Why is Tau-AB^2 not t^2 + x^2?

[BLevine1985]

TL;DR

question about proof page 21

why is Tau-AB^2 equal to t^2 - x^2 ?It seems it should be t^2 + x^2 according to the geometry of the diagram...

 

[PeterDonis]

why is Tau-AB^2 equal to t^2 - x^2 ?

Because that's how the metric is defined in an inertial frame in special relativity.

It seems it should be t^2 + x^2 according to the geometry of the diagram...

No, because the diagram is not depicting a Euclidean geometry. It's depicting a Minkowskian geoemetry. That means you cannot interpret lines and angles in the diagram as if they were Euclidean; they aren't.

 

[BLevine1985]

Thank you. That was very helpful!

 

[PeterDonis]

Thank you. That was very helpful!

You're welcome!

Also, welcome to PF!