Can Anyone Help Understand Einstein's 1905 Derivation of E=mc^2?

In summary: This is equivalent to taking the cosine of (\varphi + 180^o) instead of \varphi for the "other" light ray.
  • #1
SiennaTheGr8
492
193
Link: https://www.fourmilab.ch/etexts/einstein/E_mc2/www/

The only part I'm having trouble with is how he gets the plus sign in that 1+(v/c)cos(φ) numerator for the "other" light ray (emitted in the opposite direction of the first).

My understanding is that the φ he uses in his general Doppler equation represents the angle formed by the source/receiver line-of-sight (as seen by the source at the time of emission) and the receiver's direction of motion (as measured in the source frame). If that's right, then the plus sign in that numerator is equivalent to taking the cosine of (φ + 180°) rather than φ for the "other" light ray.

So I'm 99% sure that it's an implied symmetry argument, but I don't know how to demonstrate its validity mathematically.

Can anyone help me out?
 
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  • #2
I think maybe I've misunderstood what the angle represents, and that this is much simpler than I suspected.
 
  • #3
SiennaTheGr8 said:
Link: https://www.fourmilab.ch/etexts/einstein/E_mc2/www/

The only part I'm having trouble with is how he gets the plus sign in that 1+(v/c)cos(φ) numerator for the "other" light ray (emitted in the opposite direction of the first).

My understanding is that the φ he uses in his general Doppler equation represents the angle formed by the source/receiver line-of-sight (as seen by the source at the time of emission) and the receiver's direction of motion (as measured in the source frame). If that's right, then the plus sign in that numerator is equivalent to taking the cosine of (φ + 180°) rather than φ for the "other" light ray.

So I'm 99% sure that it's an implied symmetry argument, but I don't know how to demonstrate its validity mathematically.

Can anyone help me out?

If two photons are sent in opposite directions, and one makes an angle of [itex]\varphi[/itex] with respect to the x-axis, then the other makes an angle of [itex]\varphi + 180^o[/itex] with respect to the x axis. So for the transformation of frequency, etc., for the second photon, you just replace [itex]\varphi[/itex] by [itex]\varphi + 180^o[/itex].
 

Related to Can Anyone Help Understand Einstein's 1905 Derivation of E=mc^2?

1. What is the significance of Einstein's 1905 derivation of E=mc^2?

The equation E=mc^2 is one of the most famous and influential equations in the history of science. It established the relationship between mass (m) and energy (E), showing that they are interchangeable and can be converted from one form to the other.

2. How did Einstein come up with the equation E=mc^2?

In 1905, Einstein published his Special Theory of Relativity, which proposed that the laws of physics are the same for all observers in uniform motion. In this theory, he derived the equation E=mc^2 by combining his famous equation E=hf (where h is Planck's constant and f is the frequency of the light) with the equation for the kinetic energy of a moving object (KE = 1/2 mv^2). This led to the relationship between energy and mass, which is expressed as E=mc^2.

3. What does the "c" in E=mc^2 represent?

The "c" in E=mc^2 represents the speed of light in a vacuum, which is approximately 299,792,458 meters per second. It is a constant that is fundamental to Einstein's theory of relativity and plays a crucial role in the equation. Without this constant, the equation would not hold true.

4. Can you explain the mathematical derivation of E=mc^2?

The mathematical derivation of E=mc^2 involves using the equations for energy (E=hf) and kinetic energy (KE = 1/2 mv^2) and then applying the principles of special relativity. This includes the concepts of time dilation, length contraction, and the equivalence of mass and energy. The final equation is obtained by combining these principles and simplifying the equations to get E=mc^2.

5. How has Einstein's 1905 derivation of E=mc^2 impacted modern science and technology?

Einstein's equation has had a profound impact on modern science and technology. It has led to the development of nuclear energy and weapons, which harness the power of the atom by converting mass into energy. It has also played a crucial role in our understanding of the universe, with implications for space travel, cosmology, and the search for a unified theory of physics. Additionally, the equation has been used in various applications, such as medical imaging and GPS technology, demonstrating its wide-ranging influence on our daily lives.

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