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1

I am going to interpret your question as, it takes energy to accelerate, decelerate, and overcome friction in the other 3 dimensions, why does moving through time not require energy? The answer is very simple. In the case of moving with respect to the 3 spatial dimensions - you move. For the case of the time dimension, time "moves" with respect to you! ...


1

The three directions $x$, $y$ and $z$ are separable for the particle-in-a-box problem - the behaviour in each is independent of that in the others. Thus, each direction when separately considered only gives the contribution to the energy due to the limits of the box, or equivalently, the 'part' of the wavefunction, in that direction. In the case of $l_z ...


1

Our model for spacetime is that of a manifold, which is the mathematical term for something that looks like $\mathbb{R}^n$ in any zoomed-in patch, and where all these patches are stitched together in a sensible way. On our manifold we have $n$ coordinates -- real numbers that describe each point and vary smoothly from point to point. We also add to our ...


1

Over the real numbers, any non-degenerate quadratic form is determined (up to a change of basis) by its signature, which consists entirely of $1$s and $-1$s.


4

You can get this more "intuitively" (idiosyncratically): the flux of this force in closed surface is equal to the quantity of source inside (is a Gauss's Law). This source could be a mass or a charge. The physical picture is: the pressure applied in a closed surface by the field-force is proportional to the quantity of source inside. You can get the ...


2

Velocity is defined as distance over time, so based on that premise, you are moving through time at the rate of 1 hour per hour, or 1 minute per minute, or 1 second per second. You cannot go faster than 1 hour per hour relative to your own "clock". You are simply experiencing the one-directional "arrow of time" (Sean Carroll), whereas in space you have a ...


6

Moving through the other three dimensions necessitates energy. But why doesn't moving through time necessitate energy? Like OrangeDog and peta said, it doesn't take any energy to move through space. The Earth is moving through space, but it isn't consuming any energy. And like what ACuriousMind said, moving through time doesn't make much sense. To be blunt, ...


0

To make the math work. Ever since Einstein determined that time is actually another dimension, Physicists have used that notion to expand the conception of the Universe to include added (by not sensible) dimensions to get their math and theories to work. Of particular note is Witten's unification of string theories which "only" required the addition of yet ...


57

Moving through space at a uniform pace does not require energy, or force (Newton's 1. law), but accelerating through space does (Newton's 2. law). Similarly, moving through time at a uniform pace does not require a force, but if you're accelerating, your time will change wrt. a non-accelerating observer, so in a way you might say that you accelerate through ...


-1

It is relatively easy to imagine 4th dimension. That would be time. But time as if we had a time machine with which we we could arbitrarily move through it. Higher dimensions would be more difficult but possible as if "destinies". For example imagine that in destiny1 you see a car going from A to B in a given hour but in alternate destiny2 you see the same ...


3

In differential geometry, a space of a given number of dimensions can be curved rather than Euclidean, so for example the surface of a sphere is understood to be a 2-dimensional space in spite of the fact that we can't help but visualize the sphere sitting in a higher-dimensional 3D Euclidean space. This 3D space that we imagine the 2D surface sitting in is ...


5

The definition of dimension used here is that of a dimension of a manifold - essentially, how many coordinates (=real numbers) we need to describe the manifold (thought of as spacetime). Manifolds may carry a notion of length, and one of volume. They may also be compact or non-compact, roughly1 corresponding to finite and infinite. E.g. a sphere of radius ...



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