Four-vector: Difference between revisions

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Revision as of 19:35, 19 May 2003 editMichael Hardy (talk \| contribs)Administrators210,264 editsNo edit summary← Previous edit		Revision as of 19:45, 19 May 2003 edit undo217.158.229.230 (talk) inner product Next edit →
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	a_0 b_0 - a_1 b_1 - a_2 b_2 - a_3 b_3		a_0 b_0 - a_1 b_1 - a_2 b_2 - a_3 b_3
	</math>		</math>
	Strictly speaking, the inner product of four-vectors is not a proper inner product, since its value can be negative.		Strictly speaking, the ] of four-vectors is not a proper inner product, since its value can be negative.

	In the above, ''a''<sub>0</sub> would represent time, and ''a''<sub>1</sub>, ''a''<sub>2</sub> and ''a''<sub>3</sub> would represent space, and similarly for ''b''. The proportion between space coordinates and time coordinates should be ]; for example, if the time ] is given in ]s, the space coordinates should be given in ]s.		In the above, ''a''<sub>0</sub> would represent time, and ''a''<sub>1</sub>, ''a''<sub>2</sub> and ''a''<sub>3</sub> would represent space, and similarly for ''b''. The proportion between space coordinates and time coordinates should be ]; for example, if the time ] is given in ]s, the space coordinates should be given in ]s.

Revision as of 19:45, 19 May 2003

In special relativity, a four-vector is a vector in a four-dimensional real vector space with a scalar product between four-vectors a and b defined as follows:

ab={\begin{vmatrix}a_{0}\\a_{1}\\a_{2}\\a_{3}\end{vmatrix}}.{\begin{vmatrix}b_{0}\\b_{1}\\b_{2}\\b_{3}\end{vmatrix}}=a_{0}b_{0}-a_{1}b_{1}-a_{2}b_{2}-a_{3}b_{3}

Strictly speaking, the inner product of four-vectors is not a proper inner product, since its value can be negative.

In the above, a₀ would represent time, and a₁, a₂ and a₃ would represent space, and similarly for b. The proportion between space coordinates and time coordinates should be c; for example, if the time coordinate is given in seconds, the space coordinates should be given in light seconds.

The scalar product of two four-vectors is invariant under any lorentz transformation. The laws of physics are also supposed to be invariant under a lorentz transformation.

An object will perceive the universe as if the universe were lorentz-transformed so that the perceiving object is stationary. Stationary is defined only with respect to a chosen inertial reference frame.

Often, ds or dτ will be used, when talking about a four-vector; in that case, it will mean:

ds=d\tau =\left|da\right|={\sqrt {(da)^{2}}}.

ds will be used when talking about distances, and dτ used when talking about time. ∫dτ is the amount of time a particle will experience travelling between two points in time-space.