Mordred

Ah ok certain properties remain invariant under Lorentz boosts. One property is the speed of light. The second being the laws of physics. Certain transformations will be considered invariant here is a good explanation. According to Einstein's principle of relativity all systems moving with a uniform velocity are equivalent, the laws of physics must obey the same equations in e.g., system S and system [latex]\prime{S}[/latex]. Compare rotations in three dimensions. The laws of physics are independent from the position of one coordinate system with respect to the other. Physical quantities are described by scalars, vectors and tensors, and the physical laws are given by combination of those quantities. Since the physics is independent from the position of the coordinate system, the form of the equation is the same in each coordinate system. For example the length of a vector in four vector or four momentum is invariant. http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/vec4.html In a vector the length is the scalar quantity. Another good example is the Maxwell equations are invarient to Lorentz boosts. Another key example is the rest mass of a particle is invariant. Hence rest mass has been replaced by invariant mass. Inertial mass however is not invariant.

What do you mean?

That involves coordinate transformation rules. Ie Cartesian to polar coordinates. Not all transformations are symmetric some are antisymmetric.

That part is correct. However if you follow the math in the last link were doing the same thing. On any vector the unchanging portion is the magnitude of the vector. To translate The direction element requires showing the start position and coordinate change according to the chosen coordinate system and reference frame. For example a clock handle has rotation symmetry as it changes angles. However you can remove the clock handle and place it somewhere else in arbitrary space at the same angle and have translational symmetry. Though I see nothing in that model you posted that hints at quantifying direction as a property.

Yeah so? The model you posted simply shows rotation symmetry. ( note the model uses the same coordinate system and has the same point of reference, including starting point) not so in defining direction as an equality. Observer a measures a ball moving at 20 degrees from his position but another observer can measure a different angle. Both can be correct. Which is precisely what I am trying to get you to understand. Here perhaps this will help http://www.google.ca/url?q=http://www.springer.com/cda/content/document/cda_downloaddocument/9783642329579-c1.pdf%3FSGWID%3D0-0-45-1365411-p174888679&sa=U&ved=0ahUKEwif87rM8N_MAhUC8GMKHUSpBhUQFggcMAM&sig2=dfm5M-If42-scIMIpTsZfw&usg=AFQjCNE5MgmQRPy6mH8FlqwpGhWzeaspHQ It is a basic coverage of vector to tensor conversions.

Take for example player a throws a ball to player b. You can describe The path the ball takes in two equally accurate descriptions. 1) the ball is moving away from player a. 2) the ball is moving toward player b. In this example alone we see direction is not invarient. How do you express a direction as an equality? Good luck You can express the magnitude of change as an equality not the direction

The problem is your ignoring that tensors work regardless of coordinate system or reference point. Direction requires a reference point and a reference coordinate system. The only property of a vector that is invariant is the magnitude. Not the direction. There is a difference between Cartesian, polar, spherical vector to tensor conversions. The reason being not due magnitude but due to coordinate choice. Anytime you want to convert to tensors the first question one needs to ask is what doesn't change regardless of coordinate transformation or position. What remains invariant. For example take a fan blade. If you move the fan blade does the shape of the fan blade change? Answer is no, so you can already mathematically describe that fan blade. The other step is to describe its new position and orientation. Regardless you have symmetry. (The fan blade shape)

Ok well the first question you have to ask is what property is invariant of a vector? In other words what property will remain the same regardless of coordinate rotation or point of origin? Remember vectors has two components magnitude and direction. No I am not going to simply apply your example above. Instead I'll help guide you to the above. This question alone should provide a clue, In regards to your example above. In particular you've chosen a coordinate system. Yet ignore the arbitrary choices involved. Which does not describe the purpose and functionality of a tensor.

The last post gives you some of the tools on tensor conversion. Look through the link.

Yes but that's not the same as stating an object travels (which implies momentum).

So time doesn't pass for an object at rest? At rest meaning no momentum. How do you accurately describe an object at rest as travelling through time? It is the time coordinate that changes, yet the object doesn't move.

