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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

2 2 2 2 2 2 (△s ) = c △t − (△x ) − (△y ) − (△z ).

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 − ut x = ∘-------2--2- 1 − u ∕c

′ y = y

′ z = z

′ -t −-ux∕c2-- t = ∘ 1 − u2∕c2.

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

( ) ( ) △x ′ = ∘-x2 −-ut2-- − ∘-x1 −-ut1-- = △∘x--−-u-△t-- 1 − u2∕c2 1 − u2∕c2 1 − u2∕c2

△y ′ = y − y = △y 2 1

△z ′ = z2 − z1 = △z

( ) ( ) t2 − ux2∕c2 t1 − ux1 ∕c2 △t − u△t △t ′ = ∘-------2--2- − ∘------2--2- = ∘-------2--2. 1 − u ∕c 1 − u ∕c 1 − u ∕c

Squaring the terms yield

( ) ( ) ′2 △x − u△t △x − u△t (△x )2 − 2u △x △t + u2(△t )2 (△x ) = ∘-------2--2 ∘------2--2- = ---------1-−-u2∕c2---------- 1 − u ∕c 1 − u ∕c

(△y ′)2 = (△y )2

(△z ′)2 = (△z )2

( ) ( ) △t − u△t △t − u△t (△t )2 − 2u △x △t ∕c2 + u2(△x )2∕c4 (△t ′)2 = ∘----------- ∘------------ = ----------------------------------. 1 − u2∕c2 1 − u2∕c2 1 − u2 ∕c2

Substituting these terms into the spacetime interval gives

′2 c2((△t-)2 −-2u-△x-△t-∕c2 +-u2(△x-)2∕c4) ((△x-)2 −-2u△x-△t--+-u2(△t-)2)- 2 2 (△s ) = 1 − u2 ∕c2 − 1 − u2∕c2 − (△y ) − (△z ) .

Adding the first two terms with common denominators together yields

2 2 2 2 2 2 2 2 (△s ′)2 = c-(△t--) −-(△x-)-−-u-(△t-)-+-u--(△x--)-∕c-− (△y )2 − (△z )2. 1 − u2∕c2

Pulling out a −u2∕c2

′2 c2(△t2-) −-(△x--)2 −-u2∕c2(c2(△t-)2 +-(△x-)2) 2 2 (△s ) = 1 − u2∕c2 − (△y ) − (△z ) .

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

′ 2 (c2(△t2) − (△x )2)(1 − u2∕c2) 2 2 (△s ) = ----------1 −-u2∕c2----------− (△y ) − (△z ) .

Finally, canceling terms gives

′2 2 2 2 2 2 2 (△s ) = c (△t ) − (△x ) − (△y ) − (△z ) = (△s ) .

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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Physics Classification: 03.30.+p (Special relativity)
 03. (Quantum mechanics, field theories, and special relativity )
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