How Mass WARPS SpaceTime: Einstein's Field Equations in Gen. Relativity | Physics for Beginners

How does the fabric of spacetime bend around objects with mass and energy?

Hey everyone, I'm back with another video! This time, we're looking at the Einstein Field Equations. These are some of the most important equations found in the theory of General Relativity.

The first thing worth mentioning is that the common way to write the Field Equations is as one single tensor equation, where the subscripts added after different terms can take the values 0,1,2,3. The meaning of these subscripts is discussed in the video, but it's important to note that what looks like a single equation is actually a convenient way of writing multiple equations.

We have already mentioned tensors. The terms in the Einstein Field Equations refer to these tensors, which are rather interesting mathematical objects. They can be represented by matrices, which are just a particular way of displaying information.

The first tensor worth mentioning, labelled T_mu,nu is known as the Stress-Energy tensor. This contains information about the distribution of mass, energy, momentum, pressure, and so on, within the region of spacetime we happen to be considering. Often, this region happens to be the entire universe. Some of us may be familiar with the idea that the existence of mass and energy can cause the "spacetime fabric" around it to warp and bend. And this is exactly what the Einstein Field Equations are about.

The G_mu,nu tensor seen on the left hand side of the equation, is known as the Einstein Tensor. It contains information about the curvature of spacetime. This allows us to understand how the region of spacetime we happen to be considering is curved and warped.

In other words, the Einstein Field Equations can be simply thought of as relating a mass / energy distribution to the warping it causes to spacetime.

The final term on the left hand side of the equation is a product of lambda, the "Cosmological Constant" and g_mu,nu, the "Metric Tensor". The metric tensor simply contains information about the exact shape of the spacetime fabric. Whether it is flat, or curved, and exactly how it curves. It is a very important tensor in relativity.

The cosmological constant is simply Einstein's way to encode the fact that the universe is expanding, into these equations. In other words, the behaviour of the spacetime fabric is not only dependent on the mass / energy inside it, but also on the fact that it just seems to be inherently stretching. We have observed the accelerated expansion of the universe - galaxies very far away from us are moving away from us faster and faster, as compared with galaxies much nearer to us. The positive cosmological constant accounts for this expansion of the universe.

The cosmological constant has an interesting history. Initially, Einstein published his field equations without the constant. Then he realised it was necessary in order to combat the probable collapse of the universe due to gravity causing everything to eventually attract everything else. Then, he called it his biggest blunder. But now, we see that the universe is not just expanding, but it is expanding at an ever-accelerating rate. Therefore, the constant is indeed necessary in our mathematics.

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[johnalbers]

Excellent. Best description yet on Einstein’s Field Equations. Now we need the video on tensors!

[MyJB]

I am a retired medical physicist and my recall of Special/General relativity learned in undergraduate courses has become rather warped over time (pun intended). Moreover, my lifetime of physics experience causes me to now "dig deeper" and seek intuitive/graphical understanding of these fields (no pun intended). I deplore numerous relativity textbooks that "hide" behind equations and the algebraic gymnastics of finding their solutions, without intuitive interpretation of the main terms and their units. This is where your videos are unique by providing clear intuitive interpretations and illustrations. Thank you for helping so many viewers understand relativity conceptually. Richard Feynman would be proud of your teaching content and style! 🤓

[mobizoid]

11:43 " 8 hopefully you are familiar with "
This got me laughing 😅

[EpicMathTime]

Parth G: A physics channel trying to look like an asmr channel that's trying to look like a physics channel.

[vikramantin]

I am a retired civil engineer who is fascinated by cosmology and quantum mechanics. I am devouring these topics with a passion hitherto unseen by man! 😄 Thank you, Mr. G.

[grimley]

Aint no way I understood this video having no prior knowledge about it, best explanation I’ve seen

[drroyce]

What an amazing teacher. I have a basic mathematical understanding yet I can follow everything he says. What a talent.

