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Sangam Swadik
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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 .
There all all sorts of spaces. In some of those spaces, dimensions are not at right angles. But the great majority of people are not interested in these, because in our Universe the physical dimensions ARE at right angles. This is true of any space with the Euclidian metric of a^2 + b^2 + c^2 = d^2, where d is the distance between two points and a, b, and c are at right angles to one another.Sangam Swadik said: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".Hornbein said:The real world really does have this right angle character.
FactChecker said: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".
Hornbein said:The OP asked about dimensions, not coordinate systems.
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.micromass said: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?Hornbein said: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.
MrAnchovy said:When dealing with "real objects", in what sense does time form an angle of 90 degrees with any of the dimensions of space?
Giant said: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.
pwsnafu said: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?
MrAnchovy said:When dealing with "real objects", in what sense does time form an angle of 90 degrees with any of the dimensions of space?
Hornbein said: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.
micromass said: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?
Hornbein said: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.Hornbein said:It isn't 90 degrees, but it is orthogonal.
Hornbein said: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 won't make further arguments.micromass said:OK, how do you measure 90° angles in GR?
Hornbein said:Albert himself sometimes used that notation.
Giant said: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 won't make further arguments.
Wow. crystal example makes it a lot clear. Thanks thanks!Mark Harder said: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.
jerromyjon said:Don't the Pythagorean Theorem and calculus depend upon a right angle? Or should I say rely on...
https://en.wikipedia.org/wiki/Relativistic_Doppler_effectmicromass said:OK, how do you measure 90° angles in GR?
WWGD said:Riemannian metrics of the form ##adx^2+bdy^2+cdz^2 ## where ##a,b,c## not all ##1## refer to, or depict non-Euclidean geometries, or geometries where
##a^2+b^2=c^2## does not quite hold. But, yes, in the Calculus done in Euclidean space, the generalized Pythagorean theorem does hold. But the Pythagorean theorem does not hold in general when your Calculus is done on a general manifold.
Yes, you can see it that way; this is the Riemannian metric tensor, a quadratic form , often represented as a matrix ##g(X_i, X_j)## where the {## X_i ##} is a basis for the tangent space. The Riemannian metric tensor associated with the standard Euclidean metric is, like you said, the identity. The Riemannian metric is an inner product, defined on tangent vectors at each point in a manifold.Mark Harder said:That looks like a generalized inner product using a quadratic form:
d×(x,y,z)⋅((a,0,0),(0,b,0),(0,0,c))⋅(x,y,z)T, where (...) indicates a row matrix and ((...),(...)) is a matrix, in this case a diagonal matrix. Is this the right way to think of the metrics you refer to?
WWGD said:Yes, you can see it that way; this is the Riemannian metric tensor, a quadratic form , often represented as a matrix ##g(X_i, X_j)## where the {## X_i ##} is a basis for the tangent space. The Riemannian metric tensor associated with the standard Euclidean metric is, like you said, the identity. The Riemannian metric is an inner product, defined on tangent vectors at each point in a manifold.
Mark Harder said:I occurred to me after I wrote that post that by dx2, you meant the the square of the differential of x, not multiplication by the scalar, d. Is that correct? And so, as you imply, the vector of differentials, (dx,dy,dz) is a basis for the tangent space at (x,y,z)? Thanks for your help. I only know enough differential geometry to be dangerous, as they say...
Our perception of dimensions being at right angles is a result of our brain's interpretation of visual and spatial information. Our brain uses visual cues and past experiences to make sense of the world around us, and the concept of right angles is deeply ingrained in our understanding of geometry and spatial relationships.
While our perception may lead us to believe that dimensions are always at right angles, there are actually exceptions to this rule. In non-Euclidean geometries, such as spherical or hyperbolic geometry, the concept of right angles does not exist. Additionally, in quantum mechanics, the concept of dimensions becomes more abstract and may not adhere to our traditional understanding of right angles.
The idea that dimensions are always at right angles is a fundamental assumption in Euclidean geometry, which has been extensively studied and tested over centuries. Through mathematical proofs and observations in the physical world, we have come to accept this as a fundamental truth. However, as mentioned before, there are exceptions to this rule in other geometries.
While it may be difficult for us to imagine dimensions that are not at right angles, mathematics and physics have provided us with tools to understand and describe these non-Euclidean dimensions. For example, we can use mathematical models and simulations to visualize curved spaces, such as a sphere, where the concept of right angles does not apply.
The concept of right angles is closely related to the geometry of spacetime. In Einstein's theory of general relativity, the curvature of spacetime is described by a mathematical concept called the Riemann curvature tensor, which includes the concept of right angles. This curvature is what causes objects to move in curved paths and explains the effects of gravity.