# E=MC2 and dimensional analysis

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Energy is measured in units of Joules. We also know that 1 Joule = 1 kilogram*meter2 / second2.

Mass is measured in units of kilograms.

"c" is a velocity, which is measured in units of meter/second. c2 is therefore measured in units of meters2 / second2.

If we multiply mass by c2, we get units of (kilograms)*(meters2 / second2) = Joules, so we know that the equation is dimensionally correct because mc2 has units of energy.

I don't see how this can be correct. Shouldn't it read:

"c" is a speed, which is measured in units of meters/second?

It doesn't make sense that velocity and speed have identical dimensions. Does anyone know the definition of "dimension" that would quantify velocity and speed as being the same dimension? I don't understand how this can ever work.

Edited by steveupson

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I don't see how this can be correct. Shouldn't it read:

"c" is a speed, which is measured in units of meters/second?

It doesn't make sense that velocity and speed have identical dimensions. Does anyone know the definition of "dimension" that would quantify velocity and speed as being the same dimension? I don't understand how this can ever work.

Sure it makes sense.

Speed is the magnitude of velocity.

Velocity is a vector it gives you the movement in two (or more) directions, x and y. You can imagine this as an arrow. Each vector of that arrow will be the velocity component. The total length of that arrow is the speed, the magnitude of the velocity.

If the arrow is along one axis, say the x axis then the speed and velocity values will be identical.

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Sure it makes sense.

Speed is the magnitude of velocity.

Velocity is a vector it gives you the movement in two (or more) directions, x and y. You can imagine this as an arrow. Each vector of that arrow will be the velocity component. The total length of that arrow is the speed, the magnitude of the velocity.

If the arrow is along one axis, say the x axis then the speed and velocity values will be identical.

Maybe scalars and vectors are interchangeable when comes to dimensional analysis. Is that what you're saying? It doesn't make any sense at all to me.

But in any case, we're talking about c, the speed of light. It's a constant whereas the velocity of light is not. The gravitational lensing associated with the twin quasar shows this to a certainty.

"30 years of observation made it clear that image A of the quasar reaches earth about 14 months earlier than the corresponding image B, resulting in a difference of path length of 1.1 ly."

https://en.wikipedia.org/wiki/Twin_Quasar

By definition, the light from A has a higher velocity than the light from B, even though they have the same speed.

Edited by steveupson

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Maybe scalars and vectors are interchangeable when comes to dimensional analysis. Is that what you're saying? It doesn't make any sense at all to me.

But in any case, we're talking about c, the speed of light. It's a constant whereas the velocity of light is not. The gravitational lensing associated with the twin quasar shows this to a certainty.

By definition, the velocity of light from A has a higher velocity than the light from B, even though they have the same speed.

The length of the light velocity arrow is constant.

That length is called speed. Whilst the direction of the arrow varries.

A vector can either be a set of components or a magnitude and direction. For velocity the direction is dimensionless and the magnitude is speed.

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The length of the light velocity arrow is constant.

That length is called speed. Whilst the direction of the arrow varries.

A vector can either be a set of components or a magnitude and direction. For velocity the direction is dimensionless and the magnitude is speed.

Why does the light from A and B arrive at different times if the direction is dimensionless?

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Why does the light from A and B arrive at different times if the direction is dimensionless?

Because it's still a direction. Dimensionless in the context of dimensional analysis. Not dimensionless in the context of x,y,z,t.

The time difference is due to a path difference.

I'm a long way from a gr expert and it's been more than 10 years since I've done any but iirc you need to be careful with c only being constant locally. I'm sure others will correct me on this. But the introduction of GR concepts into an SR dimensional analysis thread might be muddying the waters a bit (unintentionally I'm sure).

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The time difference is due to a path difference.

Yes, correct, and this time difference is why the two velocities are different.