Derive 4-Velocity Components: Anyone Know How?

In summary: Derive the chain rule for derivatives of inverse functions.- Use the chain rule to derive the limit as delta t goes to zero.
  • #1
berlinspeed
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4
TL;DR Summary
Need help on a simple 4-velocity components derivation.
Anyone know how to derive ##u^0=\frac {dt} {d\tau}=\frac {1} {\sqrt {1-\mathbf {v}^2}}## and ##u^j=\frac {dx^j} {d\tau}=\frac {v^j} {\sqrt {1-\mathbf{v}^2}}##?
 
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  • #2
What textbooks have you consulted? This should be described in any introductory text on SR.
 
  • #3
Orodruin said:
What textbooks have you consulted? This should be described in any introductory text on SR.
I don't own any SR books, which ones would you recommend?
 
  • #4
By textbook I do not necessarily mean a physical textbook, it can just as well be some form of online lecture notes or at least some sort of learning material. You must be getting the information from somewhere. That somewhere should preferably be a didactic material intended to teach SR. If you do not have some sort of learning material and just look things up on, e.g., Wikipedia, that is typically not a useful or effective learning strategy. There is lots of useful information available online, but if you do not find anything else you can try my lecture notes.
 
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  • #5
Orodruin said:
By textbook I do not necessarily mean a physical textbook, it can just as well be some form of online lecture notes or at least some sort of learning material. You must be getting the information from somewhere. That somewhere should preferably be a didactic material intended to teach SR. If you do not have some sort of learning material and just look things up on, e.g., Wikipedia, that is typically not a useful or effective learning strategy. There is lots of useful information available online, but if you do not find anything else you can try my lecture notes.
Thanks so much! Gonna grind on that for a while now..
 
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  • #6
I’ll give you an outline of steps you can follow.

Write the definition of timelike interval, or proper time between events in inertial frame.

Divide by delta t.

Take the limit as delta t goes to zero.

Use the rule for derivative of inverse.

That takes care of the first question. For the second, see how to use the chain rule plus this fact.
 
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Related to Derive 4-Velocity Components: Anyone Know How?

1. What is the definition of 4-velocity?

4-velocity is a four-dimensional vector that describes the velocity of an object in four-dimensional spacetime. It takes into account both the spatial and temporal components of an object's motion.

2. How is 4-velocity related to special relativity?

4-velocity is a concept that was developed as part of Einstein's theory of special relativity. It is used to describe the motion of objects in a way that is consistent with the principles of relativity, which state that the laws of physics should be the same for all observers in uniform motion.

3. What are the components of 4-velocity?

The components of 4-velocity are the three spatial components (x, y, and z) and the temporal component (t). They are represented as a four-dimensional vector, with the spatial components represented by the first three elements and the temporal component represented by the fourth element.

4. How do you derive the 4-velocity components?

The 4-velocity components can be derived using the Lorentz transformation equations, which relate the measurements of time and space between two different reference frames. By applying these equations, the components of 4-velocity can be calculated for an object in motion.

5. Why is it important to understand 4-velocity?

Understanding 4-velocity is crucial in the study of special relativity and other areas of physics that deal with high velocities or objects moving at the speed of light. It allows for a more accurate and comprehensive description of an object's motion in four-dimensional spacetime, which is necessary for many modern theories and calculations.

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