What’s gone wrong with my derivation of the Lorentz Factor?

In summary, the article discusses common mistakes and misconceptions that can arise when deriving the Lorentz Factor, which is essential in understanding time dilation and length contraction in special relativity. It emphasizes the importance of the correct application of the principles of relativity, proper handling of geometric interpretations, and the role of light's constancy in ensuring the derivation is accurate. The piece highlights that errors often stem from neglecting the relative motion of observers or misapplying the Pythagorean theorem in spacetime diagrams.
  • #1
ryan2
4
0
Homework Statement
Trying to derive the Lorentz factor
Relevant Equations
Lorentz Factor
https://ibb.co/Wy9Pq8j

I’ve gotten something that looks almost correct, but the expression in the root is 1 + v^2/c^2 instead of minus. I understand intuitively why my answer is wrong, but don’t know where mathematically. Could someone please help me see where I went wrong? Thank you!

1722093346651.png
 
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  • #2
If any clarifications are needed, I’m more than happy to make them! I understand my work isn’t the cleanest looking.
 
  • #3
ryan2 said:
If any clarifications are needed
You might reveal which variable is in which frame of reference ...

1722087886445.png
 
  • #4
BvU said:
You might reveal which variable is in which frame of reference ...

View attachment 349017
The left hand side pertains to the stationary one, and the right to the moving one. The variable x represents the Lorentz factor(it should have been on the right). D1 is common between the two, and I simplified it to c*t.
 
  • #5
ryan2 said:
I simplified it to c*t.
What is t ?
 
  • #6
Sagittarius A-Star said:
What is t ?
The time it takes for light to make it from the bottom to the top of the vertical distance, from a stationary frame. Is it incorrect to assume it would remain the same for the moving frame?
 
  • #7
ryan2 said:
The time it takes for light to make it from the bottom to the top of the vertical distance, from a stationary frame. Is it incorrect to assume it would remain the same for the moving frame?
It is incorrect.

In your first (left) diagram, light travels a distance D at speed v [edit - that should of course say 'at speed c'] So you can find how long it takes.

In your second (right) diagram, light travels a distance bigger than D - but at the same speed (since the speed of light is constant). So the time taken is longer than for the first diagram.

Read about 'time dilation'.
 
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  • #8
ryan2 said:
Is it incorrect to assume it would remain the same for the moving frame?
Yes. The light moves with ##c## along the diagonal path. In a light-clock the light-pulse moves forth and back between the mirrors within one period of the clock, so I call 1/2 of this period of the moving clock ##t_m##. Analog for the stationary clock: ##t_s##.

##c^2{t_s}^2 + v^2{t_m}^2 = c^2 {t_m}^2## (theorem of Pythagoras)

##\gamma = t_m / t_s##
 
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FAQ: What’s gone wrong with my derivation of the Lorentz Factor?

1. What is the Lorentz Factor and why is it important?

The Lorentz Factor, often denoted by the symbol γ (gamma), is a crucial component in Einstein's theory of special relativity. It describes how time, length, and relativistic mass change for an object moving relative to an observer. The formula for the Lorentz Factor is γ = 1 / √(1 - v²/c²), where v is the object's velocity and c is the speed of light. It is important because it quantifies the effects of relativistic speeds on physical measurements and ensures that the laws of physics remain consistent for all observers, regardless of their relative motion.

2. What common mistakes can occur when deriving the Lorentz Factor?

Common mistakes in deriving the Lorentz Factor include incorrect algebraic manipulation, neglecting the effects of time dilation and length contraction, and failing to properly account for the relative motion of the observer and the moving object. Additionally, misinterpreting the geometry of spacetime or using inappropriate frames of reference can lead to erroneous conclusions. It's essential to carefully follow the principles of special relativity and ensure all assumptions are valid throughout the derivation.

3. How can I check if my derivation is correct?

To check if your derivation of the Lorentz Factor is correct, you can compare your results with established derivations found in textbooks or reputable sources on special relativity. Additionally, you can analyze specific cases, such as when the velocity approaches zero (where γ should approach 1) or when the velocity is equal to the speed of light (where γ approaches infinity). Verifying that your derivation aligns with these expected behaviors can help confirm its accuracy.

4. What role does the concept of simultaneity play in deriving the Lorentz Factor?

The concept of simultaneity is pivotal in deriving the Lorentz Factor, as it highlights how different observers may perceive events occurring at the same time differently, depending on their relative motion. This leads to the realization that time is not absolute but relative, which is a cornerstone of special relativity. When deriving the Lorentz Factor, one must account for the relativity of simultaneity, particularly when analyzing scenarios involving moving clocks and observers in different inertial frames.

5. Can the Lorentz Factor be derived from classical physics principles?

No, the Lorentz Factor cannot be derived from classical physics principles because classical mechanics does not account for the effects of high velocities approaching the speed of light. Classical physics assumes that time and space are absolute and independent, which contradicts the findings of special relativity. The Lorentz Factor emerges from the need to reconcile the constancy of the speed of light for all observers and the principles of time dilation and length contraction, which are not present in classical mechanics

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