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In another thread Naty1 said:

Basically, starting with the Lorentz transform we can notice two important things:

1) time and space are not entirely separate entities but one frame's time gets split into another frame's space and vice versa.

2) there is a notion of "distance" called the http://en.wikipedia.org/wiki/Spacetime#Space-time_intervals" which also mixes space and time together and is agreed upon by all reference frames (i.e. is invariant under the Lorentz transform).

What we would like is a convenient way to keep things organized so that we can easily keep track of the things that everyone agrees on and easily determine how any frame sees something. This is exactly what the four-vector approach accomplishes. If we take our normal space coordinates that we have all seen since introductory physics, (x,y,z), and we add a time coordinate, (ct,x,y,z), then we have four-vectors and spacetime. Now we can write the http://en.wikipedia.org/wiki/Lorentz_transformations#Matrix_form" and easily switch between reference frames. This is very useful in itself for figuring out how different frames look at times and distances.

We can also consider the spacetime interval (aka Minkowski norm) as the length of the four-vector. One important use of this approach is that in an object's rest frame all of the space coordinates are 0, so the spacetime interval is immediately seen to be the time in an object's rest frame or its "proper time". Since the interval is invariant then we see that all frames agree on the proper time along any worldline. This is important because we can take any four-vector, use the derivative wrt this proper time, and come out with another four-vector. That new four-vector will transform according to the same Lorentz transform matrix as above, and the norm of the new four-vector will also be invariant.

So, in Newtonian physics velocity is the time derivative of position. Similarly, if we start with the position four-vector, (ct,x,y,z), and take the derivative wrt proper time then we get the four-velocity. Note, the norm of the four-velocity is always c, and for an object at rest the four-velocity is (c,0,0,0). In other words, a stationary object is still "moving" through time.

Now, if we multiply the four-velocity by the rest mass we get the four-momentum. For an object at rest we have p=(mc,0,0,0). Using the famous E=mc² equation you can re-write the above as, p=(E/c,0,0,0) for an object at rest. Thus energy is seen as "momentum" through time, and energy and momentum are seen to have the same relationship to each other as time and space have. Energy and momentum are not entirely separate entites and one observer's energy gets split into another observer's momentum in the same way as space and time above, and using the same Lorentz transform matrix. The norm of this four-momentum is the invariant rest mass, and the timelike component is the energy or the relativistic mass.

I apologize for the length. I am sure that my rambling explanation left more than one question unanswered, so please don't hesitate. This is an important subject and my discovery of four-vectors is what finally made SR "click" for me, so I am certainly willing to try and help.

The Wikipedia pages on http://en.wikipedia.org/wiki/Four-vector" ) are not wonderful, but they are a good starting point. I don't know exactly how far your background in these concepts extends, so I apologize if I go over stuff you already know.Naty1 said:I just happened to reread (Richard Feynmann, SIX NOT SO EASY PIECES) that replacing in the Lorentz transformations x with p_{x}(for momentum) and replacing t with E (for energy as mc^{2}yields the four vector momentum.

Is that what you are referring to here? If so, could you elaborate a bit further as I am trying to make some sense of the four vector momentum...whereas time in Lorentz is velocity and distance dependent (u,t) now energy is velocity and momentum dependent...

So it seems like energy and momentum transform/rotate into one another...yes? Is that what your comment implies??

Basically, starting with the Lorentz transform we can notice two important things:

1) time and space are not entirely separate entities but one frame's time gets split into another frame's space and vice versa.

2) there is a notion of "distance" called the http://en.wikipedia.org/wiki/Spacetime#Space-time_intervals" which also mixes space and time together and is agreed upon by all reference frames (i.e. is invariant under the Lorentz transform).

What we would like is a convenient way to keep things organized so that we can easily keep track of the things that everyone agrees on and easily determine how any frame sees something. This is exactly what the four-vector approach accomplishes. If we take our normal space coordinates that we have all seen since introductory physics, (x,y,z), and we add a time coordinate, (ct,x,y,z), then we have four-vectors and spacetime. Now we can write the http://en.wikipedia.org/wiki/Lorentz_transformations#Matrix_form" and easily switch between reference frames. This is very useful in itself for figuring out how different frames look at times and distances.

We can also consider the spacetime interval (aka Minkowski norm) as the length of the four-vector. One important use of this approach is that in an object's rest frame all of the space coordinates are 0, so the spacetime interval is immediately seen to be the time in an object's rest frame or its "proper time". Since the interval is invariant then we see that all frames agree on the proper time along any worldline. This is important because we can take any four-vector, use the derivative wrt this proper time, and come out with another four-vector. That new four-vector will transform according to the same Lorentz transform matrix as above, and the norm of the new four-vector will also be invariant.

So, in Newtonian physics velocity is the time derivative of position. Similarly, if we start with the position four-vector, (ct,x,y,z), and take the derivative wrt proper time then we get the four-velocity. Note, the norm of the four-velocity is always c, and for an object at rest the four-velocity is (c,0,0,0). In other words, a stationary object is still "moving" through time.

Now, if we multiply the four-velocity by the rest mass we get the four-momentum. For an object at rest we have p=(mc,0,0,0). Using the famous E=mc² equation you can re-write the above as, p=(E/c,0,0,0) for an object at rest. Thus energy is seen as "momentum" through time, and energy and momentum are seen to have the same relationship to each other as time and space have. Energy and momentum are not entirely separate entites and one observer's energy gets split into another observer's momentum in the same way as space and time above, and using the same Lorentz transform matrix. The norm of this four-momentum is the invariant rest mass, and the timelike component is the energy or the relativistic mass.

I apologize for the length. I am sure that my rambling explanation left more than one question unanswered, so please don't hesitate. This is an important subject and my discovery of four-vectors is what finally made SR "click" for me, so I am certainly willing to try and help.

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