# Einstein & Dimensions

by Stu21
Tags: dimensions, einstein
 P: 18 If Eisenstein taught us the space time is one and the same thing, then how did he mathematicaly prove that. How did he put dimentions into his E=mc2 formula. The first 3 dimensions are strictly positions, positions in 3D require 3 numbers, and time is linear movement in one direction, the number being relative to whomever is making the measurement. My question is in order for Eisenstein's E=mc2 to have worked he needed to balance these factors in some how. How did he do it?
 PF Patron P: 3,923 You might spell the mans name properly. It is 'Einstein'.
PF Patron
P: 4,772
 Quote by Stu21 My question is in order for Eisenstein's E=mc2 to have worked he needed to balance these factors in some how. How did he do it?
The units do balance; the units of energy are mass times velocity squared. So mc^2 has the same units as E. What's the problem?

PF Patron
Thanks
Emeritus
P: 38,413

## Einstein & Dimensions

 Quote by Stu21 If Eisenstein taught us the space time is one and the same thing, then how did he mathematicaly prove that. How did he put dimentions into his E=mc2 formula. The first 3 dimensions are strictly positions, positions in 3D require 3 numbers, and time is linear movement in one direction, the number being relative to whomever is making the measurement. My question is in order for Eisenstein's E=mc2 to have worked he needed to balance these factors in some how. How did he do it?
You don't "mathematically prove" a physics theory. You propose a theory that fits the known experimental data and propose other experiments whose outcome would be different for different theories.

I'm not sure what your point is in saying "The first 3 dimensions are strictly positions ..."
In the formula, E= mc^2, E is energy and in the "MKS" (Meters, Kilograms, Seconds) is measured in "Joules" which is equivalent to "kg-m^2/s^2" the same as on the left side- mass times a speed (which would be meters per second) squared. Energy is "kg-m^2/s^2" because kinetic energy is "(1/2)mv^2" which has units of kg- (m^2/s^2) and "work" (equivalent to energy) is "force times distance". Force has units, in MKS, of kg-m/s, kilogram meters per second per second. Multiplying that by distance gives kg-m^2/s^2 again.
 P: 18 who is Eisenstein? :)
 PF Patron Sci Advisor P: 4,463 Did Eisenstein make science fiction films?
PF Patron
P: 5,498
Mathematicians deal with proof. Physicists deal with evidence.

 If Eisenstein taught us the space time is one and the same thing
It wasn't Einstein who 'taught us' that...it was his math teacher who helped Einstein develop the math...And the ideas originated with still others: Lorentz and Fitzgerald...
It tooks Einstein several years as I recall to adopt the idea.

ok, here is one reference:
 While spacetime can be viewed as a consequence of Albert Einstein's 1905 theory of special relativity, it was first explicitly proposed mathematically by one of his teachers, the mathematician Hermann Minkowski, in a 1908 essay[6] building on and extending Einstein's work. His concept of Minkowski space is the earliest treatment of space and time as two aspects of a unified whole, the essence of special relativity.
http://en.wikipedia.org/wiki/Spacetime

"How did he do it?"

Genius!!!! Keen physical insights first. He used the mathematics of Lorentz and Fitzgerald [for SR] but applied his own insights especially that the speed of light is fixed for all observers.

The above article explains how time and space 'switch places', are not fixed as in Newtonian physics, but vary according to observer. Your time and my time, for excample, will be different if we are in relative motion.
 PF Patron P: 5,498 Derivation of mc2 is here.... http://www.btinternet.com/~j.doyle/SR/Emc2/Deriving.htm If you search EINSTEIN ONLINE you can probably find how HE did it.
HW Helper
Thanks
P: 9,091
Oh dear - one of the problems with asking questions about things you barely know about is that the questions come out wrong and you can get dumped on. I hope this didn't put you off.
I feel this question is related to later ones so I figured I'd revisit it... some of what I will say will be repeats of what the others have said.
 Quote by Stu21 If Einstein taught us the space time is one and the same thing,
... well, the idea is commonly associated with him in popular science publications
 then how did he mathematically prove that.
He didn't - he used the mathematical description of the idea to produce results that agreed well with experiments. You can mathematically prove all kinds of things that are not true in nature. This is why physics is an empirical science.
 How did he put dimensions into his E=mc2 formula.
You mean the 4 space-time dimensions ... you can see from the formula that he didn't. But the links to the derivations the others have provided (above) should help you understand that.

