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I asked my teacher , i didnt get any satisfactory answers , can u tell me why dimensions are always at right angles .
I asked my teacher , i didnt get any satisfactory answers , can u tell me why dimensions are always at right angles .
I'm afraid I have to disagree. The real world does not measure things in coordinate systems. People do. The advantage of measuring coordinates in right angles is so great that there are few exceptions. One example is magnetic North/South versus polar North/South. Magnetic North does not line up with Polar North. So if you go in the magnetic North direction, you are (usually) also moving in the East/West direction. Because of this, Latitude / Longitude measurements are not based on magnetic North even though that is the easiest thing to determine in the "real world". But humans made that decision, not the "real world".The real world really does have this right angle character.
The OP asked about dimensions, not coordinate systems.I'm afraid I have to disagree. The real world does not measure things in coordinate systems. People do. The advantage of measuring coordinates in right angles is so great that there are few exceptions. One example is magnetic North/South versus polar North/South. Magnetic North does not line up with Polar North. So if you go in the magnetic North direction, you are (usually) also moving in the East/West direction. Because of this, Latitude / Longitude measurements are not based on magnetic North even though that is the easiest thing to determine in the "real world". But humans made that decision, not the "real world".
Well, dimensions are just numbers. So dimensions being at right angles makes no sense really.The OP asked about dimensions, not coordinate systems.
Well, dimensions are just numbers. So dimensions being at right angles makes no sense really.
This is the mathematics section of the website, hence people are answering in the context of mathematics. By that logic do you assume that the Earth Sciences forum or the Computing forum are also physics discussions just because the website is named Physics Forums?A mathematician can say that dimensions are just numbers related in arbitrary ways. That's valid, but physicists deal with real objects and measurements inside of a real universe that is overwhelmingly preferential to 3 dimensions of space and one of time, all at right angles to one another. In the context of physicsforums such may be assumed.
He's talking about relativity. It assumes a 4 dimensional space where time is another dimension. Time dilation tell us that time doesn't flow at equal rate depending on your velocity and gravitational potential. So time rate has to be measures. This is simplified by choosing a 4 dimensional space. Time does indeed flow at different rates. Experiments have been done and it's verified.When dealing with "real objects", in what sense does time form an angle of 90 degrees with any of the dimensions of space?
OK, how do you measure 90° angles in GR?He's talking about relativity. It assumes a 4 dimensional space where time is another dimension. Time dilation tell us that time doesn't flow at equal rate depending on your velocity and gravitational potential. So time rate has to be measures. This is simplified by choosing a 4 dimensional space. Time does indeed flow at different rates. Experiments have been done and it's verified.
You've got a point there.This is the mathematics section of the website, hence people are answering in the context of mathematics. By that logic do you assume that the Earth Sciences forum or the Computing forum are also physics discussions just because the website is named Physics Forums?
It isn't 90 degrees, but it is orthogonal. The Minkowski metric is x^2+y^2+z^2+it^2, which is pretty similar to the Euclidean metric. So one may loosely think of the angle as begin 90 degrees.When dealing with "real objects", in what sense does time form an angle of 90 degrees with any of the dimensions of space?
Are you sure the imaginary number ##i## should be in there?It isn't 90 degrees, but it is orthogonal. The Minkowski metric is x^2+y^2+z^2+it^2, which is pretty similar to the Euclidean metric. So one may loosely think of the angle as begin 90 degrees.
But I'm sure the OP wasn't asking about that.
Yep. Albert himself sometimes used that notation. It can also be written as x^2+y^2+z^2-t^2. So on second thought, it should have been x^2+y^2+z^2+(it)^2 with standard operator precedence.Are you sure the imaginary number ##i## should be in there?
And what about vectors for which this metric is zero. Should you interpret it as being orthogonal on itself?
I know Minkowski spacetime. I just disagree you can meaningfully talk about angles and orthogonality there. And when you move to GR it becomes even more problematic.Yep. Albert himself sometimes used that notation. It can also be written as x^2+y^2+z^2-t^2. So on second thought, it should have been x^2+y^2+z^2+(it)^2 with standard operator precedence.
The lines for which the metric is zero are null lines. These are the lines traveled by light in a vacuum. As far as light is concerned, it takes zero proper time to travel anywhere in a vacuum. So it is a pseudometric.
If you are interested, look up Minkowski spacetime. I'm sure that there are many others who can explain this better than can I.
Only if you choose a particular definition for the inner product. Coordinate systems do not need to be orthogonal, but they do need to be linearly independent.It isn't 90 degrees, but it is orthogonal.
How does this make any sense if we are to accept mathematics as the language of physics? The dimension of spacetime (Minkowski a special case) is defined as the number of parameters needed to specify an event biunivocally in a sufficiently small neighborhood. So the dimension is 4, nothing more, how is it meaningfull to mention angles here? Also, a mathematician would not say dimensions are ''numbers related in abitrary ways''.A mathematician can say that dimensions are just numbers related in arbitrary ways. That's valid, but physicists deal with real objects and measurements inside of a real universe that is overwhelmingly preferential to 3 dimensions of space and one of time, all at right angles to one another. In the context of physicsforums such may be assumed.
Ok I thought orthogonal meant 90 degrees. Sorry about that. But orthogonality test would be dot product? I'm not well read about relativity so I wont make further arguments.OK, how do you measure 90° angles in GR?
But physicists don't use it for several decades now, it's problematic - it's been discussed in Relativity forum multiple times.Albert himself sometimes used that notation.
Yes, two vectors are orthogonal if their dot product is zero, by definition. But orthogonal and 90° need not be the same. For one thing, the angle refers to a geometric situation, where the unit vectors are directions in space. Angles have no meaning in other abstract notions of vector spaces. Secondly, one could define different types of "dot" products, for which orthogonal directions are not 90°. A physical and geometric example of this would be a crystal. As you might guess from looking at different minerals that have different shapes, quartz and salt (halite) for example, not all crystals have natural coordinate systems in which the axes meet at 90°. The periodic structures within the crystal, known as unit cells, consist of just enough atoms or molecules so that when the unit cell is translated in 3 directions by the lengths of the unit cell sides in those directions, the entire crystal is generated; and these 3 directions need not be at 90° to each other. It's analogous to periodic functions, like trig functions, that repeat themselves over every period along the axis of the independent variable.Ok I thought orthogonal meant 90 degrees. Sorry about that. But orthogonality test would be dot product? I'm not well read about relativity so I wont make further arguments.
Wow. crystal example makes it a lot clear. Thanks thanks!Yes, two vectors are orthogonal if their dot product is zero, by definition. But orthogonal and 90° need not be the same. For one thing, the angle refers to a geometric situation, where the unit vectors are directions in space. Angles have no meaning in other abstract notions of vector spaces. Secondly, one could define different types of "dot" products, for which orthogonal directions are not 90°. A physical and geometric example of this would be a crystal. As you might guess from looking at different minerals that have different shapes, quartz and salt (halite) for example, not all crystals have natural coordinate systems in which the axes meet at 90°. The periodic structures within the crystal, known as unit cells, consist of just enough atoms or molecules so that when the unit cell is translated in 3 directions by the lengths of the unit cell sides in those directions, the entire crystal is generated; and these 3 directions need not be at 90° to each other. It's analogous to periodic functions, like trig functions, that repeat themselves over every period along the axis of the independent variable.