# Understanding Null Geodesics in Relativity: Insights from Dirac's Book

• exmarine
In summary, Dirac's book on relativity discusses the stationary property of geodesics and mentions "null geodesics" as a special case. However, it is not a good resource for self-teaching general relativity as it is not clear and can be confusing. In particular, the concept of "path length" is not a good parameter to use for null geodesics and a different parameter needs to be chosen.

#### exmarine

In Dirac's book on relativity, he begins and ends his section on proving the stationary property of geodesics with references to "null geodesics". His last sentence is: "Thus we may use the stationary condition as the definition of a geodesic, except in the case of a null geodesic."

What is a null geodesic? Does he mean a null interval, like for a photon, with ds^2 = 0?

Yes.

So what path does a photon take then, if not those geodesic equations?

Dirac doesn't say that light does not follow a geodesic; Dirac writes "Thus we may use the stationary condition as the definition of a geodesic, except in the case of a null geodesic."

Dirac's writing has to be unpacked very carefully. Just my personal opinion, but I think that Dirac's book is not a good book to use to teach oneself general relativity.

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George Jones said:
Dirac doesn't say that light does follow a geodesic

I think you mean "doesn't" here, correct?

PeterDonis said:
I think you mean "doesn't" here, correct?

Yes. I have edited my post to reflect this.

OK, are you saying that a photon DOES follow those geodesic equations? Please be explicit.

I see in the derivation - Hamilton's principle, stationary property, etc. - that I divided through by ds. Since ds equals zero for a photon, it seems to me that thus a photon would NOT follow the geodesic equations that result. I can't make sense of them for a photon, which is why I asked the question in the first place.

If I minimize the component time (provided there are no off-diagonal terms for the time degree of freedom in the metric, static, etc.) then I get some different geodesic equations. Haven't managed to integrate them yet to check against the empirical data, but they are different.

BTW, what is the latest and greatest / most accurate / data for the photon grazing the sun problem?

exmarine said:
are you saying that a photon DOES follow those geodesic equations?

Not in the form you're using them. See below.

exmarine said:
I see in the derivation - Hamilton's principle, stationary property, etc. - that I divided through by ds. Since ds equals zero for a photon, it seems to me that thus a photon would NOT follow the geodesic equations that result.

That's not because photons don't follow geodesics; it's because "path length" ##ds## is not a good parameter along null geodesics, because it doesn't uniquely label each point on the geodesic with a different parameter value (it can't, since ##ds^2 = 0## everywhere on a null geodesic). You have to choose some other parameter that does uniquely label each event on the geodesic with a different parameter value. (Coordinate time in a suitable coordinate chart will be such a parameter.)

## 1. What are null geodesics in relativity?

Null geodesics are the paths followed by light rays or particles with no mass in the theory of relativity. They are described by the equation of motion for a free particle in curved spacetime.

## 2. Why is understanding null geodesics important in relativity?

Understanding null geodesics is crucial in relativity because it allows us to accurately predict the paths of light and massless particles in a curved spacetime, which is necessary for understanding the behavior of objects in the universe.

## 3. How did Dirac's book contribute to our understanding of null geodesics?

Dirac's book "Principles of Quantum Mechanics" provided insights into the mathematical foundations of quantum mechanics and their connections to the theory of relativity. This helped to deepen our understanding of null geodesics and their role in the fundamental laws of physics.

## 4. What are some practical applications of understanding null geodesics?

Understanding null geodesics has many practical applications, including in the fields of astrophysics, cosmology, and GPS technology. It allows us to accurately model the motion of light and other massless particles in the presence of massive objects, such as stars and planets, and to make precise measurements of time and distance using the principles of relativity.

## 5. Are there any ongoing research or developments related to null geodesics in relativity?

Yes, there is ongoing research and developments in the study of null geodesics in relativity, particularly in the areas of gravitational lensing, black holes, and cosmology. Scientists are constantly working to deepen our understanding of these concepts and their implications for our understanding of the universe.