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explain this equation

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it is the euclidean spacetime interval equation
@agentx5 @sami-21 ...i dont know how to write this equation there are no keys of scientific operators in my keyboard

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you type \[ \text{\[ s^2 = r^2 - c^2 t^2 \] } \]
It will become \[s^2 = r^2 - c^2 t^2 \]
there are delta symbols before r and t
\[\text{ \[ s^2 = \Delta \r^2 - c^2 \Delta t^2 \] } \]
\[s^2 = \Delta r^2 - c^2 \Delta t^2 \]
oh thank you very much...
Spacetime isn't Euclidean, by the way. Did you have a question about that equation?
yes found that in wikipedia when i typed spacetime what are those equations and wat do u mean by it is not euclidean?
In Euclidean three-dimensional space, the distance between two points is defined as \[ \Delta r = \sqrt{ (\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2 } \] or equivalently, \[(\Delta r)^2 = (\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2 \]
That distance, which can be thought of as the separation between two points in space, is invariant under translation and rotation. That means that an observer moving through Euclidean space would observe the distance between two objects to be exactly the same no matter which way they were turned and no matter how they happened to be moving.
However, when we combine the dimension of time with the dimensions of space in a four-dimensional manifold called spacetime, we label our points not as points in space but as events in space and time. However, relativistically it turns out that observers moving at different speeds would not measure the distance above to be the same, and therefore \[(\Delta r)^2 = (\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2 \] is no longer invariant.
However, it was found that there is an analogous invariant interval that will be measured to be the same by all observers regardless of velocity. That interval is called s^2, and is written as \[s^2 = (\Delta r)^2 - (c\Delta t)^2 = (\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2 - (c\Delta t)^2 \] This plays the role of the invariant interval in four-dimensional spacetime.
Lastly, spacetime is not Euclidean because that invariant interval has a minus sign in it. If it were a plus sign, it would be Euclidean space, but it's not. It's usually referred to as Minkowski space.
As an FYI, sometimes the minus sign gets reversed. For reasons I'm not going to go into because it requires knowledge of special relativity, I prefer to write the spacetime interval as \[s^2 = (c\Delta t)^2 - (\Delta r)^2 \]
(Note that the choice of signs for\[s^2\] above follows the space-like convention (-+++). Other treatments reverse the sign of \[s^2\] Spacetime intervals may be classified into three distinct types based on whether the temporal separation (\[c^2\Delta t^2\]) or the spatial separation (\[\Delta r^2\]) of the two events is greater what does this mean please explain it @Jemurray3 ...thank you very much
if \[ (c\Delta t)^2 > r^2 \] then the interval is called timelike. If \[ (c\Delta t)^2 < r^2\] then the interval is called spacelike and if they're equal, the interval is called lightlike.
Events that are separated by spacelike intervals cannot be causally connected because the time it would take for information to travel from one point in space to another (at the speed of light) is greater than the separation of those two points in time. I.e. if I clapped my hands on earth and an alien in the andromeda galaxy fell out of his chair, there couldn't possibly be a causal connection between these two events because it would take years for the signal to traverse the space between us.
dont be mad at me...could you please explain what this invariant interval actually mean..if it is the distance it should be S right but it is \[S^2\] HEREwhich means \[m^2\] right
It is the spacetime equivalent of distance, yes. Take the square root if you'd like, but if s is invariant then s^2 is as well, and because it can be negative you can get into trouble. It's just more useful to refer to s^2, in general.
so no problem abt the unit
The units are consistent. Honestly the invariant spacetime interval is not of much use without the rest of special relativity, and the geometry and topological properties of flat spacetime (Minkowski space) require more mathematics to adequately understand, so I don't really think you should worry about it until you have a more thorough grounding in the math and physics.
one last question how did we incorporate that term \[c \Delta t^2\]? why do we subtract it from the variant euclidean distance why not plus it?
though its mathematical pls explain...
It didn't just pop out of nowhere. If there was a plus sign, rather than a minus sign, then the interval would change depending on the observer. Once you derive the Lorentz transformation equations of special relativity, it can be shown that the quantity s^2 as written above is the same for all inertial observers.

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