Relativity and Cosmology Einstein and Beyond
Relativity and Cosmology Einstein and Beyond PHY 312
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Date Created: 10/21/15
Lecture 7 Problems with Special Relativity and Free Fall Frames 0 So far have discussed Special Relativity Only inertial ob servers allowed We require laws of Physics to be same for all these guys We would like to generalize these ideas to cope with accelerating frames Why We actually don t know really how to find frames which just move at constant velocity in fact we know that our usual Earth bound frame is certainly not moving uniformly maybe frames of reference a long way out on space will do as examples but they are not of much practical use Worse still there are definitely accelerating FOR for which test particles move at uniform velocity Consider an elevator which is falling freely under Earth s gravity If we throw a coin in such an elevator it will appear to be moving in a straight line at constant speed just as in an inertial frame So it seems for logical consistency we should include accelerating frames of reference in our theory Will see that this necessarily involves us discussing gravity 0 Newtonian Gravity Quick recap 0 Universal attractive force of gravity acts between all bodies F GN is Newton s constant GN 667gtlt lO llmgkg1s 2 ma is the gravitational mass You should think of it as an analogous to the electric charge in electromstatics specifies the strength of the coupling between the particle and the gravita tional electric field Gravity is however a much weaker force then electromagnetism For example the ratio of gravitational to electric forces between two electrons FG F E N 1022 Thus at short distances electric forces dominate only at large scales does it become important because unlike electricity mass is always positive no cancellation or screening occurs 0 Gives a good description of our Solar system eg can compute orbits of planets using this and Newton s second law v2 GilI E Z R2 1 This leads to Kepler s law 4W2 2 GilI 5 2 o Often useful to introduce the notion of a gravitational eld Each massive object produces a field of force around it When a test mass is placed in the field it feels a force given by the field strength times its mass The field strength is just the acceleration felt by any object placed at that position 0 VVhat s wrong with this picture o Newton s theory requires an instantaneous gravitational force in con ict with special relativity which states that no physical disturbance or interaction can propagate with a speed greater than the speed of light 0 This is the most obvious reason that gravity must be left out of the special theory of relativity SR 0 Notice also an additional curiosity the gravitational mass ma is equal to the inertial mass m1 the mass that appears on the RHS of Newton s second law It is this fact that allows the free falling frame of reference mentioned above to approximate an inertial frame When we studied Special Relativity we had to neglect gravity Why well we know that if we watch free bodies from the surface of the Earth they don t move in straight lines but follow curved trajectories which we attribute to the force of gravity So frames of reference on the Earth are not inertial notice the following fact But again When we watch a body moving under gravity from a freely falling frame it obeys Newton s first law uniform velocity To see this let us imagine watching the coordinates of a particle moving under gravity from such a freely falling frame of reference Denote the coordinates of the particle in the Earth frame mpg those of the freely falling frame eg the elevator relative to the Earth mpg and the coordinates of the particle in the freely falling frame as IL pF Clearly we can use Galilean rule here since velocities are all small 3 DUFF 1TH 175 But from Newton s 2nd law 22 I FE G mF ng 22 I PE G mp ng Thus d2 DUFF 0 4 CW Where we have used the observed equality of the inertial and gravitational masses Thus although the particle would follow a curved parabolic trajectory relative to the Earth its motion from a freely falling frame looks uniform Why is this Both lift and coin are falling with the same acceleration in the gravitational field of the Earth Thus their relative motion is simple uniform in fact l This observation has nothing to do with which body I choose to observe neglecting air resistance since all bodies feel same acceleration due to gravity This is key Free falling frames are natural they are the equivalent con struct to inertial frames in SR lndeed motions of bodies can be computed using SR special relativity within a EFF free fall frame Thus the effects of gravity can be almost entirely eliminated by climbing into a freely falling frame Note how this depends on the equality of gravitational and inertial mass This equality had been long known but it was only Einstein who realized its significance This leads to the first statement of the principle of equivalence There are no local experiments which can distinguish free fall in a gravitational eld from uniform motion in the absence of a gravitational eld Turn this around Put the elevator out in empty space and make it accelerate Instead of oating observer s feet are now firmly pressed against the oor And the coin again follows a curved trajectory relative to the accelerating lift frame It looks like the lift is in a gravitational field Second statement of the Principle of Equivalence An accelerated frame relative to an inertial frame is locally identical to a frame at rest in a gravita tional eld Tidal Gravity and Locality What do I mean by locally lmagine our lift again falling in the gravitational field of the Earth Two ball bearings are released from rest at each side say take it to be 20 m across After falling for 8 secs and 315 m the balls will have moved slightly closer together 1 Check this The balls head along trajectories which are not quite parallel towards the center of the Earth This would not happen in a frame which was just undergoing uniform acceleration So there is a locally detectable difference between acceleration and gravity Real gravitational fields vary in space and this effect leads to slightly different accelerations for nearby objects so called tidal gravity It is tidal gravity that is responsible for the tides on Earth as water at different points on