# Davies & Unruh: vacuum temperature is proportional to acceleration. But $T$ is not a vector

Davies and Unruh showed that vacuum temperature is given by acceleration:

$$T = \frac{\hbar a}{2 \pi k_\mathrm B c}.$$

But acceleration is a vector, temperature is not. If vacuum temperature produces acceleration and gravity, how does it define the direction of gravity?

Even more pointed: is the direction of acceleration (or gravity) defined by the gradient of vacuum temperature? Or again: How does Verlinde's equivalence of gravity and vacuum thermodynamics determine the direction of gravity?

• I believe the main point of your question does not have to do with the Unruh effect, but rather with Verlinde's entropic gravity. In forum language, some call this the XY Problem. If I am correct, would you mind rephrasing the question in order to give more emphasis for the entropic gravity bit? As it is currently phrased, I believe Andrew's answer is the only possible one Commented Nov 14, 2021 at 5:36
• @NíckolasAlves Based on comments on my answer, I agree with your assessment. But my suggestion would be to ask a different question. Rephrasing this one so it goes from being about Unruh radiation to Verlinde gravity is such a major change in scope that I think it's not really "rephrasing", it's a new question. I see now there is an edit -- it says "or again" and asks about Verlinde, but this edit is the first time Verlinde was ever mentioned in the question. Commented Nov 14, 2021 at 14:12
• @Andrew Fair enough. A new question does seem to be a better approach Commented Nov 14, 2021 at 14:52

The Unruh temperature depends on the magnitude of the acceleration, which is a scalar.

• This is obvious. But how can/could the direction of acceleration be measured with a thermometer?
– user85598
Commented Nov 10, 2021 at 6:53
• @Christian It can't. What makes you think it can? And I don't understand why you say it's obvious when in your question you said "But acceleration is a vector, temperature is not." Commented Nov 10, 2021 at 7:00
• So let me generalize: how can one measure the acceleration vector with thermodynamics means? Is there maybe an entropy flow in direction of the acceleration?
– user85598
Commented Nov 13, 2021 at 6:50
• @Christian Why do you think you can measure the acceleration vector of an accelerating object with thermodynamics? My answer is "you can't", but obviously that's not satisfying and I feel like I am missing your point, so I'm wondering if there's some intuition behind your question we can uncover that gets to what you are really asking about. Commented Nov 13, 2021 at 8:15
• My point: researchers like Verlinde state that gravity is equivalent to vacuum thermodynamics. I want to understand this. There must be a way to describe the direction of gravity (the direction of its acceleration) with thermodynamics.
– user85598
Commented Nov 14, 2021 at 5:16