How does Einstein's equation work? Why do we need to square c?

[ineed2know]

TL;DR Summary

Why do you need to square the speed of light in E=mc^2?
And how can we warp spacetime just by going fast?
How does Einstein equation even work

If the laws of physics say that nothing can move faster than the speed of light, how can we square it in Einsteins equation? It dosent make synce to me.

How does Einsteins equation work can someone break it down for me and explain it in layman’s terms please.

How can we warp spacetime just by going close to the speed of light? How can we experience everything slower in space time just by moving faster than everything else?
I understand that your speed is relative to everything else and if you’re going in a spaceship going 90% the speed of light and you pass another space ship going 50% the speed of light. You wouldn’t view that spaceship at 140% the speed of light you would view it at 99.9999% the speed of light. But what I don’t get is why that correlates to affecting your own relitivity to how you view time yourself.

I’m a sophomore in highschool and no class teaches this subject and no teacher knows anything about this. I’m genuinely losing sleep over this and nothing I find can satisfy my need to know how this happens. I’m putting this as undergrad because I need a way deeper understanding of why this happens then at just a highschool level

 

[Sagittarius A-Star]

TL;DR Summary: Why do you need to square the speed of light in E=mc^2?

With (proper acceleration as function of coordinate acceleration) $\alpha=\gamma^3 a$ (follows from relativistic velocity addition) and if the force is in direction of movement:

relativistic kinetic energy =
$\int F \cdot ds = \int m\alpha \cdot ds = m \int \gamma^3 a \cdot ds = m \int \gamma^3 \frac{dv}{dt} \cdot ds = m\int_0^v \gamma^3 v \cdot d v = mc^2(\gamma -1)$

 

[russ_watters]

If the laws of physics say that nothing can move faster than the speed of light, how can we square it in Einsteins equation? It dosent make synce to me.

How does Einsteins equation work can someone break it down for me and explain it in layman’s terms please.

The speed of light squared isn't a speed anymore. Look at the units. And compare the equation and units to the Newtonian kinetic energy equation.

 

[Dale]

Why do you need to square the speed of light in E=mc^2?

That is for making the units match. In advanced physics $c$ and $G$ are just considered unit conversion factors.

And how can we warp spacetime just by going fast?

You cannot. If something goes fast it increases both its energy and momentum. Roughly speaking the curvature from the momentum essentially cancels out the curvature from the energy, so nothing can turn into a black hole just by going fast.

If the laws of physics say that nothing can move faster than the speed of light, how can we square it in Einsteins equation? It dosent make synce to me.

A speed squared cannot be compared to a speed. That would be like comparing a mile to an acre. Again, the speed squared is just a unit conversion factor.

How can we experience everything slower in space time just by moving faster than everything else?

You cannot.

I understand that your speed is relative to everything else and if you’re going in a spaceship going 90% the speed of light and you pass another space ship going 50% the speed of light. You wouldn’t view that spaceship at 140% the speed of light you would view it at 99.9999% the speed of light.

Actually, it would be 96.6% of $c$.

But what I don’t get is why that correlates to affecting your own relitivity to how you view time yourself.

Your own time goes at the same rate in your own inertial frame regardless of how time dilated you are with respect to anyone else. So in your frame it will always take the same amount of time to boil an egg, even if it takes you longer according to someone else’s frame.

I think you may have some misunderstandings about what happens. Probably we need to focus on that before worrying about how it happens

 

[PeterDonis]

How does Einsteins equation work can someone break it down for me and explain it in layman’s terms please.

This article might be helpful:

https://math.ucr.edu/home/baez/einstein/

 

[PeterDonis]

How can we warp spacetime just by going close to the speed of light?

We can't. That's not what warps spacetime.

All of the relativistic effects you mention (and you've gotten good responses from others about issues with what you say about them) have nothing whatever to do with the curvature of spacetime or Einstein's equation.

If you want a quick summary of what Einstein's equation says, consider John Wheeler's famous two sentence description of relativity:

Spacetime tells matter how to move. Matter tells spacetime how to curve.

Einstein's equation is the mathematical expression of the second sentence: it describes how matter tells spacetime how to curve.

 

[FactChecker]

Those are excellent questions. But they show that you have some basic misconceptions. Before you start plowing through any formal proofs or mathematics, there are a couple of things that you might enjoy reading and are no work at all. One is the short relativity chapters of George Gamow's "Mr Tompkins In Paperback". You can read it in a couple of hours. Another is Chad Orzel's "How To Teach Relativity To Your Dog". Please don't be offended by the title. I think it is entertaining to read and you can probably breeze through it. It is about 300 pages and goes deeper into the subject.
Those will at least get your understanding straight enough to start into a more serious study of the subject.

