Today HW#4 pushed back to 8:00 am Thursday Exam #1 is on Thursday Feb. 11 Bring a calculator and a #2 pencil Allowed 1 page notes (front and back) E=mc 2, General Relativity, and exam review ISP209s10 Lecture 7-1-
Comments on Exam 1 40 questions, multiple choice Bring #2 pencil and calculator Level of difficulty HW sets Extra credit opportunity if you bomb In LONCAPA, take a retake of a similar 40 point exam For each point higher than original in-class score, I will add.30 points to your original score say you got a 30/40 on the exam, and a 40/40 on the retake. Then you would get (40-30)*0.3 = 3 extra points for an adjusted score of 33/40. ISP209s10 Lecture 7-2-
Special Relativity: Time, Length, Mass Time Dilation: Moving clocks slow down Time slows down Length Contraction: Moving rulers shrink Lengths in the direction of motion are shorter than when at rest. Mass increases the faster you go (but you can never reach c since mass would be infinite) ISP209s10 Lecture 7-3-
A few words about Energy Energy is the ability to do work. It comes in two main types Kinetic energy: the energy of motion Potential energy: the energy of position Work = force x distance (it s a scalar measured in Joules, J (same as Nm)) Power is the rate work is done (it s a scalar measured in J/s (same as a Watt W) ISP209s10 Lecture 7-4-
E = mc 2 : What it means Any increase in a system s energy results in an increase to its mass! The change in mass is equal to the change in energy, divided by the square of the speed of light. The speed of light is very large, so the change in mass is undetectable in ordinary situations. This idea (that mass can change ) was very odd at the time. ISP209s10 Lecture 7-5-
E = mc 2 in action Schematic picture of a chemical or nuclear reaction: Start with some initial mass (kg) Something happens End up with some final mass (kg) Some fraction, f, is converted to energy ENERGY (Joules, J) The amount of energy is E = m converted c 2 m converted = (M initial - M final ) = M initial x fraction converted ISP209s10 Lecture 7-6-
Fraction of Energy Converted In most reactions not all the mass can be converted to energy. Actually only a very small fraction (the exact value of the fraction depends on the chemical/nuclear reaction). Reaction Matter-Antimatter Annihilation Fusion Fission Chemical Mechanical Fraction 1 0.007 0.001 1x10-10 1x10-15 Example happens at particle accelerators Power source of the Sun Nuclear power plant Burning coal Releasing a compressed spring ISP209s10 Lecture 7-7-
Some Samples A power plant generates 500 MW of electrical power and 700 MW of waste heat (plants always make more waste heat than electrical power). How many Joules of energy does the plant generate in 1 day? Data: 1 Watt = 1 Joule/s Energy (1day) = = 1200! 10 6 J s (500MW + 700MW)! seconds in a 60s! m 60m! hr 24hr! d! 1d day Electrical Energy (produced in 1day) = 1.04E14 J ISP209s10 Lecture 7-8-
More on the power plant Assume the power plant in the previous problem burns 2.2 kg of oxygen and 1 kg of carbon from coal to make 33 MJ of energy. How many kg of carbon and oxygen will the plant use in a day? total energy produced mass (kg) = = energy generated $ & kg! # % " 1.037E14 J mass (kg) = $ & 33.E6 J % ( 2.2 kg + 1.0 kg) electrical+ waste energy energy generated $ & kg! # % "! # " = 1.01E7 kg ISP209s10 Lecture 7-9-
How much of that mass was converted to energy? m E = converted m = 2 E c # mconverted = 2 c 1.04E14 J! = 1.16 " 10 converted ( m ) 2 3E8 s 3 kg But we used more than 10 7 kg (10,000 metric tons), where did it all go? Hint: The main byproduct of burning coal is C0 2. ISP209s10 Lecture 7-10-
m HW Help: How long will the Sun burn? The sun generates 3.82E24 W of power by fusion of hydrogen into helium. The fraction of mass converted for fusion is 0.007. How many kg of protons and electrons does the Sun use every second? m f E c f 3.82E24 J ( m ) 2 Get E by using 3E8 definition of power (recall, 1 sw = 16.06 J/s) E kg 0.007 2 converted burned each s = = = 9 Years Sun will last = (Total mass of the core/mass used per second) x (years/s) Note: 1 year = 3.156E+7 s ISP209s10 Lecture 7-11-
Einstein s thought experiment Here on earth, we feel the effects of gravity ISP209s10 Lecture 7-12-