Well it all boils down to coordinate and metric choice. Take for example a straight line in two coordinate systems In Cartesian coordinates its easy to visualize a straight line. However in polar coordinates this isn't so easy. A good example being the path of a photon following a null geodesic. To answer how a direction is determined you must first understand the coordinate system in use as well as the coordinate position. For example describing direction in 4d with components ct,x,y,z isn't as straightforward as simply describing a direction using just spatial components x,y,z. Either way there is no hard and fast choice of coordinate system, nor axis. The choice is made in the chosen reference point/frame and coordinate system your using. Direction must have a comparison factor. It is never the same for all observers performing the measurement. No two observers will measure the same direction unless they use the same reference point and coordinate system. For example I can choose to say the vector field is moving in the positive x direction. Then compare object a to that field. However this is a choice. As long as the comparison correctly describes the direction compared to the reference system then it's accurate. I could just as easily state the vector field flows north. Then compare. I don't know how well your math skills are but here is some of the math involved in converting Cartesian to polar coordinates. http://www.brown.edu/Departments/Engineering/Courses/En221/Notes/Polar_Coords/Polar_Coords.htm

Describing an object as travelling in time is a poor choice of wording. As time passes regardless of whether or not the object is moving.

It might help to better understand Studiots post if you look at the four theorems in vector calculus. http://mathinsight.org/fundamental_theorems_vector_calculus_summary The fundamental theorems are: the gradient theorem for line integrals, Green's theorem, Stokes' theorem, and the divergence theorem. The hyperlinks will give a basic intro into each

Why would you need to do the above? Let's take a vector field for example. You don't care which direction the "current" is travelling you only care what direction the stick moves compared to that current. (Let's assume the stick floats) You don't need to have a specific coordinate system. The above can be accurately described in 2d, 3d or 4d coordinates. This is one of the uses of tensors which describes vector and scalar relations independant on a coordinant system. Here is a good example similar enough to your above post. http://www.feynmanlectures.caltech.edu/II_31.html For example the coordinate system you described works well in Euclidean space or Minkowskii (Special relativity). However it is inadequate in curved polar coordinate metric systems such as The Schwartzchild metric. Or for example some of the dimensions in string theory where the coordinate metric is a rotating vector field. (Which is an application of circular vectors Studiot mentioned above) PS a vector isn't restricted to 2d. You can have n dimensions in a vector component.

Ok first off cosmology and universal applications involve the cosmological principle. Which essentially states no preferred direction or location. This is also true for the Einstein field equations. So in these examples direction is compared to a homogeneous and isotropic field from a coordinate. There is however metrics that involve direction compared to a bulk flow, such as direction compared to a current. Either way it doesn't change the fact that direction requires a comparison. That comparison can be a coordinate system such as a vector field, or a scalar field. The example you gave can be modelled as a vector field.

Just for the record. This site does a good job helping those who are asking honest questions. It's often difficult to gauge the skills of the OP when answering those questions. The main problem is many posters with little understanding on the subject matter will make declarations that this is how it is. Those are the posters that get into trouble. Posters that genuinely desire to learn, will ask rather than declare how something works etc.

It is a comparison, in order to define a direction you need a position to refer the direction compared to. It makes no sense in defining direction as a property. For the comparison requirement. When we describe direction of travel you need to have a point of reference. So how can you have direction as an intrinsic property when that direction value can change depending on the reference point? Properties of say for example a particle are intrinsic. Every particle of that type will have the same properties. Ie all electrons will have the same rest mass, spin etc. However the direction varies Perhaps it may help to define how physics defines the word Property. physical property - any property used to characterize matter and energy and their interactions property - a basic or essential attribute shared by all members of a class; "a study of the physical properties of atomic particles" http://www.thefreedictionary.com/physical+property direction is shared by all members of a class example electrons.

Thanks for the catch, gotta love auto corrects. Correction applied.

If your going to use Newtons law then at least show you properly understand it. Mass is resistance to inertia. The statement you made that inertia is resistance to motion makes absolutely no sense. Ether theories have been proved to be incorrect. The rest of your post is too garbled to decipher

You definitely have numerous misconceptions on Einstein vs SR. He didn't throw out SR. SR is perfectly valid in Euclidean space. It's been incorporated into GR, and is an essential aspect within GR which is designed to handle curvature.

