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Tensors and Relativity: Chapter 2

Four- velocity, momentum and acceleration

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We must express Newton's laws of motion in terms of four- vectors so that they are frame invariant and consistent with Special Relativity. The four- velocity  of a particle is the tangent to its worldline of length c [ see Figure 2.1 ]:


Figure 2.1: The worldline of a particle with four- velocity tex2html_wrap_inline1200 .

This is the most natural analogue of the three- velocity. It is clearly a four- vector since both tex2html_wrap_inline1202 and tex2html_wrap_inline1333 are invariant.

In the particle's own rest frame tex2html_wrap_inline1210 , the four- velocity is


It follows that in a general frame tex2html_wrap_inline1206 :




where tex2html_wrap_inline1339 is the particle's three- velocity.

For low velocities tex2html_wrap_inline1341 , tex2html_wrap_inline1343 and the spatial part is nearly the same as tex2html_wrap_inline1339 . If an observers own four- velocity is written as tex2html_wrap_inline1347 in tex2html_wrap_inline1206 we have


We can then write the particle's three- velocity as [Assignment 2]:


This is a spacelike vector expressed in coordinate- independent form [ although tex2html_wrap_inline1351 singles out a particular observer ].

For photons the four- velocity  is not defined since tex2html_wrap_inline1353 , i.e. there is no frame in which a photon is at rest.

The four- momentum  is defined by


where tex2html_wrap_inline1355 is the rest mass of the particle i.e. the mass in its own rest frame. The spatial part of tex2html_wrap_inline1357 [ the three- momentum ] is tex2html_wrap_inline1359 so the apparent mass exceeds the rest mass.


The time part of the four- momentum is the energy  of the particle E divided by c:


so we have


For tex2html_wrap_inline1341


Since the second term is the kinetic energy , we interpret the first term as the rest- mass energy of the particle .

In general tex2html_wrap_inline1367 implies that


where tex2html_wrap_inline1369 is the particle's three- momentum.

Since tex2html_wrap_inline1371 as tex2html_wrap_inline1373 , requiring infinite energy, we infer that the particles with non- zero rest mass, can never reach the speed of light.

For photons  traveling in the x- direction


hence photons have zero rest mass.

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