SpacetimePhysics

Spacetime Physics (Taylor and Wheeler)

  • Great book! Probably the best introduction to relativity I've seen. Close analogy between change of geometric coordinates. Primacy on the interval
  • The invariance of the interval – its independence from the choice of the reference frame – forces one to recognize that time cannot be separated from space. Space and time are part of a single entity: spacetime. The geometry of spacetime is truly four dimensional. "The direction of the time axis" depends on the state of motion of the observer, just as the direction of the y-axis depends on the angle of the observer.
  • The principles of relativity are remarkably simple. (Much simpler than the axioms of Euclid or the principles of operating a car, yet both of these have been mastered. Some of the greatest minds of the 20C struggled with the concepts of relativity. Not because it was obscure, but because man finds it difficult to outgrow established ways.
  • Principle of Relativity: All the laws of physics are the same in every inertial reference frame. Both the form of the laws and the numerical values of the physical constants that these laws contain is the same in every inertial reference frame.
  • We could go about locating events in spacetime solely by measuring the intervals between one event and the next, without any regard for the "space" and "time" coordinates individually. However we have to start as physics did before 1905 without the benefit of any concept of interval. This concept will force itself upon us as distance does for surveyors.
  • One meter of light is 3.3 nanoseconds or light goes ~0.3 m = 1ft in a nanosecond.
  • Synching clocks with a reference light flash. Can also move around a portable clock (However as we will see, the moving clock runs at a different rate, so it wont even agree with the original when it is brought back to the start)
  • The word observer is a shorthand way of speaking about the whole collection of recording clocks associated with one inertial frame of reference.
  • On deriving the interval: To say that dt2 = dxx has a standard value is to say that the altitude of this right triangle has a fixed magnitude no matter how fast the rocket is moving. The keystone was the principle of relativity. (no difference in laws of physics between reference frames) Was put to use twice in different ways: -) Used to reason that distances at right angles to the directions of relative motion are equal in the two frames. -) To deduce that their speed of light must be the same in the laboratory and rocket frame.
  • The discrepancy between clocks is caused by the character of the spacetime geometry itself.
  • Proper time is to a world line in Lorentz geometry what length is to a path in Euclidean geometry. If the path between to points is "curved" it will take a longer distance / interval. In Lorentz geometry a curved world line between two events is shorter than the direct world line.
  • Observers in different frames will all agree on the proper time along a given world line between two specified points.
  • Spacetime is the arena in which stars, atoms and people live and move and have their being. Space is different for different observers. Time is different for different observers. Spacetime is the same for everyone.
  • There is a minus sign in the formula (for the interval) that no sleight of hand can conjure away. This minus sign marks the differences in character between space and time.
  • Light Cones: interval is zero between two events that can be connected by a ray of light. The light cone is a unique feature of Lorentz geometry: no Euclidean equivalent. Defines fundamental boundaries to the basic causal structure of the universe. Light cones have existence quite apart from any coordinates that may be used to describe them. The causal structure is this independent of (and thus preserved in) all reference frames.
  • Coordinates provide a "co-variant" description of the separation of events in spacetime. vs invariant of the interval. "Co" prefix implies a "coordinated variation of the coordinates..
  • Lorentz Transformations:
    • The coefficients in the transformation are independent of the event under consideration.
    • The consideration guarantee a point at rest in the rocket frame moves with velocity beta in the lab frame.
    • They also guarantee the invariance of the interval.
  • The co-variant transformation in Euclidean geometry is analogous to the coordinate transformation in Lorentz geometry of the real physical world. The inverse slope of the axis of one coordinate system relative to the other is analogous to the velocity of one inertial reference frame relative to another.

  • The velocity parameter (like the relative slope between frame in the Euclidean analogy) is not the simplest way to describe the relation between coordinate frames. Angle is a better measure of rotation and velocity parameter eta is the most convenient measure of velocity Why is an angle more convenient than a slope? because angles are additive. Complicated addition law for slopes (analogous to the velocity addition law for relativity)

    Velocity itself is also not the simplest measure of velocity. Velocity does not satisfy a simple law of addition.

