1 The Special Theory of RelativityChapter 36 opener. An early science fantasy book (1940), called Mr Tompkins in Wonderland by physicist George Gamow, imagined a world in which the speed of light was only 10 m/s (20 mi/h). Mr Tompkins had studied relativity and when he began “speeding” on a bicycle, he “expected that he would be immediately shortened, and was very happy about it as his increasing figure had lately caused him some anxiety. To his great surprise, however, nothing happened to him or to his cycle. On the other hand, the picture around him completely changed. The streets grew shorter, the windows of the shops began to look like narrow slits, and the policeman on the corner became the thinnest man he had ever seen. ‘By Jove!’ exclaimed Mr Tompkins excitedly, ‘I see the trick now. This is where the word relativity comes in.’ ” Relativity does indeed predict that objects moving relative to us at high speed, close to the speed of light c, are shortened in length. We don’t notice it as Mr Tompkins did, because c = 3 x 108 m/s is incredibly fast. We will study length contraction, time dilation, simultaneity non-agreement, and how energy and mass are equivalent (E = mc2).

2 The Special Theory of Relativity Chapter IContradictions in physics ? Galilean Transformations of classical mechanics The effect on Maxwell’s equations – light Michelson-Morley experiment Einstein’s postulates of relativity Concepts of absolute time and simultaneity lost Chapter 36 opener. An early science fantasy book (1940), called Mr Tompkins in Wonderland by physicist George Gamow, imagined a world in which the speed of light was only 10 m/s (20 mi/h). Mr Tompkins had studied relativity and when he began “speeding” on a bicycle, he “expected that he would be immediately shortened, and was very happy about it as his increasing figure had lately caused him some anxiety. To his great surprise, however, nothing happened to him or to his cycle. On the other hand, the picture around him completely changed. The streets grew shorter, the windows of the shops began to look like narrow slits, and the policeman on the corner became the thinnest man he had ever seen. ‘By Jove!’ exclaimed Mr Tompkins excitedly, ‘I see the trick now. This is where the word relativity comes in.’ ” Relativity does indeed predict that objects moving relative to us at high speed, close to the speed of light c, are shortened in length. We don’t notice it as Mr Tompkins did, because c = 3 x 108 m/s is incredibly fast. We will study length contraction, time dilation, simultaneity non-agreement, and how energy and mass are equivalent (E = mc2).

3 Galilean–Newtonian RelativityIsaac Newton Galileo Galilei Definition of an inertial reference frame: One in which Newton’s first law is valid. v=constant if F=0 Earth is rotating and therefore not an inertial reference frame, but we can treat it as one for many purposes. A frame moving with a constant velocity with respect to an inertial reference frame is itself inertial. Relativity principle: Laws of physics are the same in all inertial frames of reference

4 Intuitions of Galilean–Newtonian RelativityWhat quantities are the same, which ones change ? Lengths of objects are invariant as they move. Time is absolute. Mass of an object in invariant in for inertial system Forces acting on a mass equal for all inertial frames Velocities are (of course) different in inertial frames (Galileo transformations) Positions of objects are different in other inertial systems (Galileo coordinate transformation)

5 Galilean TransformationsA classical (Galilean) transformation between inertial reference frames: View coordinates of point P in system S’ Figure Inertial reference frame S’ moves to the right at constant speed v with respect to frame S. Note; Inverse transformation ?

6 Galilean TransformationsIn matrix form

7 Relativity principle:The basic laws of physics are the same in all inertial reference frames Figure A coin is dropped by a person in a moving car. The upper views show the moment of the coin’s release, the lower views are a short time later. (a) In the reference frame of the car, the coin falls straight down (and the tree moves to the left). (b) In a reference frame fixed on the Earth, the coin has an initial velocity (= to car’s) and follows a curved (parabolic) path. Laws are the same, but paths may be different in reference frames

8 The domain of electromagnetism Maxwell’s equationsIntegral form Differential form Gauss Faraday Ampere/ Maxwell

9 Differential vector analysis for treating Maxwell’s equationsin Cartesian coordinates (can be done in spherical): Gradient Divergence Curl Laplacian Product (chain) rules Proof theorems on second derivatives

10 Derivation of the wave equations in vacuum (no charge, no current)Calculate:  Similarly derive:

11 Electromagnetic wave equationsNote that: is in general a “wave equation” 1855; electric and magnetic measurements Measurement of the speed of light Fizeau 1848 Foucault 1858 History of the speed of light: Maxwell:

12 Maxwell’s equations Problems:with Light is a wave with transverse polarization and speed c James Clerk Maxwell Problems: In what inertial system has light the exact velocity c What about the other inertial systems Waves are known to propagate in a medium; where is this “ether” How can light propagate in vacuum ? Laws of electrodynamics do not fit the relativity principle ?

