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[parent] spacetime interval is invariant for a Lorentz transformation (Result)

The spacetime interval between two events E1(x1,y1,z1,t1) and E2(x2,y2,z2,t2) is defined as

(s)2 = c2t2 (x)2 (y)2 (z)2.

If s is in reference frame S, then s is in reference frame S moving at a velocity u along the x-axis. Therefore, to show that the spacetime interval is invariant under a Lorentz transformation we must show

(s)2 = (s)2

with the reference frames related by The Lorentz transformation

x = x ut 1 u2 c2

y = y

z = z

t = t uxc2 1 u2 c2.

The change in coordinates between events in the S frame is then given by

x = ( x2 ut2 1 u2 c2 ) ( x1 ut1 1 u2 c2 ) = x ut 1 u2 c2

y = y 2 y1 = y

z = z 2 z1 = z

t = ( t2 ux2c2 1 u2 c2 ) (t1 ux1c2 1 u2 c2 ) = t ut 1 u2 c2.

Squaring the terms yield

(x)2 = ( x ut 1 u2 c2 ) ( x ut 1 u2 c2 ) = (x)2 2uxt + u2(t)2 1 u2c2

(y)2 = (y)2

(z)2 = (z)2

(t)2 = ( t ut 1 u2 c2 ) ( t ut 1 u2 c2 ) = (t)2 2uxtc2 + u2(x)2c4 1 u2c2 .

Substituting these terms into the spacetime interval gives

(s)2 = c2((t)2 2uxtc2 + u2(x)2c4) 1 u2c2 ((x)2 2uxt + u2(t)2) 1 u2c2 (y)2(z)2.

Adding the first two terms with common denominators together yields

(s)2 = c2(t2) (x)2 u2(t)2 + u2(x)2c2 1 u2c2 (y)2 (z)2.

Pulling out a u2c2

(s)2 = c2(t2) (x)2 u2c2(c2(t)2 + (x)2) 1 u2c2 (y)2 (z)2.

Factoring out a c2(t)2 (x)2 in the numerator

(s)2 = (c2(t2) (x)2)(1 u2c2) 1 u2c2 (y)2 (z)2.

Finally, canceling terms gives

(s)2 = c2(t2) (x)2 (y)2 (z)2 = (s)2.

Hence, the spacetime interval is invariant under a Lorentz transformation.

References

[1]   Carroll, Bradley, Ostlie, Dale, An Introduction to Modern Astrophysics. Addison-Wesley Publishing Company, Reading, Massachusetts, 1996.

[2]   Cheng, Ta-Pei, Relativity, Gravitation and Cosmology. Oxford University Press, Oxford, 2005.

[3]   Einstein, Albert, Relativity: The Special and General Theory. 1916.


"spacetime interval is invariant for a Lorentz transformation" is owned by bloftin.
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Cross-references: The Lorentz transformation, Lorentz transformation, velocity, reference frame, spacetime

This is version 2 of spacetime interval is invariant for a Lorentz transformation, born on 2006-11-25, modified 2006-11-25.
Object id is 237, canonical name is SpacetimeIntervalIsInvariantForALorentzTransformation.
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Classification:
Physics Classification03.30.+p (Special relativity)
 03. (Quantum mechanics, field theories, and special relativity )
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