    A photon shot with the speed of light form a fast rocket travels relative to the laboratory with a speed that is also the speed of light. In the language of velocity parameters this is natural: etagamma = infinity, etarocket = large. infinity + large = infinity

  • The distinction between the sign in cos2 + sin2 = 1 and cosh2 - sinh2 =1 has to do with the contrast between the length in euclidean and lorentzian geometry.
  • p58 the simplest demonstration that a Lorentz transformation preserves the invariance of the interval. Directly connected to cosh2 - sinh2 =1
  • Wheelers first moral principle: Never make a calculation until you know the answer. Make an estimation before every calculation, try a simple physical argument (symmetry! invariance! conservation!) before every derivation, guess the answer to every puzzle. Courage: no one else needs to know what the guess is. Make it quickly by instinct. A right guess reinforces the instinct. A wrong guess brings the refreshment of surprise. In either case life is more fun!

  • No one has ever found a change in velocity without something to cause it. Therefore a force can be said to be some form of interaction. -) When the presence of A brings about a change in the velocity of B, the presence of B also causes an alteration in the velocity of A

    Thus instead of talking about the forces between particles we can talk about their changes in momentum. Indeed to discuss both momentum and forces in the context of relativity is to complicate the story. In this account we will focus on momentum.

  • What makes classical momentum useful is that it is conserved in low-velocity collisions. Observations show that mv is not conserved in high-velocity collisions. We must choose to either abandon the definition of momentum or the law of conservation of momentum.
  • We will start with the law of conservation of momentum and from it derive the expression for the momentum defined as that vector quantity which is conserved in all frames of reference. We will use it three times each will produce a revolution in our way of looking at nature. -) Will use glancing elastic collision in two dimensions to derive the relativistic expression for momentum -) applied to 1D collisions to derive the expression for the relativistic energy of a particle. -) conservation will be applied to an inelastic collision to derive the equivalence of mass and energy.

  • All the laws and theories of physics have a deep subtle character, in that they both define for us the needed concepts and make statements about those concepts. Contrast with the statement "Define your terms before you proceed"!

    Physics provides a way to harmonize the experimental facts. No single experiment suffices to establish a conservation law. At least two are needed, one to establish the definition of the conserved quantity, the other to verify that this quantity is really conserved.

  • In an inertial frame, space is the same in all directions, isotropic. So the only unique direction associated with the motion of the particle that moves is a straight line is the direction in which the particle is moving.

    We assume that the momentum is determined uniquely, both magnitude and direction, by its momentum.

    Conclusion, the momentum vector of a particle points in the direction of its velocity.

    requiring that the total momentum be conserved in an elastic collision, coupled with invariance of the interval in Lorentz geometry will be enough to show that the Newtonain expression for momentum p = mass x (displacement per unit time) will have to be replaced by the relativistic formula p = mass x (displacement per unit of proper time)

  • Figure 82) if the collision pictured is elastic, then each ball changes its direction of motion but not (except temporarily) its speed; the overall effect of the collision is to rotate the velocity vectors of both particles.

    The x-direction of the reference frame can be chosen to lie in a direction such that the x component of the velocity of each particle does not change in the collision.

    The y direction of each particle reverses direction.

    We will study the total y-component of momentum and its conservation in this collision. To do this most simply observe the collision in a frame in which ball A moves only in the y direction.

    Objective is to find the momentum of a particle – which may be moving near c – from what we know from Newtonian physics about the momentum of a particle with low velocity. To do this look at a glancing collision where the struck particle has an arbitrarily low velocity before and after the collision and therefore the Newtonian expression holds p = mv

    From the symmetry of the diagram: the momentum transferred is twice the momentum before it is struck 2 m dy/dt. This momentum transfer is from a change in direction of the momentum vector, not from a change in magnitude.