13 Maxwell’s equations do not obey Galilei transformSimple approach: Consider light pulse emitted at time t=0; at time t>0 {x,y,z,t} in frame So: {x’,y’,z’,t’} In the moving frame Apply Galilei transform

14 Maxwell’s wave equation transformedApply it to the wave equation in (x,t) dimensions – calculate differentials (difficult ?): Calculate field derivatives using the “chain rule”: Spatial part Then also second Temporal part

15 Maxwell’s wave equation transformed IIInsert in Maxwell wave equation This is not an electromagnetic wave equation

16 The Michelson–Morley ExperimentAlbert Michelson Edward Williams Morley Nobel 1907 "for his optical precision instruments and the spectroscopic and metrological investigations carried out with their aid" Albert Abraham Michelson Questions: What is the absolute reference point of the Ether? In which direction does it move ? How fast ? Ether connected to sun (center of the universe) ? Figure The Michelson–Morley experiment. (a) Michelson interferometer. (b) Boat analogy: boat 1 goes across the stream and back; boat 2 goes downstream and back upstream (boat has speed c relative to the water). (c) Calculation of the velocity of boat (or light beam) traveling perpendicular to the current (or ether wind). Motion of the Earth Should produce an Observable effect }

17 The Michelson–Morley ExperimentNote: we adopt the classical perspective axis Figure The Michelson–Morley experiment. (a) Michelson interferometer. (b) Boat analogy: boat 1 goes across the stream and back; boat 2 goes downstream and back upstream (boat has speed c relative to the water). (c) Calculation of the velocity of boat (or light beam) traveling perpendicular to the current (or ether wind).

18 The Michelson–Morley Experimentaxis Figure The Michelson–Morley experiment. (a) Michelson interferometer. (b) Boat analogy: boat 1 goes across the stream and back; boat 2 goes downstream and back upstream (boat has speed c relative to the water). (c) Calculation of the velocity of boat (or light beam) traveling perpendicular to the current (or ether wind). Transversal motion always account for the “stream”

19 The Michelson–Morley ExperimentInterferometer: If v=0, then Dt=0 no effect on interferometer Figure The Michelson–Morley experiment. (a) Michelson interferometer. (b) Boat analogy: boat 1 goes across the stream and back; boat 2 goes downstream and back upstream (boat has speed c relative to the water). (c) Calculation of the velocity of boat (or light beam) traveling perpendicular to the current (or ether wind). If v≠0, then Dt≠0 a phase-shift introduced But this is not observed (actually difficult to observe)

20 The Michelson–Morley ExperimentRotate the interferometer Numbers: v~3x104 m/s v/c~10-4 l1~l2~11 m Approximate: Visible light: l~550 nm  f~5 x 1014 Hz Phase change (in fringes) Then: Figure The Michelson–Morley experiment. (a) Michelson interferometer. (b) Boat analogy: boat 1 goes across the stream and back; boat 2 goes downstream and back upstream (boat has speed c relative to the water). (c) Calculation of the velocity of boat (or light beam) traveling perpendicular to the current (or ether wind). Should be observable ! Detectability: 0.01 fringe

21 Conclusion: The Michelson–Morley ExperimentThis interferometer was able to measure interference shifts as small as 0.01 fringe, while the expected shift was 0.4 fringe. However, no shift was ever observed, no matter how the apparatus was rotated or what time of day or night the measurements were made. The possibility that the arms of the apparatus became slightly shortened when moving against the ether was considered by Lorentz. Hendrik A Lorentz Nobel 1902 "in recognition of the extraordinary service rendered by their researches into the influence of magnetism upon radiation phenomena" Lorentz contraction

22 Possible solutions for the ether problemThe ether is rigidly attached to Earth Rigid bodies contract and clocks slow down when moving through the ether 3. There is no ether

23 A new perspective Albert Einstein

24 On relativity Albert Einstein

25 Postulates of the Special Theory of RelativityThe laws of physics have the same form in all inertial reference frames Light propagates through empty space with speed c independent of the speed of source or observer This solves the ether problem – (there is no ether) The speed of light is the same in all inertial reference frames

26 Simultaneity One of the implications of relativity theory is that time is not absolute. Distant observers do not necessarily agree on time intervals between events, or on whether they are simultaneous or not. Why not? In relativity, an “event” is defined as occurring at a specific place and time. Let’s see how different observers would describe a specific event.

