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The dimension of force is $\text{mass} \cdot \text{distance} / \text{time}^2$ and this is independent of the dimension of the space (as $\vec F = m \vec a$ holds in any dimension). The relations $\text{distance}^{-d+1}$ for gravitation in $d$ dimension has a different reason. The fundamental relation holding in all dimensions is $$ \nabla \cdot \vec F_G = ...


-1

Anit-Matter could be a negative dimension, like how a 3d shape like a cube could be imitated by anti matter. If they go into each other, Who knows?


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In the FLRW model which is used today normal (baryonic) matter and dark matter together add up to matter. It dilutes with the growing radius to the third power (because volume is proportional to radius³). So the ratio matter : dark matter is and was always the same, at least in the plot you showed and the model that was used there. Edit: Maybe the 5.25 to ...


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The ratio of dark to baryonic matter is 5.25 in the first diagram and 5 in the second diagram, but I don't think the difference is significant. We don't know the densities with absolute certainty, especially near the Big Bang, and the small difference between the ratios is probably just down to the uncertainties in the densities. We would expect the ratio ...


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As it happens, you are absolutely correct. The velocities we encounter in everyday life are 3D velocities that are vectors defined as: $$ \vec{v} = \left(\frac{dx}{dt}, \frac{dy}{dt}, \frac{dz}{dt}\right) $$ In special relativity we use a 4D velocity called the four-velocity, and this is a four-vector defined as: $$ \vec{v} = \left(c\frac{dt}{d\tau}, ...


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It is easiest if you consider a rectangular box in 3D, with length $L_x$ in the $x-y$ plane, and length $L_z \ll L_x$ in the $z$ direction. Now, $L_x$ is very large, and you are only interested in the physics deep within the bulk, far from the boundaries. Therefore it is of little importance exactly which boundary conditions you choose. It is often easier to ...


1

First off your premise makes no sense. There's no reason to believe that something that exists in 3 dimensions necessarily needs to perceive in 3 dimensions (although one would certainly expect that to offer an evolutionary advantage). So in that sense your question doesn't make sense. However, I'll attempt to answer the question of why is it that we sort ...


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It's because our eyes have a 2D array of photo-sensitive receptors. Using two eyes you can 'detect' the third (depth) component. If you consider each eye to be a single point (for simplicity), the distance between the eyes themselves is a given 'constant'. A distant object in your vision, seen by both eyes, will be in a different position for each eye. You ...


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Tegmark has a nice discussion of this. He essentially argues that more than 1 timelike dimension would remove our ability to predict the future from the present. In math terms, more than one timelike dimension would render our differential equations ultrahyperbolic, and ultrahyperbolic PDEs are not well posed for initial data given along spacelike ...


3

Dimensional analysis only gives results that are correct to first order and up to a constant. So really there is just some qualitative guessing on physical behaviour with dimensional analysis.


3

Yes it can! However, the term dimension analysis needs to be seen in its/a context. Buckinghams Pi-Theorem did not just emerge out of the blue. And, the reason it works is not pure luck. There is a well grounded physical/theoretical basis for it. And it is the basis which allowed for dimesion analysis. Three scientists come to my mind. Sophus Lie wrote ...


4

As an addendum to @CuriousOne answer, also in mathematical analysis the dimensional analysis (or more properly scale transformations) can be used to guess a priori estimates and useful results, that anyways has to be proven in a rigorous fashion by other means. Two relevant examples may be the Sobolev and Strichartz estimates, whose admissible indices can ...


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Dimensional analysis can help to "guesstimate" the form of many important results but it can, for instance, not produce general solutions to equations of motion. It's an invaluable tool to understand the structure of physical theory, including quantum mechanics and relativity, and to check results for consistency, but it can rarely replace complex ...



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