  • Momentum is like displacement in that the transverse components of both vectors are unaffected by the motion of the observer. Good reason for this! momentum is obtained from displacement by multiplication with (m d/dtau), which has the same value in all inertial frames.
  • Note: we attribute all the difference between Newtonian and Relativistic momentum in terms of the d/dt -> d/dtau not any change in m. (Some older discussion talk as if m changes) However the most useful quantities in physics are often those, such as m and dtau that are invariants.

  • Newtonian momentum = m Beta = m tanh(eta) Relativistic momentum = gamma m Beta = m sinh(eta)

    The difference (gamma) is the same factor as the time dilation factor between laboratory time and proper time.

    The presence of this factor implies that a particle can transport an arbitrarily large amount of momentum into a collision process if it travel arbitrarily close to the speed of light. One would not expect this if use used the incorrect, Newtonian, formula. where the momentum would be bounded by mc.

  • Despite the arbitrariness in the means that we use to express a four vector AB, the 4-vector itself is well defined. Not only does the interval AB have the same magnitude in all frames. More significantly, the locations of events are we well defined as the location of two town gates, regardless of what coordinate frame one uses, or even if you use coordinates at all.

  • The connection of the energy-momentum 4-vector with the displacement four-vector is direct.

    1. The displacement four vector is (dt, dx, dy, dz)
    2. The unit vector obtained by dividing the four vector by the proper interval dtau = sqrt(dt2 - dx2 - dy2 - dz2) gives (dt/dtau, dx/dtau, dy/dtau, dz/dtau)
    3. the energy-momentum 4-vector is obtained by multiplying by the constant m.

    The original four-vector remains a four-vector after multiplication by invariants 1/dtau and m.

  • Why can we call the first component of the momentum four-vector "Energy"
    1. Has the right units
    2. The total time component is conserved in all collisions

  • p115 We draw three conclusions from this analysis

    -) we can assign to any particle of mass m a "relativistic energy" E=m(dt/dtau) -) when there are several particles in free motion, the relativistic energy of the system is the sum of the relativistic energies of the separate particles. -) when these particles collide and separate, which changes in the individual energies of the particles, the total relativistic energy of the system is conserved

  • It is impossible to have the laws of conservation of momentum and energy upheld in all inertial frames unless the rest energy is included in the bookkeeping of the energy in every frame.
  • E = m(dt/dtau) = m (gamma dt/dt)
  • The increase in rest mass measures precisely the energy that has gone into heat and whirling and any other forms of internal excitation of the final system.

  • The velocity and the velocity parameter are almost never used in problems involving the momentum and energy of particles moving at relativistic speeds. It is inconvenient to deal with beta since it enters the expressions for momentum and energy in the square root. More important very small change in speed can lead to a an immense change in the momentum and energy of a particle moving close to c.

    If you need to get beta out Beta = p /E

    1 - beta = m2 / 2E2 (from (m/E)2 = 1 - beta2 = (1-beta)(1+beta) ~ 2(1-beta)

  • p150 Radiation from one side of closed box to the other

    The simplicity of this derivation and the importance of the result makes this analysis one of the most interesting in all of physics. The mass equivalence of radiant energy implies the mass equivalence of thermal energy and by extension - of other forms of energy.

    Whatever the details of the mechanism by which light is emitted and absorbed the net effect is the transfer of heat energy from one end of the box to the other. To say that mass had to pass down the length of the box when radiation went from one wall to the other then implies that mass moves when thermal energy changes location.

  • In 1923, Arthur Compton showed that X-rays scattered by free electrons have less energy after the scattering process than before. His experiment is rated by many as the most important done in the 1920s At the time there was great resistance to the idea that when the electron is "shaken" by the electric field of the photon at one frequency it should scatter this radiation at a lower frequency.

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