27 Simultaneity Thought experiment: lightning strikes at two separate places. One observer believes the events are simultaneous – the light has taken the same time to reach her – but another, moving with respect to the first, does not. Figure A moment after lightning strikes at points A and B, the pulses of light are traveling toward the observer O, but O “sees” the lightning only when the light reaches O.

28 Simultaneity Who is right ? From the perspective ofboth O1 and O2 they themselves see both light flashes at the same time From the perspective of O2 the observer O1 sees the light flashes from the right (B) first. Figure Thought experiment on simultaneity. In both (a) and (b) we are in the reference frame of observer O2, who sees the reference frame of O1 moving to the right. In (a), one lightning bolt strikes the two reference frames at A1 and A2, and a second lightning bolt strikes at B1 and B2. (b) A moment later, the light from the two events reaches O2 at the same time. So according to observer O2, the two bolts of lightning strike simultaneously. But in O1’s reference frame, the light from B1 has already reached O1 whereas the light from A1 has not yet reached O1. So in O1’s reference frame, the event at B1 must have preceded the event at A1. Simultaneity in time is not absolute. Who is right ?

29 Simultaneity Here, it is clear that if one observer sees the events as simultaneous, the other cannot, given that the speed of light is the same for each. Figure Thought experiment on simultaneity. In both (a) and (b) we are in the reference frame of observer O2, who sees the reference frame of O1 moving to the right. In (a), one lightning bolt strikes the two reference frames at A1 and A2, and a second lightning bolt strikes at B1 and B2. (b) A moment later, the light from the two events reaches O2 at the same time. So according to observer O2, the two bolts of lightning strike simultaneously. But in O1’s reference frame, the light from B1 has already reached O1 whereas the light from A1 has not yet reached O1. So in O1’s reference frame, the event at B1 must have preceded the event at A1. Simultaneity in time is not absolute. Conclusions: Simultaneity is not an absolute concept Time is not an absolute concept

30 Time Dilation Clocks moving relative to an observer run more slowly na) Observer in space ship proper time Clocks moving relative to an observer run more slowly b) Observer on Earth speed c is the same apparent distance longer n Light along diagonal Figure Time dilation can be shown by a thought experiment: the time it takes for light to travel across a spaceship and back is longer for the observer on Earth (b) than for the observer on the spaceship (a). This shows that moving observers must disagree on the passage of time.

31 The Special Theory of Relativity Chapter IIRelativistic Kinematics Time dilation and space travel Length contraction Lorentz transformations Paradoxes ? Chapter 36 opener. An early science fantasy book (1940), called Mr Tompkins in Wonderland by physicist George Gamow, imagined a world in which the speed of light was only 10 m/s (20 mi/h). Mr Tompkins had studied relativity and when he began “speeding” on a bicycle, he “expected that he would be immediately shortened, and was very happy about it as his increasing figure had lately caused him some anxiety. To his great surprise, however, nothing happened to him or to his cycle. On the other hand, the picture around him completely changed. The streets grew shorter, the windows of the shops began to look like narrow slits, and the policeman on the corner became the thinnest man he had ever seen. ‘By Jove!’ exclaimed Mr Tompkins excitedly, ‘I see the trick now. This is where the word relativity comes in.’ ” Relativity does indeed predict that objects moving relative to us at high speed, close to the speed of light c, are shortened in length. We don’t notice it as Mr Tompkins did, because c = 3 x 108 m/s is incredibly fast. We will study length contraction, time dilation, simultaneity non-agreement, and how energy and mass are equivalent (E = mc2).

