounding space-time. – When an object moves, the surrounding warp of space and time moves to readjust to the new position. • These readjustments produce ripples in the overall geometry of space-time. • The ripples that travel outward from the gravitational sources at the speed of light are gravitational waves. • First detected in 2016 by LIGO • Yes, we’re just proving some things about relativity…. NOW

Relativistic Phenomena • Gravitational Waves – Every object has mass, and therefore makes a bump or depression in the surrounding space-time. – When an object moves, the surrounding warp of space and time moves to readjust to the new position. • These readjustments produce ripples in the overall geometry of space-time. • The ripples that travel outward from the gravitational sources at the speed of light are gravitational waves. • First detected in 2016 by LIGO • Yes, we’re just proving some things about relativity…. NOW

Relativistic Phenomena • Gravitational Waves – Requires VERY large bodies to move and produces the weakest waves known in nature – First detection was between two 30 solar mass black holes merging 1. 3 billion light years from Earth • Solar mass = mass of sun • Gravitational waves may move a detector 10 -18 m – tough to observe!

Relativistic Phenomena • Procession Of Planetary Orbits – Based on relativity, the orbits of planets around the sun were different than originally predicted – Einstein recalculated orbits based on relativity, which were similar to Newton’s original calculations – Mercury is the only planet close enough to the sun to produce a measurable difference, which has been observed

Relativistic Phenomena • Gravitational Red Shift – Einstein predicted clocks run at different speeds based on their proximity to large bodies, such as planets and stars • Time runs slower closer to the surface of a planet or star – stronger gravity, slower time – Light travelling against gravity has a slightly lower frequency (and energy), which is referred to as a red shift (as opposed to a blue shift)

Gravitational Red Shift • Bigger star = stronger gravity = tougher for light to escape = less energy http: //faculty. humanities. uci. edu/bjbecker/Exploringthe. Cosmos/redshift 5 b. jpg