- #1
motoroller
- 29
- 0
I've tried proving the invariance of the spacetime interval from Lorentz transformations 3 times now, but every time I end up with two extra terms that don't cancel! Could I have some help?
haushofer said:I'm trying to visualize your calculations via my paranormal abilities, but somehow I fail.
motoroller said:My LaTeX isn't so good, but substituting:
[tex]x'=\gamma(x+vt)[/tex]
[tex]t'=\gamma(t+\frac{vx}{c^2})[/tex]
into
[tex](x')^2+(y')^2+(z')^2-c^2(t')^2[/tex]
trying to get
[tex]x^2+y^2+z^2-c^2t^2[/tex]
i.e. invariant interval
haushofer said:Did you manage to do the calculation?
Fredrik said:All of these things get easier when you're used to working with matrices. We want to prove that [itex](x-y)^T\eta(x-y)[/itex] is invariant, i.e. that it's equal to [itex](\Lambda(x-y))^T\eta\Lambda(x-y)[/itex]. So we stare at it for two seconds and realize that the equality follows immediately from the definition of a Lorentz transformation and a trivial fact about the transpose of a product.
I have said before that I think you can learn SR and matrices in less time than you can learn just SR, and I still think that's right.
The invariance of spacetime interval refers to the idea that the measurement of distance and time between two events in the universe is the same for all observers, regardless of their relative motion. This concept is a fundamental principle in the theory of special relativity.
The invariance of spacetime interval is important because it is a key principle in understanding the nature of space and time in the universe. It helps us to reconcile the differences in measurements of time and distance between different observers and allows for the development of the theory of special relativity.
The invariance of spacetime interval is expressed mathematically using the equation: Δs² = Δx² + Δy² + Δz² - (cΔt)², where Δs is the spacetime interval, Δx, Δy, and Δz are the spatial distances, c is the speed of light, and Δt is the difference in time between two events.
No, the invariance of spacetime interval applies to all objects regardless of their speed. However, it becomes more significant and apparent for objects moving at speeds close to the speed of light. This is because at these speeds, the differences in measurements of time and distance become more noticeable.
The concept of invariance of spacetime interval challenges our intuitive understanding of time and space because it suggests that our perception of time and space is relative and depends on the observer's frame of reference. It also blurs the distinction between space and time, as they are both considered part of a unified concept of spacetime.