32 Time Dilation Calculating the difference between clock “ticks,” we find that the interval in the moving frame is related to the interval in the clock’s rest frame: Dt0 is ther proper time (in the co-moving frame) It is the shortest time an observer can measure with then

33 On Space Travel 100 light years ~ 1016 mIf space ship travels at v=0.999 c then it takes ~100 years to travel. But in the rest frame of the carrier: The higher the speed the faster you get there; But not from our frame perspective !

34 Twin Paradox Question:On her 21st birthday an astronaut takes off in a rocket ship at a speed of 12/13 c. After 5 years elapsed on het watch, she turns around and heads back to rejoin with her twin brother, who stayed at home. How old is each twin at the reunion ?

35 Twin Paradox Solution:The traveling twin has traveled for 5+5=10 years so she will be 31. As viewed from earth the traveling clock has moved slower by a factor: So the time elapsed on Earth is 26 years, and her brother will be celebrating his 47th birthday. Note that the traveling twin has really spent only 10 years of her life. She has not lived more, her clock ticked slower. Time really has evolved slower.

36 Twin Paradox Where is the real paradox?Think about the problem from the perspective of sister who sees the Earth moving in her frame of reference, with the consequence that the time in her brothers frame should evolve more slowly. Why isn’t the brother “younger” ? The two twins are not equivalent ! The sister is not in an inertial frame of reference ! She is not a stationary observer. The space ship turns around which requires acceleration.

37 Length Contraction Distance between planets is: Time for travel:Earth observer Time dilatation Space craft observers measure the same speed but less time

38 Length Contraction Only observed in the direction of the motion.No contraction, or dilation in perpendicular direction Solution: a. The captain is in the rest frame of the painting, and sees it as 1.00 m tall and 1.50 m wide. b. The height is unchanged; the length is shortened to 0.65 m.

39 The Barn and Ladder ParadoxExcercise The Barn and Ladder Paradox There once was a farmer who had a ladder too long to store in his barn. He read some relativity and came up with the following idea. He instructed his daughter to run with the ladder fast, such that the ladder would Lorentz contract to fit in the barn. When through the farmer intended to slam the door and hold the ladder fixed inside. The daughter however pointed out that (in her frame of reference) the barn, and not the ladder would contract, and the fit would be even worse. Who is right ?

40 Lorentz TransformationsIn relativity, assume a linear transformation: g as a constant to be determined (g=1 classically). Inverse transformation with v  -v Consider light pulse at common origin of S and S’ at t=t’=0 measure the distance in x=ct and x’=ct’ : fill in Transformation parameter

41 Lorentz TransformationsSolve further: Leading to the transformations: Time dilation and length contraction can be derived From these Lorentz transformations

42 Test invariance of Maxwell equations under Lorentz TransformationsCoordinates Partial Derivatives Spatial, first Spatial, second Evaluate similarly the temporal term and test invariance of Maxwell’s wave equation

43 The addition of velocities in reference frames I. longitudinalExcercise The addition of velocities in reference frames I. longitudinal Observer in frame S determines speed ux of object in S’ (x’, t’,ux’) Derivatives: of coordinates Then:

44 The addition of velocities in reference frames II. TransversalExcercise The addition of velocities in reference frames II. Transversal Observer in frame S determines speed ux of object in S’ (x’, t’,ux’) Derivatives: Then: So also uy and uz transform; this has to do with the transformation (non-absoluteness) of time

45 Lorentz TransformationsExcercise Lorentz Transformations Calculate the speed of rocket 2 with respect to Earth. Figure Rocket 1 moves away at speed v = 0.60c. Rocket 2 is fired from rocket 1 with speed u’ = 0.60c. What is the speed of rocket 2 with respect to the Earth? Solution: Using the relativistic formula for velocity addition gives u = 0.88c. This equation also yields as result that c is the maximum obtainable speed (in any frame).

