Physics 1600 10. Special relativity I

• Theory of Relativity

• To summarize briefly,

  • Laws of physics are “frame invariant”
    • This can be explained by assuming that the speed of light is constant.
  • Ah, indeed, this is the theory of relativity. I understand the name now. (blu3mo)(blu3mo)

• Galilean Transformation

  • It’s about considering a new frame of reference.
    • Constant Galilean velocity.
  • This is a straightforward calculation in vector algebra.

• However, this actually breaks down.

  • 

• From another perspective, the Laws of Electromagnetism

  • Maxwell Equations
    • The terms in the equations involve the speed of light, c.
  • This contradicts the principle of “laws don’t change when the frame is changed” in Newtonian physics (Galilean Relativity).
    • I see! I understand now.
  • To resolve this, Lorentz Transformation and others were developed.
    • However, the reasons and theoretical background for this were not understood.

• => Einstein: Galilean Relativity holds even with Maxwell Equations.

  • i.e. the “law of constant speed of light.”
  • He demonstrated the reason for this.

• Transformation of Frames

  • As a premise, it is necessary to have Linear Transformation.

    • It means that once you change and then revert, you return to the same frame.

  • Let’s consider what can be derived under the conditions $t^{\prime }=At+Bx{x}^{\prime }=Ct+Dx$.

    • What was the reason for setting up this equation?
      • Why was it possible to exclude the possibility of dependence on $x^2$?
    • From the condition of Galilean Transformation $x=v_{G}t$, these can be derived.
      • Files／Screenshot 2022-12-13 at 10.56.40 AM.png
      • Files／Screenshot 2022-12-13 at 10.59.17 AM.png
    • From the observational fact that $c$ remains invariant after transformation, these can be derived.

      • Files／Screenshot 2022-12-13 at 11.08.22 AM.png

      • 

      • Pay attention to the difference between C and c.

      • Oh, interesting!

    • As a result, B, C, and D can be expressed in terms of A.
      • $t^{\prime }=A(t-\frac{v_B}{c^2}x)$
        • What is $v_B$ again?#question
      • $x^{\prime }=A(t-v_G)$
    • Then, let’s consider Condition 4: $t^{\prime \prime }=t$.
      • This is a condition that it would be strange if it didn’t return to the original after transformation and inverse transformation.
    • Finally, the Lorentz Transformation can be obtained.
      • Files／Screenshot 2022-12-13 at 11.27.58 AM.png
    • B: Boost
      • It represents a change in frame.
    • y=1/sqrt(1-x^2)
      • Ah, it makes sense that it asymptotes to 1 for both x=1 and y=1.

  • 8430 mistake

  • 

  • $c^2{t}^2-x^2$ is invariant

    • This result is actually important.
    • 
      • Here, is τ the length dimension?
        • As an image, τ = the distance between oneself and the light moving away from oneself
          • If x at a certain t in a frame, then x’ at a different t’ in another frame.

  • 

  • From this, it is necessary to understand that the magnitude of the velocity of any particle must always be less than the speed of light.

  • Hmm, I still don’t quite understand the feeling of the graph.

    • What does “timelike” and “spacelike” mean?
    • terminology - Space-like and time-like: where do the names come from? - Physics Stack Exchange
      • It doesn’t quite fit, but there may be no need to force it.
        • Maybe I’ll understand it later.
    • Relativistic Interval

10.4 Transformation of velocities

• I’m not sure about this part, so I’ll review it. (blu3mo)(blu3mo)
  • Why does β’x become 1 when βx = 1? (blu3mo)(blu3mo)
    • It’s obvious when you look at it, but it’s important to derive it.
  • What about when βB → 1?
    • If βx was not originally 1, it approaches -1.
    • If βx was originally 1, does it become 0?
  • Or rather, can boost exceed 1?
    • Considering γB, if it goes beyond the range of (-1,1), the denominator becomes imaginary.
      • I see! I understand now. (blu3mo)

• Important: there is no prohibition in SR against relative velocities having magnitudes greater than unity.

  • However, the maximum is 2c, right?
• In the case of perpendicular velocities,- The LT of Δy does not cause any change.
  • Since there is no change in the y-direction of the frame, γ=1.
• The LT of Δt changes due to the boost in the x-direction.
• After rearranging the equations, it takes the form of “まあせやな” (a local expression).

10.5.1 Observations, measurements, “clocks and rulers”

• How to make measurements:
  • When moving between frames, both t and x change.
  • Event -> (t, x, y, z)
    • When capturing an event, all four parameters are necessary to avoid ambiguity.
  • For example, when considering a time interval, x also needs to be taken into account.
    • Vice versa.
  • The concept of local position/local time:
    • While the occurrence of an event can be agreed upon between frames, the location and time cannot be agreed upon.

  • It is important to understand that the differences in time and length scales between inertial frames do not result from changes in dimensionful scales associated with fundamental interactions.

    • ?(blu3mo)
    • Physical values such as the vibration frequency of a crystal are measured in that specific frame.
    • Therefore, they are frame-dependent and change when the frame is changed. Is it to caution against the misconception that “it’s just a change in the unit of measurement on the observer’s side”?

  • 

    • Timelike and spacelike:
      • ❓What do these terms mean?
• Time dilation:
  • ❓In the rest frame of the particle => Δx = 0, is it a frame where the particle appears to be stationary?
• Lorentz Contraction:
  • ❓(No further information provided)

10.5.5 Simultaneity and lack thereof

• The paradox of a barn fitting into a pole depending on the frame.

  • The pole is Lorentz-contracted.

• The important point here is that simultaneity changes depending on the frame.

  • ❓I don’t understand it at all, so let’s move on when I understand it.

• Causality:

  • Objects that do not have the same t and x in one frame do not influence each other.
    • ＝ Information is not transmitted faster than the speed of light?
  • This contradicts gravitational force and other phenomena.
  • 
    • ❓I don’t understand the meaning of this.
    • 
      • When there are two 4-vector positions, if the difference in their Lorentz-invariant length is less than or equal to 0.
      • I still don’t understand it, I want to confirm.

Reexamining the LT

• 

10.7.3 Particle rapidity

• Is rapidity (y) additive in a normal sense?

  • When there are two transformations, the result of performing both of them and the result of adding their rapidities are the same.
  • That’s interesting. (blu3mo)

• 

  • These resemble the Lorentz transformations of time and space.
    • ❓I want to understand it. (blu3mo)

• It would be nice if t and x could be treated as quantities of the same dimension.

  • ❓Why would that be nice? (blu3mo)

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• HW1
  • 
  • Why…?
    • At t=0, there is no boost or anything, so why is it determined?
      • Boost is not about the movement of the object, but about the movement of the frame (i.e., a problem of reference frames), so it doesn’t matter.