So, we know from GR that the relativistic theory of gravity that should obey special relativity is a theory of curved space-time. The mathematics that comes into play is differential geometry. In extrinsic way, we can embed a curved space into a higher dimensional flat space, define a position vector from the arbitrary origin to any point on the surface and then differentiate with respect to the different cartesian coordinates. The curvature of space implies coupling of coordinates.

Example: a curved 2D-space can be embedded in flat 3D space where the position vector will have components x1, x2, x3 ( x1, x2). We can differentiate with respect to x1, x2 and obtain tangent vectors. From there we can define new mathematical entities and then differentiate again to get yet more mathematical entities. This cycle repeats as many times as needed to model a given physical phenomenon. This way we define tangent vectors, reciprocal tangent vectors (co-vectors), co- and contra-variant metric tensors, normal vectors, Christoffel-symbols (connection-coefficients), covariant-derivative, Riemann-curvature tensor, Ricci tensor and scalar etc.

The metric tensor is the object we seek to obtain by solving the Einstein-equation which equates the Einstein tensor + cosmological constant term with the energy-momentum tensor which is just the 4-flux of each component of 4-momentum. The metric tensor then allows us to solve for the length of an arc and to solve the geodesic equation.

Unfortunately, the effects of GR and thus curvature of space-time is only noticeable for huge amounts of energy and matter; therefore we only perceive them on a macroscopic scale. Unlike the other 3 (?) known forces (electromagnetic, weak and strong), GR is not a quantum theory and is thus incompatible with the quantum field theoretical descriptions of the other forces as excitations of bosonic fields interacting with fermionic fields through a coupling. All these bosonic fields are special vector fields, namely gauge fields which are themselves connection fields.

The idea of a connection is that we want to compare neighboring physical objects corresponding to different space-time points / events. However, the difference may be due to a change of basis vectors/reference frame etc. so just differentiating would be nonsensical. Example: Imagine a homogeneous vector fields on 2D flat surface with covariant basis vectors on the polar coordinate lines. These basis vectors will obviously point in different directions as we move through space. The same invariant vector can in general have different components on different space-time points despite the field being homogeneous. Just differentiating the components alone is therefore nonsensical as one will conclude that the vector field is not homogeneous due to changing components. To account for the change of reference frame (different basis vectors here), we introduce the covariant derivative which essentially yields the additional Chriftoffel-term we get for free when considering product rule for differentiation on our vector fields.

The Christoffel-symbols alone do not imply any curvature themselves but the commutator of covariant derivatives acting on a vector is in general non-zero and the result is what is known as the Riemann curvature tensor which is THE object for measuring curvature. We can understand this by considering the "parallel transport" of a vector around a loop. The following is no rigorous proof but if your space is flat then you can always switch to cartesian coordinates and the induced vector basis is orthonormal and constant, the metric tensor is simply the Kronecker-delta (the identity matrix). Thus, the Riemann-Christoffel-tensor will be zero as it contains derivatives of Christoffel symbols and just Christoffel symbols which themselves are all functions of derivatives of the metric. Since zero times the Jacobi-tranformation term yields zero again, we conclude that the space is flat. Likewise, if the Riemann-tensor is non-zero, we cannot transform to cartesian coordinates, hence space must be curved.

Now let's turn to QFT where we have Lagrangian-densities to understand the EOM's for our fields. One of the most important Lagrangian is the Dirac-Langrangian which describes the behavior of spin1/2 spinor fields. All three generations of quarks and leptons are described by spinors. We look for symmetries in our Lagrangian and we find that there is a U1 symmetry (complex phase in form of e^i"theta") That is a global symmetry. In order to promote it to a local symmetry, the theta must be dependent on the coordinates. We find that the LG (Lagrangian) is not invariant under said transformation. We introduce something we also call covariant derivative that transforms just like the field does. We have to introduce an additional field which turns out to be a connection field A whose transformation is prescribed to fit the above condition. Having introduced a new field, we need an equation of motion for it. We need a quadratic kinetic term to add to our Lagrangian which are terms including the Faraday-tensors/field tensors of EM, essentially the Maxwell-Lagrangian or Proca-Lagrangian for m=0. Now, it turns out that we can express the field tensor in terms of the commutator of these covariant derivatives which make it formally look similar to how the commutator of covariant derives in GR give rise to Riemann curvature tensor. As such, people started giving them the term of curvature as well. So the EM Faraday-tensor is a curvature tensor. Similarly, for Su2 (weak) and Su3 (strong), we find field tensors which can be interpreted as curvature as well.