46 Faster than the speed of light ? Cherenkov radiationA blue light cone Cf: shock wave Pavel Cherenkov Nobel Prize 1958 Application in the ANTARES detector (example of “good thinking”) Particle travels at Waves emitted as (spherical) Emittance cone:

47 The Special Theory of Relativity Chapter IIIRelativistic dynamics Momentum and energy E=mc2 Relativistic particle scattering Chapter 36 opener. An early science fantasy book (1940), called Mr Tompkins in Wonderland by physicist George Gamow, imagined a world in which the speed of light was only 10 m/s (20 mi/h). Mr Tompkins had studied relativity and when he began “speeding” on a bicycle, he “expected that he would be immediately shortened, and was very happy about it as his increasing figure had lately caused him some anxiety. To his great surprise, however, nothing happened to him or to his cycle. On the other hand, the picture around him completely changed. The streets grew shorter, the windows of the shops began to look like narrow slits, and the policeman on the corner became the thinnest man he had ever seen. ‘By Jove!’ exclaimed Mr Tompkins excitedly, ‘I see the trick now. This is where the word relativity comes in.’ ” Relativity does indeed predict that objects moving relative to us at high speed, close to the speed of light c, are shortened in length. We don’t notice it as Mr Tompkins did, because c = 3 x 108 m/s is incredibly fast. We will study length contraction, time dilation, simultaneity non-agreement, and how energy and mass are equivalent (E = mc2).

48 Relativistic MomentumExcercise The formula for relativistic momentum can be derived by requiring that the conservation of momentum during collisions remain valid in all inertial reference frames. Note: that does NOT mean that the momentum is equal in different reference frames Figure Deriving the momentum formula. Collision as seen by observers (a) in reference frame S, (b) in reference frame S’. Result Go over this and derive !

49 Relativistic Force Newtons second law remains valid (without proof)1) For every physical law it has to be established how they transform in relativity (under Lorentz transformations) 2) Quantities (like F) not the same in reference frames

50 Relativistic accelerationThe force vector does not point in the same direction as the acceleration vector Note: in case of acceleration g is not constant

51 Relativistic Mass From the momentum:Gamma and the rest mass are combined to form the relativistic mass:

52 Relativistic Energy Work done to increase the speed of a particle from v=0 (i-state) to v=v (f state): because use So: Kinetic energy of the particle is 1) Amount of kinetic energy depends on inertial frame 2) Amount of kinetic energy reduces to classical value at low v 3) Note

53 Mass and Energy The kinetic energy Can be written as the total energy:Where the difference is the rest energy: The last equation is Einstein famous equation implying that mass is equivalent to energy The energy of a particle at rest. Note that mc2 is the same as seen from all reference frames; It is an invariant upon frame transformation

54 Mass, Energy, Momentum Energy Momentum Combining these relations givesHence also the following Is an invariant under Lorentz transformations

55 Mass, Energy, Momentum for light particlesLight particles have no “rest” mass (m=0), but they have energy hence Energy in the quantum picture Hence momentum

56 Compton scattering Before collision After collision photon electronA photon (is a light particle) collides with an electron and its energy (so its wavelength) must change !! Before collision photon electron After collision photon electron Write the momentum conservation equations along the x-coordinate and along the y-coordinate. Write the energy conservation equation. Then solve the equations and determine the wavelength l’ for angle f.

57 Compton scattering Conservation of energy Conservation of momentumAlong x: Along y: Three equations with 3 unknowns, eliminate v and q Compton scattering:

58 the effect named after him"Compton scattering Arthur Compton The Nobel Prize in Physics 1927 "for his discovery of the effect named after him" Note that lC ~ nm So the effects is not so well visible with visible light Compton performed his experiment with x-rays

59 Doppler Shift for LightThe Doppler shift for light for c=constant in all inertial frames. Two effects in relativistic Doppler 1) Moving waves + 2) Time dilation!! 1) 2) When source moving toward observer

60 Doppler Shift for LightHence, one can derive the observed frequency and wavelength: If the source and observer are moving away from each other, v changes sign. Remember: higher pitch, blue shift when moving toward each other

61 Doppler Shift for LightSpeeding through a red light. A driver claims that he did not go through a red light because the light was Doppler shifted and appeared green. Calculate the speed of a driver in order for a red light to appear green. l = 500nm ; l0 =650 nm v = 0.26 c

62 Aspects of the General Theory of Relativity Chapter IVHow does gravity act Cosmological redshift Gravitational redshift Black holes Chapter 36 opener. An early science fantasy book (1940), called Mr Tompkins in Wonderland by physicist George Gamow, imagined a world in which the speed of light was only 10 m/s (20 mi/h). Mr Tompkins had studied relativity and when he began “speeding” on a bicycle, he “expected that he would be immediately shortened, and was very happy about it as his increasing figure had lately caused him some anxiety. To his great surprise, however, nothing happened to him or to his cycle. On the other hand, the picture around him completely changed. The streets grew shorter, the windows of the shops began to look like narrow slits, and the policeman on the corner became the thinnest man he had ever seen. ‘By Jove!’ exclaimed Mr Tompkins excitedly, ‘I see the trick now. This is where the word relativity comes in.’ ” Relativity does indeed predict that objects moving relative to us at high speed, close to the speed of light c, are shortened in length. We don’t notice it as Mr Tompkins did, because c = 3 x 108 m/s is incredibly fast. We will study length contraction, time dilation, simultaneity non-agreement, and how energy and mass are equivalent (E = mc2).