Einstein s Simple Question (Deep in outer space) Would you be able to tell the Difference between this situation (where you NOT in a gravitational Field) and accelerating upwards at 9.8 m/s 2 from when you are On earth experiencing the earth s Gravity? ISP209s10 Lecture 7-13-
Another one of Einstein s simple questions The equivalence principle: No experiment performed inside a closed room can tell you whether you are at rest in the presence of gravity or accelerating in the absence of gravity. ISP209s10 Lecture 7-14-
Another one of Einstein s simple questions Scott throws a ball with an initial horizontal Velocity. The ball follows the curved path due to gravity till it hits the ground ISP209s10 Lecture 7-15-
Another one of Einstein s simple questions Throw a ball on a rocket in space that is uniformly accelerating at 9.8 m/s 2 (= 1g) Ball follows same Trajectory it did when I threw it on earth. ISP209s10 Lecture 7-16-
The Equiv. Principle => Gravity Bends Light! NOTE: You would need a HUGE acceleration >> 1g to see this effect. This is why we don t see flashlight beams falling to the floor! ISP209s10 Lecture 7-17-
General Relativity continued Main Postulate: Acceleration in one direction is like gravity in the other direction. It is not possible distinguish the two. Equivalence Principle What we perceive as gravity is really acceleration resulting from space curved and stretched by mass Mass warps space Space and time are combined into a 4-dimensional space-time ISP209s10 Lecture 7-18-
Pictorial Gravity is actually the result of warped space. What we perceive as acceleration (and hence say is due to a force) is really just stretched and curved space. ISP209s10 Lecture 7-19-
Einstein Equation 8 R # 1 Rg # " g =! ij ij ij 2 c 4 A rank-2 tensor equation that describes how spacetime is influenced by mass. Approximately, the left side is the curvature and motion of space and the right side is the location and motion of mass and energy. Rij is the Ricci tensor (curvature of spacetime), g is the metric of space-time (specifies how space and time are interrelated), G is the same constant in Newton s equation, etc. G T ISP209s10 Lecture 7-20- ij
Einstein Equations predict Black holes A huge mass in a tiny spacetime region warps it so strongly that it has infinite curvature aka a spacetime singularity develops. A hole so deep that even light can not escape. Growing experimental evidence Einstein didn t believe in them. He thought it was an artifact of the mathematics. www.ifa.hawaii.edu/~barnes/ast110_06/bhaq.html ISP209s10 Lecture 7-21-
Einstein Equation also predicts Wormholes This could be the basis for a time machine. Wormholes emerge as mathematical consequence of Einstein s Equations. Unlike Black Holes, there is ZERO evidence or hint that they are real. ISP209s10 Lecture 7-22-
Gravitational Time Dilation Mass stretches space, but since space and time are connected (space-time) it also affects time. Mathematically, this interconnection of space and time is described by the metric of spacetime. Near a mass, time runs more slowly. On the surface of the Earth this affect is only 10-9 s, but near a black hole it could be infinite! Why? As you travel through space you travel through time. Where space is stretched, time is stretched. ISP209s10 Lecture 7-23-
Proof of General Relativity I Bending of star light the gravitational field of the Sun bends star light by 1.75 arcseconds. This was observed by A. Eddington in 1919 during an eclipse. ISP209s10 Lecture 7-24-
Proof of General Relativity II Gravitational Lensing: Routinely observed and used to measure the mass of distant clusters of galaxies. ISP209s10 Lecture 7-25-
Real picture from the Hubble Telescope Abel galaxy cluster ISP209s10 Lecture 7-26-
Paradoxes in Time Travel If time is a dimension like the other three, can we move back and forth in time? Forward, yes! Think of the Twin Paradox. What about backwards in time? If we can travel back in time, it would be possible for us to influence things so that we are not born. Two theories to resolve the paradox Travel back in time is not possible. End of story There are a very large number of parallel universes (string theories) ISP209s10 Lecture 7-27-