ADDED: Don't let the above make you think that the math of Special Relativity is advanced. In fact, it is surprisingly basic. The Pythagorean Theorem is about all that is required.
Another good introductory book is Epstein's "Relativity Visualized".

 

[Nugatory]

Einstein's equation is the mathematical expression of the second sentence: it describes how matter tells spacetime how to curve.

Judging by the thread title and initial post, there’s a pretty good chance that by “Einstein’s equation” @ineed2know means $E=mc^2$, not the Einstein Field Equations.

 

[Ibix]

For those of us who aren't American, what age is "sophomore in high school"?

 

[topsquark]

For those of us who aren't American, what age is "sophomore in high school"?

circa 16.

-Dan

 

[jbriggs444]

For those of us who aren't American, what age is "sophomore in high school"?

Roughly speaking..

5 years old gets one into kindergarten which is more or less public school day care and socialization. Sometimes half days.

6 through 11 is elementary school, grades 1 through 6
12 through 14 is junior high school, grades 7 through 9
15 through 17 is high school, grades 10, 11 and 12. Also known as sophomore, junior and senior.

So sophomore would be 15 years old, give or take.

Some school systems do four year high school, but sophomore is still grade 10 regardless.

 

[Ibix]

Thanks.

OP - I would strongly suggest learning about Minkowski diagrams. They are a more sophisticated version of displacement-time diagrams that you will likely have come across in physics, and they let you draw different frames' perspectives on events. They help a lot to clear up confusion atound relativity. They won't directly answer your question about $E=mc^2$ but they will start you in the right direction.

 

[pervect]

TL;DR Summary: Why do you need to square the speed of light in E=mc^2?
And how can we warp spacetime just by going fast?
How does Einstein equation even work

We know we need to square the speed of light by a simple dimensonal analysis of units.

https://en.wikipedia.org/w/index.php?title=Dimensional_analysis&oldid=1254137729

In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantities (such as length, mass, time, and electric current) and units of measurement (such as metres and grams) and tracking these dimensions as calculations or comparisons are performed. The term dimensional analysis is also used to refer to conversion of units from one dimensional unit to another, which can be used to evaluate scientific formulae.

A dimensional analysis for instance says that we measure volume in cubic meters, not meters. Similarly the dimensions of energy are (kg) (meters^2) / (seconds^2) in MKS units. Or more generally mass * distance^2 / time^2. We can see these dimensions from the Newtonian equation $E = m\,v^2$, and any theory using standardized units will use the same dimensions, so that is why the dimensional analysis doesn't need the underlying theory to know why we need the c^2.

We do not warp space-time simply by moving fast. I'm guessing that you are imagining that moving fast makes an object more massive, and that that mass warps space-time more. However, Einstein's field equations do not use mass, they use a concept called the stress-energy tensor instead. The stress-energy tensor conceptually does not change when you change velocity, though the particular numbers comprising it do change.

As I read this, I think this is unsatisfying, so I'll say that from the point of a particular observer, space-time is always warped around a massive object, but that "length contraction" and "time dilation" of special relativity make the description of the warp different for a moving observer. A better but graduate level explanation can be found in Olson & Guarino's paper, https://ui.adsabs.harvard.edu/abs/1985AmJPh..53..661O/abstract , "Measuring the active gravitational mass of a moving object". Be warned - while I like this paper a lot, it's citation count is not all that high, meaning a lot of people may not have heard of it.

To go further than that, you'd need to know what tensors are, and I am expecting from your previous questions that you lack that background. But I'll say a few things anyway.

In tensor notation, Einstein's field equations can be written (in my favorite units, which are confusingly enough not standardized MKS units):

$$G_{\mu \nu} = 8 \, \pi \, T_{\mu \nu}$$

You can see what it looks like in standard units in wiki, which uses a slightly different notation and introduces the cosmological constant Lambda, which I did not do because among other reasons it wasn't in Einstein's original formulation.

https://en.wikipedia.org/w/index.php?title=Einstein_field_equations&oldid=1258815940

Sorry if this is a bit rambling and parts of it are too advanced, but it's the best I can do. Good luck. I will add that it's be best to learn special relativity before General relativity. Certain approaches to GR based on "sector models" are somewhat accessible at the high school level, but even those really require understanding special relativity first.

The problem of the gravity of a massive object requires GR - but an understanding of the SR treatment of the momentum of a rapidly moving object is the domain of SR.