63 General Relativity: Gravity and the Curvature of SpaceA light beam will be bent either by a gravitational field or by acceleration (outside observer): Definition of a straight line; The line that a light ray follows

64 An outlook on General RelativityGR deals with: Gravitation Acceleration Principle of equivalence: it is impossible to distinguish a uniform gravitational field and a uniform acceleration. Another way to put it: mass in Newton’s first law is the same as the mass in the universal law of gravitation.

65 General Relativity: Gravity and the Curvature of SpaceThis can make stars appear to move when we view them past a massive object: Note: The bending of light can in principle be explained by Newtons law (Soldner in 1801) The difference is quantitative; a factor of 2, measured by Eddington in 1919.

66 Gravity and the Curvature of SpaceGravitational lensing This bending of light as it passes a massive object (star or galaxy) has been observed by telescopes: Fermat’s principle in optics: light traveling between points chooses the shortest track

67 Gravity and the Curvature of SpaceEinstein’s general theory of relativity says that space itself is curved – hard to visualize in three dimensions! This is a two-dimensional space with positive curvature: Not known what the overall curvature of the universe is (but close to zero) NB; most curvature is local !

68 Gravity and the Curvature of SpaceSpace is curved around massive objects: Fundamental notion of GR: gravity is not a force but deformation of space calculation is difficult, because of non-Euclidean geometry

69 Cosmological Red ShiftEdwin Hubble Scale factor: Cosmological redshift Galaxies moving away from each other Expansion of the universe Interpretations: Change of the underlying metric in expanding universe Interpretation as a Doppler shift Note: Redshift does not have dispersion

70 Expansion of the Universe“Hubble measurements” H = 71 km/s/Mpc = 22 km/s/Mly How much time did it take for galaxies to be separated at distance d Assuming they depart with Hubble speed v = Hd This corresponds to ~14 billion years Note: accelerated expansion of the Universe

71 Gravitational Red ShiftIn General Relativity it is time that depends on the gravitational dependence. This is at the heart of an explanation of gravitational redshift – it is a gravitational Time dilation. But it can be understood as an “energy loss in a gravity field”. At the surface of a heavy object and hn M R Photons loose energy when traveling “uphill” Photons shift their energy to the red

72 Simple (Newtonian) view on a Black HoleEscape from a distance Rs with an escape velocity Requirement: Kinetic energy must beat the gravitational potential Rs Ve Take c for the escape velocity (of course not correct but some approx) Schwarzschild radius (also valid in GR) Laplace (1795): “possibly the greatest luminous bodies are invisible”

73 Search for varying constantsIntermezzo Search for varying constants in the early Universe Compare the absorption spectrum of H2 in different epochs Each line is redshifted Lab today QSO 12 Gyr ago Redshift & time nm ~ nm Spectral lines of a molecule depend on :

74 Search for varying constantsIntermezzo Search for varying constants in the early Universe Astronomical spectra For high z Laboratory spectra For z=0 Make a comparison

75 Quasar Q1441+272 ; the most distantIntermezzo Quasar Q ; the most distant At zabs = 4.22 ; 1.5 Gyrs after the Big Bang spectrum Result Important: Knowledge from Molecular Physics Ki values different for all spectral lines Molecules are sensitive for the fundamental constants

76 A Stringent Limit on a Drifting Proton-to-Electron Mass Ratio Effelsberg Radio Telescope PKS “molecular factory” at z= (7.5 Gyrs look-back) A Stringent Limit on a Drifting Proton-to-Electron Mass Ratio from Alcohol in the Early Universe Bagdonaite, Jansen, Henkel, Bethlem, Menten, Ubachs, Science 339 (2013) 46 Intermezzo K=-33 K=-1 K=-7