The latter part is describing null geodesics which apply to photons. Photons don't interact with gravity. Look at the interactions list on the wiki page I posted earlier. Massless particles aren't dragged per se. Geodesic relations can be tricky when verbally describing them. For example wiki describes it as "Certain types of world lines are called geodesics of the spacetime straight lines in the case of Minkowski space and their closest equivalent in the curved spacetime of general relativity." That description applies to geodesics in general. There being three main types. https://en.m.wikipedia.org/wiki/Spacetime#Spacetime_intervals_in_flat_space PS another thing to keep in mind is often verbal descriptions can be misleading and can vary upon coordinate systems being described. An excellent article describing numerous artifacts of coordinates is the lecture notes by Mathius Blau. http://www.blau.itp.unibe.ch/newlecturesGR.pdf "Lecture Notes on General Relativity" Matthias Blau This last link is roughly 900 pages, but details numerous coordinates. In many ways it's near a textbook in style.

A handy statement was made in this Master-Geodesic article. Geometry=energy. http://www.google.ca/url?q=http://www.physics.usyd.edu.au/~luke/research/masters-geodesics.pdf&sa=U&ved=0ahUKEwjJkf3tuNHMAhVB52MKHVpUAtwQFggRMAA&sig2=FXfhVBHku5zK1fjW-tkXWw&usg=AFQjCNEr4WEHhcvoL-LVhqBLVIcgBRFdkQ Its an excellent article but takes a bit to understand the math behind it. An old relativity perspective is the "River model of blackholes". This particular article provides me numerous clues when I was first studying relativity. http://www.google.ca/url?q=http://arxiv.org/pdf/gr-qc/0411060&sa=U&ved=0ahUKEwiz1KubvNHMAhUD4WMKHXNRD0MQFggZMAI&sig2=KMwrVt094ZmEpor3wKz94w&usg=AFQjCNFI8GTbcqya0j0qAbhzraDu9VmS6w Here is the River model of space. http://www.google.ca/url?q=https://arxiv.org/pdf/1204.0419&sa=U&ved=0ahUKEwiR0Iaxv9HMAhUT42MKHYPbBg0QFggRMAA&sig2=U3lt-9TLseeVW-NNxLJLZA&usg=AFQjCNE5KQpW-SbSZ4Sqg_jkHLGfemxynQ Keep in mind these papers are helpful but do not imply an eather

The diagonal term is detailed somewhat in equation 5.4. I have a better breakdown in one of my textbooks Sort of the weak force is mediated byThe w+,w- and z bosons. Those bosons interact with the Higgs field which has four components. [latex] \phi_1,\phi_2,\phi_3,\phi_4[/latex] The interaction essentially uses up the last three components leaving the first. This results in the non zero Higgs field Now look at the interactions of Say a photon compared to a neutrino. https://en.m.wikipedia.org/wiki/Photon keep in mind the photon mediates the electromagnetic field but does not have a charge. (No binding energy) or doesn't couple to the electromagnetic field. Neutrinos interact with the weak force via the weak gauge bosons. So indirectly they gain mass via the Higgs field but in an indirect manner. The majority of the mass of objects is electromagnetic mass. Ie your table. The strong force loses strength as a function of radius extremely quickly so it's influence is limitted to within composite particles. Ie protons and neutrons. Inertial mass is essentially energy gained due to inertia which also correlates to a mass gain. So when your calculate say the mass of a proton, what your calculating is how strong the particle couples to its interaction field. For a proton 1% roughly is the Higgs field the other 99% is its coupling strength to the strong force. Now if gravity is a force then the mediator boson would be the graviton. However gravity may very well be just the result of curvature relations. We still don't know for sure as we can't fully quantize gravity at the particle level. It's influence one particle to another is too weak

Keep in mind I posted a single field example above. When it comes to mass you must involve all fields and their coupling constants that are present. For example you can have electromagnetic mass or mass due to the strong force interaction. The stress energy tensor is the term that describes the energy and momentum relations in the EFE. Tensors take some considerable time to learn. Each position has its own unique derivative. Which will depend on what that is being related to ie the curvature tensor. I'll dig up some examples once I unpack my textbooks( just finished moving) However in the meantime this article may help. http://www.google.ca/url?q=http://mathreview.uwaterloo.ca/archive/voli/2/olsthoorn.pdf&sa=U&ved=0ahUKEwjY1ceOqdHMAhVU3WMKHegsCW8QFggUMAE&sig2=cKOZEEemRIw0wylOMc0lYQ&usg=AFQjCNEOA2zinwqJc_O4wdiLvAirH1GfqQ