the Earth s surface responds to slightly different gravitational pulls from the Moon Thus ifl have sensitive instruments 1 can detect such a difference and say hey this is gravity not just acceleration But notice that if I drop the lift for a shorter time or put the ball bearings closer together this effect is smaller indeed I can always find a region of spacetime over which these tidal effects can be neglected This is what we mean by the equivalence being local over sufficiently small regions of spacetime the tidal effects can be neglected and we can always get rid of gravity by jumping to a locally freely falling frame of reference 0 Now place ball bearings vertically apart and drop again you should fine that they are 2mm apart at end Tidal gravity stretches things vertically and squashes them horizontally Consequences of the principle of equivalence 0 Light is bent by gravity lmagine shooting a light ray in the accelerating lift its trajectory will be bent since the lift moves up at increasing speeds as it travels horizontally But by the POE this means that light is bent in a gravitational field Does this make sense Well in SR we know energy of photon can be thought of as mass so indeed we would expect that light can be bent by gravity For a light ray grazing the Sun GIMS A N R502 5 0 Observed one of the first tests of GR 0 Clocks are slowed in a gravitational field Imagine a clock which is placed in a frame of reference which is accelerating at a con stant rate For a small interval of time dt in our frame we can treat the velocity of the clock as constant and find that the ratio of the small element of elapsed proper time d r time measured on the clock to the time elapsed in our frame is given by the usual time dilation result dr 2 dt 1 009202 Lets assume vt 2 at constant acceleration and expand the square root factor for small 00 We can now add up all the elements of proper time for a finite time interval by integration We get T2a2 39r t 1 602 But the distance gone by the clock in the nonrelativistic limit is just A5 aT2 so we nd Now by the principle of equivalence we should be able to swap the inertial acceleration for a gravitational acceleration 9 Thus we would expect that a clock at rest in a gravitational eld should be slowed down by an amount 1 But what is As To gure this out lets think of the gravitational eld of the spherical body like the Earth Then at radius R we expect 9 GillR Furthermore from our considerations of locality we know we can only trade acceleration for gravity if the distance gone is small compared to the distance scale over which the gravitational eld varies Thus As 3 R Thus the slowing of clocks in such a gravitational eld we expect to be governed by approximately the formula r Gili t c2 B This turns out to be roughly right Gravitational red shift Consider a photon emitted from from distance R from the center of a spherically symmetric gravita tional eld like the Earth Since it has a mass by virtue of E c2 it has an initial potential energy ln escaping from this eld it must lose an equal amount of kinetic energy Thus we expect that AE N Gili c2 R Notice the same factor of Gridc2 For a photon E h f where f is the frequency and h is called Planck s constant Thus we expect that the frequency of such a photon is reduced by an amount which depends on the gravitational acceleration at the point at which it is produced This result is consistent with our earlier result concerning the slowing of clocks since I can think of the temporal oscillations of a light wave as consistuting a type of clock and if clocks slow so should the oscillations of light waves Notice that the light is shifted to longer wavelengths hence the term red shift since the red part of the visible spectrum is at longer wavelength than the rest of the visible light spectrum Gravity curves spacetime 0 We have seen using the principle of equivalence that light appears bent by gravitational fields and that clocks are slowed by gravity We have also seen that freely falling frames of reference remove the gross effects of gravity leaving only tidal gravity as a physical phenomenon These tidal gravitational effects can be measured by watching the motion of test particles in a sufficiently large volume of spacetime Furthermore these tidal motions do not depend on the nature of our test particle mass charge etc etc We might ask whether there is some other way of picturing these tidal effects which makes the action of gravity natural A couple more thought experiments will lead us in the right direction indicating that I can think of gravity not only dilating time but also deforming space This will then allow us to think of tidal gravity purely in terms of a deformation of spacetime Lets see how this goes Consider a disk rotating at high speed relative to an inertial frame An observer outside the disk would measure its diameter to be d and its circumference to be C using a ruler at rest He would observe that the ratio of C to d was 71quot in accord with ordinary Euclidean geometry However a man on teh disk would disagree using the same ruler he would read off the same diameter d but would see the circumference as less than C since the ruler would be contracted in the direction of its velocity which is tangential He would see that C was not 7rd l This might indicate to him that the geometry of the disk was not Euclidean at but curved Similarly the two observers would disagree on the time for the disk to rotate decause of the usual time dilation effect But by the principle of equivalence this accelerating disk should just be like a regular disk in a gravitational field Thus gravity in uences space and time in particular it can cause space to be nonEuclidean or curved l How can curved spacetime be like gravity Consider an analogy the surface of a sphere for example the Earth If I draw a circle on the surface of such a sphere I would indeed find that C d were less than 71quot It is an example of a curved or nonEuclidean geometry Lets see what happens when we try to move freely over this surface Two observers start a small distance EVV apart on the equator and head N As they go further they notice that their EVV dis tance is decreasing Each observer is desperately trying to keep going in a straight line but inexorably they nevertheless approach each other eventually colliding at the North Pole this is just