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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)
- Laws of physics are “frame invariant”
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Galilean Transformation
- It’s about considering a new frame of reference.
- Constant Galilean velocity.
- This is a straightforward calculation in vector algebra.
- It’s about considering a new frame of reference.
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However, this actually breaks down.
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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.
- Maxwell Equations
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=> Einstein: Galilean Relativity holds even with Maxwell Equations.
- i.e. the “law of constant speed of light.”
- He demonstrated the reason for this.
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Transformation of Frames
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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.
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Let’s consider what can be derived under the conditions .
- What was the reason for setting up this equation?
- Why was it possible to exclude the possibility of dependence on ?
- From the condition of Galilean Transformation , these can be derived.
- From the observational fact that remains invariant after transformation, these can be derived.
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Pay attention to the difference between C and c.
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Oh, interesting!
- As a result, B, C, and D can be expressed in terms of A.
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- What is again?#question
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- Then, let’s consider Condition 4: .
- 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.
- 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.
- What was the reason for setting up this equation?
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8430 mistake
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is invariant
- This result is actually important.
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- 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.
- As an image, τ = the distance between oneself and the light moving away from oneself
- Here, is τ the length dimension?
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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.
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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.
- It doesn’t quite fit, but there may be no need to force it.
- Relativistic Interval
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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)
- Considering γB, if it goes beyond the range of (-1,1), the denominator becomes imaginary.
- Why does β’x become 1 when βx = 1? (blu3mo)(blu3mo)
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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.
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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”?
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- Timelike and spacelike:
- ❓What do these terms mean?
- Timelike and spacelike:
- 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
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The paradox of a barn fitting into a pole depending on the frame.
- The pole is Lorentz-contracted.
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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.
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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.
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- ❓I don’t understand the meaning of this.
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- 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.
- Objects that do not have the same t and x in one frame do not influence each other.
Reexamining the LT
10.7.3 Particle rapidity
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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)
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- These resemble the Lorentz transformations of time and space.
- ❓I want to understand it. (blu3mo)
- These resemble the Lorentz transformations of time and space.
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It would be nice if t and x could be treated as quantities of the same dimension.
- ❓Why would that be nice? (blu3mo)
- 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.
- At t=0, there is no boost or anything, so why is it determined?
- Why…?