Unraveling The Minkowski Metric: Intuitive Explanation

[aaaa202]

yeh well, I once understood this, but now looking at it today I can't get it to make sense intuitively.

The quantity:

dx²+dy²+dy²-c²dt²
is the same for every intertial frame in SR - just like length is the same in all inertial frames in classical mechanics. Now I am not sure that I understand intuitively why the above quantity is invariant. Can someone explain to me?
For me it should rather be something that expresses that light travels the same distance in every same dt, so maybe rather cdt = cdt' but then again dt and dt' are in general different right? So long since I did SR

 

[omega_minus]

You're right t does not equal t' but then again x does not equal x' for velocities along x. The length contraction will make ds smaller while the time dilation makes ds longer by the same amount (gamma). Recall that SR is a 4-D volume preserving theory. If a cube of 1m^3 comes into existence for one second then vanishes again, all observers will agree on how much spacetime was enclosed by it. They will disagree on how long it was spatially (and hence its 3-D volume) and how long it existed, but take the euclidian norm sqrt(x^2+y^2+z^2-(ct)^2) and everyone agrees on how much spacetime was inside the box. In summary: we disagree on how much space there was between two events and we disagree on how much time was between two events, but we ALL agree on how much spacetime was between two events.

 

[Muphrid]

Where's the closest hospital to you? Let's say it's 5 miles away from you. It might be 3 miles north and 4 miles east, and you'd use a distance formula to turn those two components into the full distance. But something you implicitly understand is that it doesn't matter whether you break the distance down into north and east components specifically. The distance is the same regardless of coordinate system.

The distance formula in special relativity is what you have. Even if you rotate or change the coordinate system, the spacetime distance (the "interval" is what we call it so we don't get confused with conventional distance) is the same between two points (events).

 

[pervect]

yeh well, I once understood this, but now looking at it today I can't get it to make sense intuitively.

The quantity:

dx²+dy²+dy²-c²dt²
is the same for every intertial frame in SR - just like length is the same in all inertial frames in classical mechanics. Now I am not sure that I understand intuitively why the above quantity is invariant. Can someone explain to me?
For me it should rather be something that expresses that light travels the same distance in every same dt, so maybe rather cdt = cdt' but then again dt and dt' are in general different right? So long since I did SR

If you look at the Lorentz interval between two points on a light beam, it will always be zero, because the distance^2 will equal (ct)^2.

So the constantcy of the speed of light for all observers is equivalent to saying that a Lorentz interval of zero in one frame is zero in all frames.

IT's not intuitively obvious how non-zero Lorentz intervals should transform (it is intuitively obvious via the constancy priciples how zero Lorentz intervals must transform). But I don't believe it has to be " intuitively obvious".