[Primitarian]

This is fantastic! I've subscribed to a lot of physics channels, but you have achieved the following: 1) Seized the bull by the horns by the explaining the equations themselves; 2) Explained them in a way so intuitively, precisely and compactly that even an amateur like me feels he understands them (instead of just getting eyestrain, as used to be the case).

[cristianpaulnitulescu]

Thank you I was the one to suggest this I remember, and you replied you would like to do it .

[edhead]

I'll be honest, when I first saw the equation, I was quite intimidated, but I truly appreciate the explanation in micro-chunks with visuals. Thankfully, I learned tensors as part of TensorFlow package while learning to program machine learning. This has given me all the knowledge and confidence to take me to that next level of understanding! Thank you so much!!

[omsingharjit]

Your video explaining it in much much simpler way than others

[adriannuske]

I like your pace and the fact that all of what you say is information, and closely related to what you intend to explain. Much, much appreciated!! Congratulations!

[poke]

Great video. Just a quick note that it looks like you forgot to increment your Mu index in the matrix / tensor description at 6:30 and 6:50. They are all showing first index = 0, but as an example your pressure terms should be (1, 1), (2.2), and (3, 3).

[RafiqueEdu]

wow, you are so genius and well prepared i wish that this video should be in billions seriously i lov it

[grzegorzbrzeczyszczykiewic]

i am literally watching ur vids like 5hrs straight just because you tell something that is in another vid im instantly switching to it
great work dude

[Eagle_View_degree]

Hey parth... also please explain

1.Why we are adding cosmological constant
2. Why we divide with C^4
4. Why C^4
4. Why there is 8pi..

Explain all these stuffs

[triqpham]

Thank you thank you. I’ve been watching a few of these concerning the field equations but yours is the absolutely best explained.

[elenabodna]

Thanks for the explanation. Physics is my passion, this equation is beautiful linking geometry to energy.

[thoughtfuloutsider]

It is so exciting to find your videos. I'm a fan after only watching a couple. I love physics especially Quantum mechanics and Relativity, but sadly I never formally studied it or maths because of various complexities in my personal history.

I have a question that I've been posing to a variety of great communicators of physics, especially on relativity. Your title suggested you would answer it but you didn't. So far no one else has been able to answer it. I loved the way you explained the equations of General Relativity and it was a great help and it confirmed that it doesn't answer my question even though in your explanation you use words that suggest you are answering my question. You say these equations tell us "how mass bends space-time." But they don't.

They tell us that mass causes space-time to curve or bend. Brilliant! They help us MEASURE the effect of Mass on space-time very accurately. Brilliant!
But not what CAUSES the curvature that would be the true HOW of the curving effect.

It is as if you were saying that measuring the current in amps tells us what electricity is and how it works. Our capacity for measuring current is very accurate, it tells us a lot that is useful and important, but it's not quantum mechanics which helps us understand electricity (and its related topics of electromagnetism and matter) in profound ways.

It seems to me if we truly practically understood the mechanism for how matter interacts with space-time, we'd get closer to the much sought so-called "theory of everything". In your video, on the basic postulates of quantum mechanics, you talked about the "kinetic theory of gases". You said that if you change the size of the enclosing space of gas then the standard theory falls over. We know that a gas under pressure requires a different explanation as it is liquified and the temperature changes. Is this analogous to what happens in a black hole? More importantly, is there a way of talking about how space-time works inside matter, whether talking about a planet or a lump of metal? How do state changes in matter affect space-time? Since there is a change in the space (at least) occupied by matter in different states. Further different types of matter can occupy the same size space but have different effects, such as weight, on gravity, therefore space-time.

[joj]

I've been on a GR binge lately, and I'm so happy you never used the John Archibald Wheeler quote

Yes, as a matter of fact I DO know that spacetime tells matter how to move, and matter spacetime how to curve. I think it sunk in somewhere around the eightieth time I heard it.