Thus, all forces except the Higgs field (not sure whether you consider it a force or not since it is technically a bosonic field) can be perceived as some sort of curvature in an underlying abstract mathemathical spaces. However, unlike space-time whose curvature can be understood visually from an extrinsic point of view, I've never seen any visualizations of the "gauge" curvatures. I'm not even sure what exactly is curved. The people I've asked talked about the principle U1 bundle etc. What I am looking for is a concise and simple geometrical visualization of what is curved and how it is curved. Curvature is by nature a very visual concept and see hundreds of such visualization when you google GR but the same cannot be said for gauge fields… Why is that? Please enlighten me or provide a good reference.

I was recently fortunate enough to both be accepted to Northwestern for a Physics PhD and get off the waitlist at Duke. Before this waitlist result I was 100% set on northwestern but now I am questioning myself. I want to do high energy theory, and at Northwestern I would already have a professor I connected with and a department that I got to know/became familiar with. However, I think Duke is slightly better ranked in this field, and i’m not sure if that slight difference in prestige will make a difference for academia and post doctoral positions. Any advice from someone who had to make a similar choice or who knows the field would be appreciated.

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If you need to make an important decision regarding your future, or want to know what your options are, please feel welcome to post a comment below.

A few years ago we held a graduate student panel, where many recently accepted grad students answered questions about the application process. That thread is here, and has a lot of great information in it.

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I play a game called Airsoft it's similar to paintball. The projectile we fire can have different weights and I am told that the heavier the projectile the further it goes. I would have thought that since the energy transferred from the spring to the bb is the same, so since the bb is heavier the muzzle speed will decrease and therefore travel less further. Why is this not correct?

If Einstein said that E=mc², why do photons have energy if they don't contain any mass?

Hello Reddit,

I'm considering my options for my PhD, focusing on either Quantum Field Theory in Curved Space-time (QFTCST) or Quantum Gravity (QG) through Quantum Information.

I have two potential theoretical Master's thesis topics to choose from:

1. General Relativity: This project aims to study spin precession in a binary black hole system.

2. Weak Gravity Conjecture: Explore composite magnetic monopoles and develop an extended model for multiple U(1) gauge symmetry.

My current specialization in my Master's program includes Astrophysics and Quantum Field Theory, and I'm deeply interested in cosmology as well. I'm wondering if choosing the first thesis topic would better prepare me for my extended interests in QFTCST or QG for my PhD.

My plan is to leverage my coursework in QFT and my thesis work in GR to bridge my knowledge gap and pursue more advanced topics in my PhD.Additionally,

I've chosen Astrophysics as a backup interest (and also to understand BH Mechanism Astrophysically), with the possibility of later transitioning to Astroparticle Physics or Cosmology-related topics.

I'd appreciate any opinions or advice on how to best align my Master's thesis choice with my future PhD goals.

Thank you!

I am a 3rd grade university student and I’d like to start reading articles apart from the textbooks used in class. Most recent articles are either too specific or not that interesting for someone who still has a small knowledge. Any recommendations?

Hi i am currently ending pref-graduate schools and i was wondering if there is a program that is capable of doing some simulations of basis of physics such a throwing something in gravitational field or something falling in graviton field/force. Is this there an any program that is easy to use and so universal for any field of this study? Sorry for bad English, not an native speaker.

Specifically posed to people who do some variety of spectroscopy on atoms and molecules : considering that the residuals of fitting Voigt/Lorentzian curves to spectroscopic data is very much non-Gaussian, is it mathematically appropriate to even use least square fits to extract information from the data? I've seen some papers that specifically call out this issue and use something like MLE, and yet I think in our labs we are more than happy to mindlessly use non-linear least squares.

What do you think?

Im talking about programs, frameworks,...I feel like computational tools are nowdays everywhere. Is there a branch where this is missing? Do you have something where you have no python framework or whatever to work with?

Disclaimer: this might not be strictly about physics but I don’t know where else to post this.

Today was the tipping point for me. I went to a lecture about differential geometry to get a better grip on Lie groups (QFT) and GR. I was really motivated and wanted to give listening to lectures another try. But the docent just wrote down definition after definition. Examples that didn’t contribute to the bigger picture in any way and Exercises that were only solvable in the one minute break by geniuses or people who already knew the subject. Worst of all, the lecture was 1:1 taken from the script. So why even bother. This is pretty much the experience I had both in physics and math lectures over and over again.

There are so many better ways to hold lectures. I understand the importance of formality but why not give the students the technical stuff to learn beforehand (for example some chapters from a good book) and then use the lecture to put all of this into a bigger picture with minimal formality. Think in the style of a (less elaborate) 3b1b video.

