Deriving Energy in Special Relativity: The Principle of Extremal Aging

In summary: This means that the ratio of the frame time to proper time is equal to the ratio of energy to mass, which is a constant of motion. This was derived using the Principle of Extremal Aging, which states that the path taken by a particle between two points in spacetime maximizes the proper time experienced by the particle. This equation is important in understanding the effects of time dilation in special relativity.
  • #1
H-bar None
45
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I'm reading Taylor and Wheeler's, Exploring Black Holes.

I was doing okay until I reached their derivation of energy in Special Relativity.

They arrived at this equation:

[tex] \frac{t}{\tau} = \frac{E}{m} [/tex]

Tau is proper time, t is the frame time, E is energy and m is mass.

The authors used the Principle of Extremal Aging to derive the equation. How did they arrive at E/m as a constant of motion?
 
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  • #2
H-bar None said:
They arrived at this equation:

[tex] \frac{t}{\tau} = \frac{E}{m} [/tex]

Tau is proper time, t is the frame time, E is energy and m is mass.

t is the time-component of the position 4-vector with magnitude [tex]\tau [/tex].
[tex]t=\gamma \tau [/tex]

E is the time-component of the momentum 4-vector with magnitude [tex]m [/tex].
[tex]E=\gamma m [/tex]
 
  • #3
:confused:

I sort of understand the 4-vector part. How does that relate to "E/m"?
I'm going to do some more reading check back with you later on in life.

Could go into a litte more detail, maybe I'm missing something.
Thanks for the response.
 
  • #4
Since [tex]t=\gamma \tau [/tex], we have [tex]\frac{t}{\tau}=\gamma[/tex].
Since [tex]E=\gamma m [/tex], we have [tex]\frac{E}{m}=\gamma[/tex].

Thus, [tex]\frac{t}{\tau}=\gamma=\frac{E}{m}[/tex].
 

1. What is the equation for energy in special relativity?

The equation for energy in special relativity is E=mc^2, where E represents energy, m represents mass, and c represents the speed of light.

2. How does the concept of energy change in special relativity?

In special relativity, energy is no longer a constant quantity but is dependent on the frame of reference. This means that different observers will measure different amounts of energy for the same object.

3. Can energy be converted into mass in special relativity?

Yes, according to Einstein's famous equation E=mc^2, energy and mass are equivalent and can be converted into each other. This has been demonstrated through nuclear reactions such as nuclear fusion and fission.

4. How does the concept of energy affect time in special relativity?

In special relativity, time is relative and can be affected by the amount of energy present. As an object's speed increases, time slows down for that object relative to a stationary observer.

5. Can energy be created or destroyed in special relativity?

No, energy cannot be created or destroyed in special relativity. It can only be converted from one form to another, such as kinetic energy to potential energy. This is known as the principle of conservation of energy.

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