The mass-energy relation is very famous but it is not the only relation concerning relativity.
 The first 3 dimensions are strictly positions, positions in 3D require 3 numbers, and time is linear movement in one direction, the number being relative to whomever is making the measurement.
Technically the three position coordinates are also relative to whoever is doing the measuring so there is a similarity there. It is clear that we can express where and when something is by three position coordinates (x,y,z) wrt some origin (0,0,0) and one time coordinate t wrt whenever we started the stopwatch.

But it is clear that time, like this, is quite different - eg, the distance to a point (x,y) in space is $\sqrt{x^2+y^2}$ but the distance to point (x,t)? You can't do: $\sqrt{x^2+t^2}$ can you?

I have a feeling that this is what you are actually asking about.

In order to make time play well with the position coordinates, we have to measure time in units of length. That's not as silly as it sounds. We turn a time into a length by multiplying it with a speed ... unfortunately the speed of something is relative to the observer: if only there were some speed that was the same for all observers, then we could describe the space-time coordinates as (x,y,z,ct) - oh hey! :) Speed of light to the rescue!

Notice that the c in "ct" acts as a constant of proportionality relating one way of looking at something to another. It is as valid to think of time and distance as the same thing as it is to think of mass and energy as the same thing.

The trick with this sort of representation is you have to change the way a distance measurement is done ... the "distance" to an event in space-time is $r=\sqrt{x^2+y^2+z^2-(ct)^2}$

This should give you an idea of how time can get included as a dimension of space.

My notation is a bit sloppy - you should read more around the subject of space-time to get the conventions down properly. The idea here is to point you in a helpful direction to do your own searching, not to provide complete answers.
 P: 526 I think (though I may be wrong) that the OP's question is how the view of space-time came about. In ordinary 3 dimensional space, we can perform certain operations on objects within this space, such as a vector. Most important is a transformation called a rotation, in which components of the vector in one direction become components in another direction. If you're moving with a certain velocity in the x direction, I can give you a rotation that will distribute your velocity between the x and y (or z) directions. So, we write concepts like velocity as three vectors, and treat the three dimensions as one space, because of the fact that we can rotate one component into another. In special relativity, the Lorentz transformation can be thought of a rotation through space and time - so, time receives a role that is dependent on space. Hence, 'space-time'.
Emeritus
P: 7,204
 Quote by Stu21 If Eisenstein taught us the space time is one and the same thing, then how did he mathematicaly prove that. How did he put dimentions into his E=mc2 formula. The first 3 dimensions are strictly positions, positions in 3D require 3 numbers, and time is linear movement in one direction, the number being relative to whomever is making the measurement. My question is in order for Eisenstein's E=mc2 to have worked he needed to balance these factors in some how. How did he do it?
If you can answer the following question for yourself:

"Why do we consider north-south and east-west distances part of a larger, unified structure, rather than two independent dimensions"

then you might be ready to read http://www.eftaylor.com/pub/spacetim...tEdThruP20.pdf for a rather interesting analogy, called "The Parable of the Surveyors". This is a publically available download of the intro to EF Taylor's textbook, "Space time Physics", a college level textbook on relativity. It suggests that the Lorentz transforms provides the same justification for regarding space as unified with time as geometry does for assuming that north-sout is unified with east-west.

It is college level, though, so if you aren't quite ready to mathematically justify why north-south and east-west distances are part of a larger structure, it may be too advanced for you. My apologies if I'm condescending, or whatnot, but I have absolutely no idea of what your background is.
 HW Helper Sci Advisor Thanks P: 9,091 I think the problem here is that it is not quite clear what OP is trying to ask about. We are each replying from some interpretation ... what's needed not is feedback from OP to make the matter clear.
PF Patron
Thanks
Emeritus
P: 38,413
 Quote by PAllen Did Eisenstein make science fiction films?
Eisenstein also gave a criterion for a polynomial to be reducible over the rational numbers:
http://en.wikipedia.org/wiki/Eisenstein's_criterion

No, he did not make science fiction movies but he did make movies- "The Battleship Potemkin", "Ivan The Terrible", and "Alexander Nevsky".

(I suppose it is possible that these were two different guys!)
PF Patron