This thread is a dedicated thread for you to ask and answer questions about concepts in physics.

Homework problems or specific calculations may be removed by the moderators. We ask that you post these in /r/AskPhysics or /r/HomeworkHelp instead.

If you find your question isn't answered here, or cannot wait for the next thread, please also try /r/AskScience and /r/AskPhysics.

Whether it's from your undergraduate or graduate days, can you recall a particularly memorable final exam? Maybe it was extremely challenging, or perhaps you left the exam room feeling triumphant, like a champ. I'm surrounded by my own study chaos (there's a kid that keeps pressing one of those annoying buttons) and am curious to hear about others' experiences

Any thoughts on this response to the investigation into the high-Tc superconductivity claims concerning the C-S-H and Lu-N-H compounds?:

Response by Prof. Dias to Rochester Investigation

This is a thread dedicated to collating and collecting all of the great recommendations for textbooks, online lecture series, documentaries and other resources that are frequently made/requested on /r/Physics.

If you're in need of something to supplement your understanding, please feel welcome to ask in the comments.

Similarly, if you know of some amazing resource you would like to share, you're welcome to post it in the comments.

For background here, I'm a high school student.

I've recently been reading a lot about how it's like to solve math problems at the research level. They just pick interesting questions in their field and work on them. It's just problem solving but a pretty high level. However, it's been very hard to find a physics equivalent answer to this question. All the posts I have been reading about are about experimentation and collecting data and no seems to discuss what it's like to solve problems at the research level. What do you theoretical physicists work on?

I read Walter Isaacson's biography on Albert Einstein and from what I saw, he was doing much the same thing in physics too. He was asking what are the implications if speed of light is constant in any frame? Is this a special case or is research level theoretical physics the same?

I would really love it if you guys could explain with an example which you have worked on.

P.S : If there are any sources where I could read more about this I would be grateful :) .

This is a dedicated thread for you to seek and provide advice concerning education and careers in physics.

If you need to make an important decision regarding your future, or want to know what your options are, please feel welcome to post a comment below.

A few years ago we held a graduate student panel, where many recently accepted grad students answered questions about the application process. That thread is here, and has a lot of great information in it.

Helpful subreddits: /r/PhysicsStudents, /r/GradSchool, /r/AskAcademia, /r/Jobs, /r/CareerGuidance

I was doing a campus visit recently where another perspective student was handing out puzzles from her undergrad research project. 1000 pieces, 20"x30", and all the black swirly portions really sucked to put together. It is truly a testament to my commitment to silly things.

This is the actual puzzle fully assembled. The picture has more of a red tint than the real thing, but shows the glossiness well.

A digital picture and an accompanying article can bee seen here with more details on the research project. Along with some more info on SOFIA specifically. She's essentially a telescope strapped to a 747 and flown around the globe, taking measurements of the near, mid, and far-infrared radiation coming from different sources.

I'm going to build a frame for this and hang it in my office, so let me know if you have any ideas on the frame design/color!

Hi all, first post here and I am not sure if it is appropriate or not. I am a lecturer/researcher in a not so advanced uni in some 3rd world country, with background in control engineering > pivoted to Density Functional Theory for material simulations in masters > currently working on functional surfaces (liquid on hydrophobic surface) for PhD. Due to possible focus on national research interest, I most likely will have to pivot again to Computational Fluid Dynamics and related stuffs. I have tried reading on the current state of the field, but since I do not really have extensive background in Fluid Dynamics, I might have miss quite a bit of things, especially with the requisite maths needed for it.

One bit of puzzle that intrigues me at the moment is one of so called Navier-Stokes existence and smoothness problem. I am aware that the problem is currently more of a Mathematics problem than Physics problem, but I am not sure to what extent. A lot of literature suggests that since we do not have a proper solution to Navier-Stokes smoothness, a computational fluid simulation may blow-up some times. Is there any way to predict on when a simulation will blow-up, and parameters that help to avoid it. I presume there would be some kind of "best practices" in the community that with some kind parameters/setups to avoid or to use. If I allude to some similarities in DFT simulations it would be the choice of basis sets (plane wave or Gaussians), exchange-correlation functions for different type of materials, adding van der Waals parameters/terms and so on which reasons are not exactly intuitive to someone who is new to the field. I wonder if it will be the same for Fluid Dynamics due to Navier-Stokes smoothness issue, or is it an issue that only happen on the edge cases? Sorry if my exposition is not succinct enough to explain my case.