# Dimensional equivalence

#### SW VandeCarr

Given two units of measurements with the same physical dimensions (LMT), are they redundant? For example, Einstein's stress-energy tensor of GR uses units of pressure and energy density which are clearly distinguished in most interpretations. However both pressure and energy density have the same LMTdimensions: (L^-1)M(T^-2). In what way are they distinguished?

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#### SW VandeCarr

Why no responses? It's it's fundamental question. Every first year course in physics teaches LMT systems (MKS, cgs, etc). Reducing derived quantities to a set of fundamental dimensions can be very useful. It turns out that energy density (Joules/meter^3) is dimensionally identical to pressure. In Einsteins General Theory of Relativity's stress-energy tensor, energy density and pressure are treated as distinct quantities. You don't need to know relativity to answer the question: Are two quantities which are dimensionally equivalent redundant or are there good reasons for distinguishing them?

#### m00npirate

I don't know anything about stress energy tensors but you can certainly come up with two things which are dimensionally equivalent but are very different. Think of all the combinations of quantities that become dimensionless (like length per length for a slope). All of them are very different, but have identical dimensions.

#### SW VandeCarr

I don't know anything about stress energy tensors but you can certainly come up with two things which are dimensionally equivalent but are very different. Think of all the combinations of quantities that become dimensionless (like length per length for a slope). All of them are very different, but have identical dimensions.
I agree, but zero dimensions is a special case. Energy has the LMT dimensions M(L^2)(T^-2). If you want to express energy density you would divide by L^3 and get
M(L^-1)(T^-2). Force has the dimensions ML(T^-2). Pressure is force per unit area or
M(L^-1)(T^-2) which is identical to energy density.

I already raised this issue in the cosmology forum and didn't get a satisfactory answer. Essentially the problem arises by setting the speed of light 'c' to unity. Then c^2 also equals unity. This materially changes the dimensions of the units because 'one' simply drops out of the equations. I think this leads to fundamental misrepresentation of what the units mean. Like I said, this is not an issue of relativity but how fundamental concepts of units of measurement are used and interpreted. Am I wrong?

#### robphy

Homework Helper
Gold Member
Work and Torque have the same units.
Work comes from a dot-product. Torque comes from a cross-product.

#### Dale

Mentor
Essentially the problem arises by setting the speed of light 'c' to unity. Then c^2 also equals unity. This materially changes the dimensions of the units because 'one' simply drops out of the equations.
A dimensionful quantity with value of 1 in some system of units is not at all the same as a dimensionless 1. E.g. 1 m ≠ 1 m² ≠ 1.

#### SW VandeCarr

Work and Torque have the same units.
Work comes from a dot-product. Torque comes from a cross-product.
Good example. Would this apply to energy density vs pressure? Energy density is a scalar but pressure, as a force acting on a surface, would be a vector. I guess that's the answer I was looking for. Thanks.

#### SW VandeCarr

A dimensionful quantity with value of 1 in some system of units is not at all the same as a dimensionless 1. E.g. 1 m ≠ 1 m² ≠ 1.
I know, but if x=1 then x^2=1 regardless of the units assigned to 'x'.

#### Dale

Mentor
I know, but if x=1 then x^2=1 regardless of the units assigned to 'x'.
No, if x = 1 then x is dimensionless and cannot have any units assigned to it. If x = 1 U where U is any unit then x² = 1 U² regardless of the unit U assigned to x.

#### SW VandeCarr

No, if x = 1 then x is dimensionless and cannot have any units assigned to it. If x = 1 U where U is any unit then x² = 1 U² regardless of the unit U assigned to x.
Measured in meters/sec (m/s) c=3x10^8 m/s and c^2=9x10^16 (m/s)^2.

Set 'c' to unity and call this unit a 'lux', then c=1 lux and c^2=1 lux^2. The units then for E=mc^2=m (kg)(lux^2). The problem is that the units may drop in calculations so that you get a result E=m in terms of the numbers. In any case, as robphy pointed out, the problem is solved. Quantities can be dimensionally equivalent and still be different if they are of different tensor ranks or use cross products vs dot products.

#### Dale

Mentor
The problem is that the units may drop in calculations so that you get a result E=m in terms of the numbers.
I don't see the problem. If you drop the units then you have made a mistake. Don't make the mistake of dropping units, then setting c = 1 lux or whatever is not a problem.

"Dimensional equivalence"

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