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You need a constant with proper dimensions, because $\sqrt{-g} \;d^4 x \sim \mathrm{L}^4$ (an "hyper volume"), while $R \sim \mathrm{L}^{-2}$ (a "curvature"). Then, the integral has dimensions $\mathrm{L}^2$ (like an area). Yet, the action shouldn't have any dimensions, in natural units (or if you set units such that $\hbar = 1$). You then need a constant ...

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The best way is to solve for $r_2$ algebraically and only then evaluate it by putting in values with units.

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The proportionality constant will always have units of $A/B$, but these units can often be written in more than one way when expressed using derived units. For instance the proportionality constant $k$ in the electric force \begin{align} \vec F=k \frac{q_1q_2}{r^2} \end{align} can be expressed either in $F\cdot m^{-1}$ (farads per meter) or $N\cdot m^2\cdot ... 3 One argument goes as follows: Recall that the path-integral/partition function$Z[J]$is the generator of all Feynman diagrams in the source picture. Similarly$W_c[J]$is the generator of all connected Feynman diagrams in the source picture. All the Feynman diagrams in$Z[J]$and$W_c[J]$have mass-dimension zero. Now we want to find the mass-dimension ... 3 This statement is correct. What is a Feynman diagram? The most naive way to look into it is to say that it's a diagram that dictates all possible ways how a set of initial particles can transform into a set of final particles. The intermediate steps(the loops and internal lines) basically show one of the plausible ways allowed by the theory(these steps may ... 3 That particular multiplicative constant$c^4/16\pi G$in the Einstein-Hilbert gravitational action is necessary to produce the proper constants on the right side of the Einstein field equations, $$R_{\mu\nu}-\frac12Rg_{\mu\nu}=\frac{8\pi G}{c^4}T_{\mu\nu},$$ when one varies the complete action $$S=S_\text{EH}+S_\text{MR}$$ including a matter-and-... 3 We get many questions on here that ask, "why is this physical quantity defined in this way?" And the answer is always the same: because it's useful. There are many instances where we don't really care about the actual area used to determine the current. For example, if someone has already determined the resistance of a resistor, and I want to know what the ... 2 One might describe the terms as: Current is "the amount of charge passing through some chosen surface". Current density is "the amount of charge passing through some chosen surface divided by the area of that surface". There's nothing inherent in the definition of current about the amount of surface area involved. When defining a current, you don't have to ... 2 Think of it this way: Although$c=1$the standard relation$c = s/t$still holds. Hence,$s = c * t$. The dimension of the gravitational constant is given by$[G]= [F/m \cdot r^2] = m^3 / (kg \, s^2)$. Solving for "length" we obtain$m = kg\,s^2/m^2 \cdot [G] = kg \; [G/c^2]$. Hence, you always take the original definition of the constants and solve ... 2 since apples are never equal to pears k has always to have the dimension of A/B 2 If the metric gμν is dimensionless and gravitons are quantum excitations of the metric does that mean that gravitons themselves are dimensionless? It has dimensions , at least look here to a particular metric , the matrix elements have the dimensions of meter square. Is graviton energy included in the stress-energy tensor Tμν? In phsyics there are ... 2 Radians are a funny unit. They are not a "real" unit like meters or kg. For example, you would never put any other unit inside of a trigonometric function like$cos(5{\rm m})$or an exponential$e^{i10{\rm kg}}$. But radians go there with no problem. Part of the reason is that radians are more of a ratio than a proper unit. Radians are the "natural" unit of ... 1 Let’s check the units in your question: $$\dfrac{1}{2}I\,\left( \dfrac{d\varphi }{dt}\right) ^{2}=\tau \varphi$$ The Inertia has the unit$[N\,m\,s^2]$, torque has the unit$[N\,m]\,,\varphi$has no unit and time unit is$[s]$thus you get $$[N\,m]=[N\,m]$$ as it should be. 1 Distances may be defined by the radar method, which reduces measurement of distance to measurement of time. This is a fundamental principle in special relativity. It has been said that all measurements can be reduced to measurement of time and position, and I am not aware of a counter example. Certainly most measurements can reduced to a set of measurements ... 1 The Standard Model of modern physics includes over a hundred constants that just have to have their numerical values measured in the lab and plugged in to the equations, we have little or no idea why they are what they are. Using these measured values in our equations, we can run complex and sophisticated computer simulations which accurately model many ... 1 Quite simply, current$I$is current density$\rho$integrated over a surface: $$I = \iint_S \vec{\rho} \cdot d \vec{S}$$ So when one speaks of the current$I$, the surface$S\$ is implicit in the above definition. Typically the current density is very localized (ie a current carrying wire), which is why it’s not necessary to explicitly define the surface ...

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