 

[PeterDonis]

Judging by the thread title and initial post, there’s a pretty good chance that by “Einstein’s equation” @ineed2know means $E=mc^2$, not the Einstein Field Equations.

Possibly, but if so, the term "warp spacetime" in the OP is completely out of place, since $E = mc^2$ has nothing to do with warping spacetime.

 

[PeterDonis]

I’m a sophomore in highschool and no class teaches this subject and no teacher knows anything about this.

If so, where are you getting the information that is prompting your questions? Knowing that might help us in responding.

 

[pervect]

The best two recommendation I have for special relativity, which I think is the OP's main interest would be Taylor & Wheeler's "Space time physics", and Bondi's "Relativity and common sense". One can find a full text download of the first book (albeit an older version) on Taylor's website at https://www.eftaylor.com/special.html, The download, broken into parts, is available from the link that says "Full text download" on this page, which is the author's website. Bondi's book seems to be in the internet archive, https://archive.org/details/relativitycommon0000bond, so it's also available for legitimate download.

The treratment in Taylor's book is somewhat advanced (college level), but the author has tried an informal "chatty" style that I think makes for interesting reading and I hope will be somewhat accessible before college level. I don't recall offhand if it uses much calculus - it's hard for me to think back to the high school days - or if it has other roadblocks of needed math background for a sophmore in high school. Apologies if it's a bit advanced. Bondi's "Relativity and Common sense", which is a very old book (60 years old, now, written in 1964) is at a lower level, I recall reading it in high school. However, Bondi's book won't talk about momentum and energy at all.

I think we have other threads on book recommendations at the high school level on PF, but I don't recall exactly where.

Taylor's book will not immediately get to E=mc^2.. Momentum, energy, and mass are discussed in chapter 7 of the book. Thus it will be a while before they are talked about, patience is required. Skipping directly to chapter 7 is unlikely to work very well. The book will start about talking about space and time and why we unify them into space-time. Bondi's book never talks about momentum and energy, as I mentioned. It talks about an approach based on doppler shifst which is named k-calculus (though no calculus is involved).

 

[Sagittarius A-Star]

TL;DR Summary: Why do you need to square the speed of light in E=mc^2?

This is the related article of A. Einstein "Does the Inertia of a Body Depend on its Energy Content?" from 1905:
https://www.fourmilab.ch/etexts/einstein/E_mc2/e_mc2.pdf

 

[robphy]

Here are some additional suggestions (I thought Taylor-Wheeler and Bondi were helpful when I was learning). These all develop the spacetime viewpoint.

General Relativity from A to B by Bob Geroch
(It might read a little slow.... but it is surprising deep.
It develops the various spacetime models according to a pseudo-historical development Aristotle->Galilean-Minkowski->GeneralRelativity. It was written for a nonscience major course at UChicago.)
https://www.amazon.com/General-Relativity-B-Robert-Geroch/dp/0226288641?tag=pfamazon01-20
(more advanced notes are at: https://home.uchicago.edu/~geroch/Course Notes )

The Wonderful World of Relativity: A precise guide for the general reader by Andrew Steane
https://www.amazon.com/Wonderful-World-Relativity-precise-general/dp/0199694613?tag=pfamazon01-20
(He has two more-advanced texts: Relativity Made Relatively Easy: Volume 1 and 2.)

A Traveler's Guide to Spacetime by Tom Moore
(This came before "Six Ideas that Shaped Physics - Unit R".)
https://www.amazon.com/Travelers-Guide-Spacetime-Thomas-Moore/dp/0070430276?tag=pfamazon01-20

Special Relativity Primer (incomplete draft)
by William L. Burke and Peter Scott
https://scott.physics.ucsc.edu/pdf/primer.pdf

 

[Mister T]

TL;DR Summary:
How does Einstein equation even work

A better way to write the Einstein mass-energy equivalence is $E_o=mc^2$. It's really just a statement that rest energy $E_o$ is equivalent to mass $m$. The way you use it is to convert something that's measured in units of mass to something that's measured in units of energy. For example, a mass of $2.0 \ \mathrm{kg}$ is equivalent to approximately $1.8 \times 10^{17} \ \mathrm{J}$ of energy, where I have used $c^2=9.0 \times 10^{16} \ \mathrm{J/kg}$ as the conversion factor, which is the appropriate conversion factor in that particular set of units.

It's possible to define a set of units where $c=1$, and therefore the same unit is used for both energy and mass. In that case the Einstein mass-energy equivalence is simply $E_o=m$, and the factor of $c^2$ doesn't even appear.