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moatasim23
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Why in deriving invariant distance in space time we use Pythagorean Theorem with a negative sign?
Because it matches the results of physics experiments.moatasim23 said:Why in deriving invariant distance in space time we use Pythagorean Theorem with a negative sign?
See also History of Lorentz transformations.
Many physicists, including George FitzGerald, Joseph Larmor, Hendrik Lorentz and Woldemar Voigt, had been discussing the physics behind these equations since 1887.[1][2] Larmor and Lorentz, who believed the luminiferous ether hypothesis, were seeking the transformation under which Maxwell's equations were invariant when transformed from the ether to a moving frame.
A spacetime interval between two events is the frame-invariant quantity analogous to distance in Euclidean space. The spacetime interval s along a curve is defined by:
where c is the speed of light (see sign convention). A basic assumption of relativity is that coordinate transformations have to leave intervals invariant. Intervals are invariant under Lorentz transformations.
The spacetime intervals on a manifold define a pseudo-metric called the Lorentz metric. This metric is very similar to distance in Euclidean space. However, note that whereas distances are always positive, intervals may be positive, zero, or negative. Events with a spacetime interval of zero are separated by the propagation of a light signal. Events with a positive spacetime interval are in each other's future or past, and the value of the interval defines the proper time measured by an observer traveling between them. Spacetime together with this pseudo-metric makes up a pseudo-Riemannian manifold.
I thought it is negative because time has an inverse relationship with length.guss said:The -c2 appended onto that term in that equation acts as a correction factor. The negative is there because time is imaginary. The c2 is there to act as a correction factor for the way we measure time. If we were to measure things very fundamentally, time and space would be measured in the same way. But we don't, so we must add the c2.
Since time is unlike any other dimension, that equation calls it imaginary to make it clear that the time dimension is different than space dimensions. There are probably deeper reasons than that, though.nitsuj said:I thought it is negative because time has an inverse relationship with length.
What does "time is imaginary" mean?
It's not the way "we" measure it. It's the way it is measured. The physics of measuring time includes distance doesn't it?
In any case we do measure time and space the same with intervals.
guss said:...imaginary to make it clear that the time dimension is different than space dimensions.
guss said:The c*t in (ct)2 can be viewed sort of like velocity*time = distance.
Instead of thinking of it like -c^2t^2 think of it as -(ct)^2. Here, you can see the c*t is referring to velocity*time, so we get a distance. So, just like the other terms in that equation, we need to square it.nitsuj said:Does the squared in c2 come into play because in velocity*time = distance, time is being used twice? Once in velocity and again in duration of velocity.
I guess I am asking the why is c squared there.
nitsuj said:Maybe I need to ask it differently,
Why does c have to be squared (why are the terms squared in the first place)?
I'm pretty sure I've heard it has to do with the units (physical units I assume). I am just trying to piece that together.
In introductory SR you use only inertial coordinate systems and generally simple algebra. Once you are a little more advanced you combine space and time into a single geometric structure, spacetime. There are several ways to do this.nitsuj said:"Imaginary time works in inertial frames in SR, but doesn't generalize to non inertial frames or curved space times."
I don't know GR at all, does that statement mean equivalence isn't so simple in GR? Are there no inertial frames in GR (i interprut curved spacetime as acceleration of somesort)? Said differently does GR ruin this awsome symmetry thing? (specifically complicates it).
nitsuj said:Cool stuff, thanks guss.
I still can't see it though.
lets say the value for cordinate x is 3 and y is 5. What is the "function" of this part; (3+3+3)+(5+5+5+5+5).
Asked differently,
Why does the value of length along a dimension have to be squared in order to calculate the distance to an other value of length along a different dimension. (oohhh triangulating )
Maybe if I play with right angle triangles looking for this it'd make it more clear.
Glad to help. For me, the most intimidating part was just the weird tensor notation with the Einstein summation convention, so I tried to avoid that and just use more standard matrix multiplication. Eventually you cannot easily avoid it, but I didn't go that far.nitsuj said:That funky looking equation/metric doesn't seem as intimidating now that you explained what it represents. And rather clearly.
I thought it is negative because time has an inverse relationship with length.
Naty1 said:What do you mean?
To me it is not logical if one were starting out from scratch, but makes sense within the context of the necessary math...as the quotes I posted above imply.
The sign is negative because that's what keeps transforms Lorentz invarient.
moatasim23 said:Why in deriving invariant distance in space time we use Pythagorean Theorem with a negative sign?
Naty1 said:Here is how Sean Carroll explains the space time interval:
http://ned.ipac.caltech.edu/level5/March01/Carroll3/Carroll1.html
Then he writes the spacetime interval equation. He does not really explain how he got there. Or do i not understand what he says?An event is defined as a single moment in space and time, characterized uniquely by (t, x, y, z). Then, without any motivation for the moment, let us introduce the spacetime interval between two events:
The Pythagorean Theorem is a mathematical principle that states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
In space-time, the Pythagorean Theorem is used to calculate the distance between two points that are moving in different directions and at different speeds. This is because in space-time, distance is not measured in a straight line, but rather in a curve. By using the Pythagorean Theorem, the distance can be calculated as the hypotenuse of a triangle formed by the space and time components.
Invariant distance in space-time is a concept in physics that refers to the distance between two points in space-time that remains constant regardless of the observer's frame of reference. This is important in understanding the relationship between space and time, as it allows for the measurement of distance in a consistent and objective manner.
Deriving invariant distance in space-time is important in order to accurately describe and understand the behavior of objects in motion. It allows for the calculation of distance in a way that is consistent and independent of the observer's perspective, which is crucial in the study of relativity and other areas of physics.
Yes, there are many real-world applications of deriving invariant distance in space-time. This concept is used in various fields such as astrophysics, navigation, and telecommunications to accurately measure and predict the movement of objects in space and time. It is also crucial in the development of technologies such as GPS and satellite communication systems.